The Superfluid Phases of Helium 3

The Superfl uid Phases of Helium 3 DIETER VOLLHARDT Rheinisch-westfä lische Technische Hochschule Aachen, Federal Repub...

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The Superfl uid Phases of Helium 3

DIETER VOLLHARDT Rheinisch-westfä lische Technische Hochschule Aachen, Federal Republic of Germany

PETER WOLFLE Universität Karlsruhe Federal Republic of Germany

Taylor & Francis London o New York 1990

o

Philadelphia

UK

Taylor and Francis Ltd, 4 John St, London IWCLN 2ET

USA

Taylor and Francis Inc., L900 Frost Road, Suite 101, Bristol, PA 19007 Copyright @) D. Vollhardt and P. Wölfle

All righæ reserued. No part of this publication may

be

reproduced, stored in a retrieual system or transmitted, in any form or by any means, electronic, electrostatic, magnetíc tape, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

British Library Cataloguing in Publication Data Vollhardt, Dieter The superfluid phases of helium L. Helium I. Title II. WöIfle, Peter

3

546',.751599

ISBN 0-85066-412-8 Library of Congress Caúaloging-in-Publication Data Vollhardt, Dieter. The superfluid phases of helium 3 WöIfle.

/ Dieter Vollhardt, Peter

p.

cm. Includes bibliographical references ISBN 0-85066-412-8 1.. Helium-Isotopes. 2. Liquid helium.

WöIfle, Peter. II. Title. phases of helium three. QC145.45.H4V65 1990 530.4'2-4c20

I.

3. Superfluidity.

III. Title: Superfluid

90-10762

CIP

Cover design by Jordan and Jordan, Fareham, Hampshire, UK Typeset in l0lL2 point Times by The Universities Press (Belfast) Ltd. Printed in Great Britain by Burgess Science Press, Basingstoke on paper which has a specified pH value on final paper manufacture of not less than 7.5 and is therefore 'acid free'.

{ß#,+å V 3 trå

"4.È

Preface

)

I

a

.qet

The superfluid phases of 3He were discovered in 1971, i.e. almost 20 years ago. Since then, around 2000 papers have been published on this subject. There is now consensus in the low temperature physics community thai the basic physics of the superfluid phases of 3He is well understood. Indeed, there are many examples where the theory has been refined to the point that excellent quantitative agreement with experiments is achieved. We therefore feel that the time has come tô write a book that collects not only the ideas and concepts on superfluid 3He that have been developed in these years, but also the methods of theoretical physics used and the results obtained. We have attempted to arrive at a coherent presentation with little redundancy, while avoiding gaps or leaps in the argumentation. The enormous amount of material (the true extent of which we came to realize only after we had started the project) made it inevitable, however, that we treat some of the more technical theories, which require a longer exposition, as well as some of the more specialized topics, in a differ"n-t *uy thul we should have liked given infinite book space ãnd working time. The need for a bogk on superfluid 3He has not been feli for a long time because several excellent reviews were available, in particular the early articles by Wheatley (1975) and Lee and Richardson on experi*"nt 1f^lZS¡ and by Leggetr (1975a) and Anderson and Brinkman (ns, tgls) on theory. The more recent reviews on superfluid 3He have aiready started to specialize; these include the excellent articles by Brinkman and iross (197g) on spin and orbital properties, by volovik (19g4b) on the superfluiá properties of the A phase, by Hall and Hook (19g6) on hydrodynamics, and by Fetter (1986) and Salomaa and Volovik on rotating superfluid irgsíl -He. A general discussion of all aspects of superfluid 3He cleaily requires the form of a book This book is on the theory of superfluid 3He, although contact with experiment is made in many specific cases. It is intended for readers with a background in quantum mechanics and statistical mechanics, but it does not assume previous knowledge of the theory of superfluid aHe or of superconductivity, or indeed of any such more speciálized subject in condensed

vi

preface

matter theory. In order to keep the entrance barrier as low as possible, we have decided not to make use of advanced theoretical techniques such as Green's functions or functional-integral methods, since ,.r.i techniques require the reader to have considerable training in many-body theory- In our view the limitations caused by this self-imposed restriction are still tolerable. Nevertheless, we have also attempted to address those topics requiring more elaborate techniques, so as to provide the reader with the information that is necessary to find his/her own way in the riterature. The first part of the book.(chapters 1-4) is designed . to provide an introduction to the subject, while the later chapters are intended to give an exposition of the important new concepts that grew out of the more conventional theories in the course of time. Quite naturally, some of the topics-in particular those where we have ourselves been actively involved-have been covered in more detail than others. In general we have tried to illuminate the basic concepts rather than to stress the details of the actual calculations or the detailed comparison with experiment. We have gone to great pains to do justice to most of the theory papers that have been written on the subject and to at least reference ttre woit if space has not permitted a somewhat longer discussion. The selection of a particular theoretical formulation in cases where several exist in the literature is necessarily subjective. We apologize to those who feel that their work has not been represented adequately, and we ask them to blame this on our ignorance and inability to do better, rather than on evil intention ot presumptuousness. In any case, we welcome any feedback, be it positive or negative.

Superfluid 3He is one of the most fascinating condensed matter systems yet discovered. It is a test system for most of thJideas of modern theoretical physics: (i) the simultaneous occurence of several broken symmetries, including non-Abelian gauge symmetries; (ii) the existence of a multitude of topological defects of the order-parameter field; (iii) the possiblity of massless and massive collective modes of the order parameter. On the other hand, superfluid 3He is the prototype system for anfuotropic superfluidity. It has helped to shape our underitanding of neutron stars, anisotropic superconductors such as may be realtzed iñ heavy-fermion ,yrt".r, or even ceramic high-?: superconductors. we hope thai this book can serve as a source of education and information on superfluid 3He, not only for those working in this very field, but also in particular for those who want to carry over the concepts and methods to other systems. The book is divided into twelve châpters. chapter 1 introduces the subject with a discussion of the qualitaiive propeities of liquid ,He as compared with liquid 4He and other systems. The historical development that finally led to the discovery of the superfluid phases of 3He is sketched. In addition, Y" present an extended elementary discussion of the properties of superfluid 3He the concepts involved. We believe that this will be of -and help to those readers who are not familiar with the subject at all. Chapter Z

Preface

vll

of normal liquid 'He in as much as it is needed for the later development of the theory of the superfluid phases. It covers the phenomenological Fermi-liquid theory as well as recent attempts to calculate the effective interaction leading to superfluidity. Chapter 3 deals with presents the theory

the standard weak-coupling pairing theory of p-wave paired superfluids. The order parameter of such a superfluid is introduced. The ground state and

single-particle excited states as well as thermodynamic properties are calculated explicitly for two model states, the so-called ABM and BW states. Chapter 4 provides a summary of important experimental observations as well as a sketch of the development of experimental techniques. The experimental data are found to support the identifications of the two observed superfluid phases with the two model states mentioned above, in spite of the fact that the weak-coupling theory of Chapter 3 predicts only one stable state, the BW state. This calls for a discussion of corrections to weak-coupling theory, presented in Chapter 5. The general Landau-Ginzburg representation of the free energy given there also allows for a convenient discussion of the effects of a uniform static magnetic field on the phase diagram and the order-parameter structure. Chapter 6 addresses the symmetry properties of the superfluid phases of 3He. Broken symmetries and remaining symmetries are discussed in detail. A general classification scheme for p-wave order parameters is presented. In the final part of this chapter the effect of internal and external fields in orienting the (degenerate) preferred directions of the two order-parameter structures is considered. This sets the stage for a detailed discussion of spatially varying order-parameter conñgurations, so-called textures, and superflow in Chapter 7. After the derivation of the gradient free energies and bending free energies, superflow is discussed as a first application. Then the nature of defects in the two phases is discussed, first for an infinite system and later for finite geometries. Symmetry considerations and topological properties play an important role in this chapter. The properties of linear defects, or vortices, are studied in detail, with an apptication to rotating superfluid 3He, as well as those of planar defects, or solitons. The effect of superflow on the textures and the superfluid state itself is also discussed. The remaining chapters deal mainly with dynamical phenomena. In Chapter 8 the theory oÌ nuclear magnetic resonance in superfluid 3He is presented. A complete solution of the weakly excited spin system is given for uniform as well as nonuniform textures. A few important nonlinear resonance phenomena are discussed in detail. The effect of relaxation processes is also considered. Chapter 9 contains a complete derivation of the phenomenological hydrodynamic equations, based only on symmetry principles. The independent thermodynamic derivatives and transport coefficients in these equations are identified. They are calculated from kinetic theory in Chapt"i L0, which contains a rather detailed exposition of the kinetic theorylof anisotropic fermi superfluids. Chapter 11 addresses the collective modes of the system, both in the hydrodynamic and coltisionless regimes. The theory here is

viii

Preface

based on the hydrodynamic equations and the kinetic theory, and again symmetry considerations are of value. Finally, in Chapter l1we take another look at macroscopic quantum coherence in superfluià tHe and discuss some rather subtle consequences of broken parity and time-reversal symmetry in the two superfluid phases of 3He. We began writing this book in 1983 when both of us were in Munich, namely at the Max-Planck-Insitut fiir Physik and Astrophysik (DV and pWj and the Technische Universitet München (Pril). Since itt"n, uòtn of us have moved'to different institutions: DV to the Technische Hochschule Aachen, and PW first to the University of Florida, Gainesville, and then to the Universität Karlsruhe. We gratefully acknowledge the support by these institutions for our book project. In the course of writing we have had numerous discussions with colleagues and friends around the world, which have improved our understanding of the subject. These include many of our co[eågues at the institutions with which we were associated during that period õf time. In particular, we have profited from discussions with A. Bùas, D. Einzel, G. Eska, I. Fomin, G. G. Ihas, p. Kumar, A. J. Leggett, M. Liu, K. Maki, R. Peccei, P- Ramond, H. saller and L. stodolsky. special thanks go to D. Einzel, T. Kopp and M. Liu for critically reading parts of the manuscript and for suggesting explicit improvements. Above ãf, *" are grateful to A. L. Fetter, H. E. Hall, J. R. Hook, A. J. Leggett and D. Raiñer, who have been so generous with their time in aiding us with their criticism and comments on the completed manuscript. Our cordial thanks go to Mrs Wiltrud (Willi) Huber at the Max-Planck-Institut für physik und"Astrophysik, Munich, for her selfless effort and wonderful spirit in typing the taigest part of rhe manuscript between 19g3 and rgg7, urrã fo, ueing å n"tpruì i' every respect- 'We are also extremely grateful to Mrs G. Kram-p-SalecÊer for typing the final parts of the book and for her invaluable assistance in preparing and organizing the completed manuscript. Furthermore, we should like to thank Mrs I. Kuchenbecker at the Technische Universität München, as well as Mr H. Mattke and Mr R. Roeger at the Technische Hochschule Aachen fot their careful drawing of the figor"r. Lastly, we are grateful to Andrea Wölfle and Jutta Muttenhammer-Vollhardt for iheir help with the references. We also acknowledge the smooth collaboration with the editorial staff of Taylor and Francis, in particular with Julie Lancashire. Finally, and above all, we thank our wives Jutta and uschi and our families for their endless support and unselfish sacrifice of many leisurely weekends during the past five years. Without them, this book would have hardly been finished.

D.

Vor-r-HARDr, P. Wölrr-e

Contents

PREFACE

LIST OF FREQUENTLY USED SYMBOLS INTRODUCTION

1.1

The helium liquids

1..2 Early history of superfluid tHe

1.3

L.4 1.5

Elementary discussion of superfluid 3He L.3.1 The internal structure of Cooper pairs 1.3.2 Broken symmetry and the order parameter 1.3.3 Orientational effects I.3.4 Textures 1.3.5 Superfluid mass currents in ,He-A 1.3.6 Dynamic properties Relation to other fields Reviews and introductory articles on superfluid 3He 1.5.1 General reviews 1.5.2 Reviews of specific topics 1.5.3 Discussions in books on related subjects L.5.4 Introductory articles

THEORY OF NORMAL FERMI LIQUIDS

2.L The quasiparticle concept 2.2 Thermodynamic properties

2.3

2.2.1 Entropy and specific heat 2.2.2 Spin susceptibility 2.2.3 Compressibility Transport properties 2.3.1 Hydrodynamic equations 2.3.2 Kinetic equation 2.3.3 Quasiparticle lifetime 2.3.4 Local equilibrium 2.3.5 The linearized collision integral 2.3.6 Calculation of the transport coefificients

v xvi

I 1

5

7 8

9 12 15

t6 17

2t 22 22 23 24 24

25 25 28 28 30 30

32 32 33 34 37 38 40

x

Contenß 2.4 2.5

Collective modes Theories beyond the Fermi-liquid model 2.5.1, Microscopic theories 2.5.2 Polarization potential theory 2.5.3 Extended Fermi-liquid theory 2.5.4 Lattice-gas model 2.5.5 Phenomenological approaches 2.5.6 Pair interaction Further reading

PAIR CORRELATIQNS IN THE MAK_COUPLING LrndIT

42 46

6

48 49

52 55

57 59

6l

Cooper instability G ener alned pairing theory 3.2.1, Generalized BCS wave function 3.2.2 Diagonalization of the mean-field Hamiltonian 3.2.3 Single-particle excitations 3.2.4 Singlet versus triplet pairing 3.3 Pairing theory at finite temperature 3.3.L Effective Hamiltonian 3.3.2 Gap equation 3.3.3 Critical temperature 3.3.4 Free energy 3.4 Thermodynamic properties of model states 3.4.1, p-wave pairing; the Brù/ and ABM states 3.4.2 Gap parameter 3.4.3 Specific heat 3.4.4 Normal-fluid density 3.4.5 Spin susceptibility 3.5 Expectation value of two-particle quantities Further reading

90 92

BASIC EXPERIMENTAL PROPERTIES

93

3.1 3.2

4.1 Experimental techniques for attaining ultralow temperatures 4.2 Thermodynamic properties 4.2.1 4.2.2

4.3 4.4 4.5

4.6

Phase diagram

Specific heat Magnetic properties Sound propagation Hydrodynamic properties and superfluidity 4.5.L Normal-fluid density and viscosity 4.5.2 Persistent currents Rotating superfluid 3He Further reading

6L

& 65

67 69

70 7L

72 73 75

76 80 80

82 83

84 87

93 95 95

98 100

102 106 106

106 109 111

Contents

XI

PÀTN CORRELATIONS BEYOND WEAK COUPLING

5.1 General Ginzburg-Landau expansion of the free energy 5.2 Phenomenological classification of model states 5.2.1 Unitary states 5.2.2 Relative stability of model states

5.3 Spin fluctuations and the stability 5.4 Effects of a magnetic field

of the ABM state

Further reading

113 118 118 L22

t23 129 136

BROKEN SYMMETRIES AND MACROSCOPIC ORDER

137

6.1, Broken symmetries and group theory

t4t

6.1,.1 The symmetry group describing superfluid 3He 6.1.2 A two-dimensional model of superfluid 3He 6.1.3 Remaining symmetries of the tHe-A, -A, and -B order

7

t13

r43 144

parameters 6.2 Symmetry and order-parameter structure 6.2.1, Continuous symmetries 6.2.2 Discrete symmetries 6.2.3 Symmetry and stationary points of a free energy 6.2.4 Symmetry reduction due to a magnetic field or spinorbit coupling 6.2.5 Symmetry classification of an exactly solvable case: d-wave pairing 6.2.6 Broken symmetries in high-energy physics 6.3 Orientation of the order parameter by internal residual interactions and external fields 6.3.1. Dipole interaction 6.3.2 Orientation induced by a magnetic field 6.3.3 Orientation induced by an electric field 6.3.4 Effect of superflow 6.3.5 Orientational effects in the B phase due to magnetic and electric fields and superflow 6.3.6 Surface energies and boundary conditions Further reading

183 185 188

SUPERFLOIV AND TEXTURES

189

7.1 7.2

190

Superfluidity Gradient free energy 7.2.1, The A phase 7.2.2 The B phase 7.2.3 Healing lengths

1.47

149 150

L62 165

L69 170 L72

t74 174 180 181

L82

194 195 L99 200

xii

Contents

7.3

Supercurrents 7.3.1 The A phase 7.3.2 Finite normal density at T :0 in the A phase 7.3.3 Quantization of circulation 7.4 Topological investigation of defects 7.5 Linear defects 7.5.1 Topological stability 7.5.2 Topological properties of B-phase vortices 7.5.3 Topological properties of A-phase vortices at small distances properties of A-phase vortices at large Topological 7.5.4 distances 7.5.5 Symmetry classification of vortices 7.5.6 Axisymmetric vortices 7.5.7 Energetics of vortices 7.5.8 Vortices in the A phase 7.5.9 Vortices in the B phase 7.5.L0 Magnetic properties of vortices in the B phase 7.6 Rotating superfluid 3He 7.6.L Isotropic superfluid 7.6.2 Superfluid 3He 7.7 Point defects 7.7.t The B phase 7.7.2 The A phase 7.8 Planar defects 7.8.1 Topological classification 7.8.2 Planar solitons in the A phase 7.8.3 Planar solitons in the B phase 7.9 Surface-induced textures 7.9.L 'He-B in a slab 7.9.2 tHe-B in a cylinder 7.9.3 'He-A in a slab 7.9.4 'He-A in a cylinder 7.9.5 'He-A in a sphere 3He-A 7.10 Stability of superflow and related textural transitions in 7.10.L Stability of superflow in the bulk liquid 7.10.2 The helical instability of the uniform I texture 7.t0.3 Textures and superflow in the presence of boundaries 7.10.4 The effect of superflow on domain walls 7.10.5 Flow-induced dynamical textures P air-breaking critical currents 7 7.11..1 The B phase 7.11.2 The A phase 7.12 Dissipation of superflow 7.12.1 Superfluid aHe 7.12.2 The B phase 7.L2.3 The A phase Further reading

.ll

202 203 206 208 zLT

2L9 2L9 220 223 231.

233 235 238 240 245 252 257 257 259 263 264 265 267 267

n2 n8 280 28L ?Å3

286 288 292

294 295 302 312 313

3t9 321 323 326 326 327 329 331.

337

xlll

Contenß

E

SPIN DYNAMICS

339

8.1 8.2

340

8.3

8.4

Derivation of the equations of motion Nuclear magnetic resonance under linear spatially homogeneous conditions 8.2.1 Limit of zero magnetic field 8.2.2 Effect of the magnetic field on NMR Nonlinear NMR phenomena in uniform textures 8.3.1 The Leggett equations for the A and B phases 8.3.2 Spin dynamics in zero magnetic field 8.3.3 Nonlinear spin dynamics in the A phase 8.3.4 Nonlinear spin dynamics in the B phase Texture-induced magnetic resonance phenomena 8.4.1 Tilted uniform textures 8.4.2 NMR shifts induced by superflow 8.4.3 General theory of texture-induced NMR shifts in the

342

344 345 348 348 349 355 3s9 362 363 364

A

phase

8.4.4 Dynamics of solitons in the A phase 8.4.5 NMR signature of solitons in the A phase 8.4.6 General theory of texture-induced NMR shifts in the B phase

8.5 8.6

9

NMR in rotating superfluid 3He 8.5.1 The A phase 8.5.2 The B phase Spin-relaxation phenomena Further reading

HYDRODYNAMIC THEORY 9.1

General principles Thermodynamic identities 9.L.2 Standard procedure for deriving hydrodynamic equations 9.1.3 Equations of motion for the symmetry variables Hydrodynamic equations for the B phase 9.2.1 The thermodynamic identity and equilibrium conditions 9.2.2 Hydrodynamic equations for the symmetry variables 9.2.3 Hydrodynamic equations for the conserved quantities Hydrodynamic equations for the A phase 9.3.I The concept of an intrinsic angular momentum 9.3.2 Equations of motion for the syÞrmetry variables 9.3.3 Thermodynamic identities and equilibrium conditions 9.3.4 Entropy production 9.3.5 Derivation of the hydrodynamic currents 9.3.6. Further consequences of an intrinsic orbital angular momentum

9.11

9.2

9.3

365 376 377 383 389 389 390

394 403

405

406 406 407

409

4tt 4t2 4L3

414 4L6

4t7 418 420 423 425 427

xlv

Contenß

9.4

Hydrodynamics for finite magnetic field 9.4.1, The A phase in a magnetic field 9.4.2 The B phase in a magnetic field

Further reading

10

429 430 432 434

TRANSPORT PROPERTIES

435

10.1 Kinetic

437

equations

10.1.1 Matrix kinetic equation 10.1.2 Conserved quantities and conservation laws 10.1.3 Kinetic equation for Bogoliubov quasiparticles 10.1,.4 Gauge transformation of the kinetic equation 10.1.5 Gauge-invariant densities and currents 10.2 Collision integral 10.2.1 Matrix operator for binary collision processes I0.2.2 Conservation properties of the collision integral 10.2.3 Collision integral for Bogoliubov quasiparticles 10.2.4 Relaxation-time approximation 10.2.5 Bogoliubov-quasiparticle relaxation rate 10.3 Transport coefficients 10.3.1 Transport coefficients of the B phase 10.3.2 Intrinsic spin relaxation 10.3.3 Transport coefficients of the A phase 10.4 Flow in restricted geometries 10.4.1 Slip correction to hydrodynamics L0.4.2 Andreev reflection 10.4.3 Poiseuille flow L0.4.4 Vibrating-wire experiments 10.4.5 Sound propagation and other flow problems 10.4.6 Superfluidiry in tHe films 10.4.7 Ion mobility Further reading

LT COLLECTIVE MODES Ll,.L Hydrodynamic

modes

1L1.1, Sound modes

ll.2 rl.3

11.1,.2 Spin-wave modes Other hydrodynamic modes Symmetry classification of order-parameter modes in the

ll.I.3

collisionless regime 11,.2.1 The B phase 11.2.2 The A phase Time-dependent mean-field theory in the colrisionless regime Collective modes in the B phase LL.3.2 Collective modes in the A phase 11.3.3 Effect of residual interactions

Il.3.t

438

MO 442 445

M7 450 450 453 454

456 457 461,

464 475 479

484 484 486 489 491 491.

494 496 499

501

502 502 50s 511 511

512 515 519 523

526 533

Contenß

Ll.4

xv

tl,.3.4 Observability of collective modes Collisionless sound 11.4.1, Phenomenological model 11,.4.2 Sound propagation in the B phase 1,1.4.3 Sound propagation in the A phase 11.4.4 Nonlinear effects of sound propagation 11.4.5 Transverse sound and spin waves Further reading

s34 535

s36 538 s43 546 546 547

T2 AMPLIFICATION OF \ryEAK INTERACTION

EFI.ECTS DUE TO MACROSCOPTC QUANTUM COHERENCE

12.1 Properties of the BCS pair wave function 12.2 The permanent orbital magnetic moment of the A phase L2.3 The permanent electric dipole moment of the B phase Further reading

s49 550 553

555 560

REFERENCES

56r.

AUTHOR INDEX

595

SUBJECT INDEX

6Ut

List of Frequently Used Symbols

Meaning

Svmbol

Defrnition

Latin symbols

A

prefactor of A-phase

A, Ap¡ (1.t, j

:"1.,2,3)

CN j : I,2,3)

C* (r) (p, *:0,

*, -)

d(k), dr@) Qt,:"1.,2,3) d, dp¡ (lr,

free

below (7.L52) and in (7.233) (5.1a)

normalized orderparameter matrix and components thereof specific heat in the normal (2.t4) phase

Co C, C,j (i,

energy

j

:'/.., 2, 3)

d ê, Î @: e xÎ)

coefficient in A-phase supercurrent tensor coefficient of Aphase bending energy and components thereof normalized orderparameter matrix in basis of spherical harmonics order-parameter vector in spin space and components thereof order-parameter matrix

Li:"ïï:,Tilï',"0

(7.2Lc), (7.39) (7.23b), (7.37b)

(7.86)

(3.34b)

(3.64) (3 68)

direction in spin of the A phase

ffiad of unit vecrors

in

(5.27) and

Frequently Used Symbols

spin space energy of Bogoliubov quasiparticle in state /c

Ek

ß*, F*np(u,

Fì,

þ:1,

J)

Fi

9,8¡Q:t,2,3) It

8"'

8"(T)

H

H*

j j,

j" jíQt:'1.,,2,3) kF

I ,îr, ñ

(î:

,ît x fr)

m*

ms

ñ rrk,

nknp(n, þ

-

xvlt

1, J)

pair-amplitude matrix (offdiagonal distributionfunction matrix) in spin space and components thereof Landau parameters of Fermi-liquid interaction momentum density and components thereof momentum densities of normal and superfluid components dipole coupling constant magnetic field characteristic magnetic field for orientation of Aphase order parameter particle number current density energy current density heat current density spin current density for spin component S, Fermi wavenumber "Îvector": preferred direction of A-phase order parameter in orbital space triad of unit vectors in orbital space effective mass of quasiparticles in the normal phase "spin" effective mass "ñ vector": axis of spinorbit rotation in B phase diagonal distributionfunction matrix in spin

below (6.15) (3.32) (3.43b)

(2.6)

(2.2s) (3.8s), (7.14b)

(6.101)

(7.28c)

(2.ztb) (2.23) (e.73) (2.24)

(2.r) (3.70)

(3.68) and below (6.15) (2.3)

(10.44) (6.106)

Q.a3a)

space NF

density of quasiparticle states of both spin components in the normal phase at the Fermi energy

Q.aa)

Frequenþ Used Symbols

xviii N(o) P R,

- åNe

Rp¡

(lr,

i : I,2,3)

RD ^s

s

T

r, r.

Up

% un

Yo(T)

density of states for one spin component thermodynamic pressure spin-orbit rotation matrix and components thereof dipole torque spin density entropy density temperature Fermi temperature transition temperature to superfluid state Fermi velocity superfluid velocity normal-fluid velocity averaged Yosida function

(2.3) (7.s), (7.10) above (3.84) (3.101c)

coefifrcients of fourth-order

(s.4)

(3.7)

(3.67) (8.7) (2.e)

(2.r3) below (2.14) (3.48)

Greek symbols

þt(i:1,.-.,5) L*, A*np@,

^Q) AoQ)

þ:

1,

l)

invariants in free-energy functional energy-gap matrix in spin space and components thereof average energy gap maximum value of energy

(3.I7), Q3aa) (3.s7) (3.68a), (3.6e)

gap

^ A¡, Au At,

Att

eL AD

ht l.t

EQ)

complex unit vector in orbital space (A phase) components of order parameter in a magnetic freld (A phase) components, of order parameter in a magnetic freld (B phase) Leggett angle dimensionless dipole ;t.

couplingparameter ¡;j* coupling constant of pair interaction chemical potential temperature-dependent coherence length

(7.IZa) (s.s0)

below (5.66b)

below (6.107) (6.e8)

(2.e7) (2.3) (7.18c)

Frequently Used Symbols ç

5o þ

5k 9D þA ¡B Çn' 9H

zero-temperature coherence length quasiparticle energy in the normal state dipole coherence length magnetic healing lengths

(7.18b)

(3.4) (7.27a) (7.28a), (7.29a)

_inAandBphases

,

Þs

fl, II,¡ (i, j :1,,2,3) p

P, P'

P2

P3

op T¡,

rR

xrx

(r)

XN

a)v

(2o, d),

surface healing length stress tensor and components thereof mass density normal-fluid density tensor (with and without Fermi-liquid corrections) superfluid density tensor (with and without Fermiliquid corrections) ¡^lth Pauli spin matrix lifetime of quasiparticle in state ft quasiparticle lifetime at the Fermi level in the normal state Pauli spin susceptibility of normal liquid 3He

Larmor frequency longitudinal NMR frequencies in A and B phases

(7.31a) (2.22) below (2.L0) (3.86), (3.87b)

(7.1sb), (7.23a)

below (2.6) (2.37)

(2.38)

(2.18)

below (8.16) (8.20b), (8.24)

1

lntroduction 1.1

THE HELIUM LIOUIDS

There are two stable isotopes of the chemical element helium: helium 3 and helium 4, conventionally denoted by 'He and aHe respectively. The existence of the heavier one, aFIe, had already been establishèd indirectly in 1871' by its characteristic line in the solar spectrum (hence the name "helium"). Then in L895 Ramsay succeeded in obtaining an actual sample of aHe gas by heating the uranium ore cleveite. It was later found that aile is also _part of the Earth's atmosphere, but only at a fraction of about 2 x L}-s . Kamerlingh onnes \¡vas the first to liquefy aHe in 190g. For a fascinating account of the history of early helium physics and of lowtemperature physics in general, we refer the reader to the books by Keesom (L942) and Mendelssohn (1977). The discovery and identification of the lighter isotope 3He was only made muclr later (Oliphant et al. 1933). In fact, at that time it was thought that the 3He isotope was unstable and that tritium (tH) was stable. ùsing a cyclotron as a mass spectrograph, Alvarez and Cornog (1939a,b) then showed that 3He is indeed a stable isotopic constituent of "ordinary" helium. However, in the atmosphere 3He constitutes only one part in a million of the total helium content, which is so small apyway. The quantities of 3He needed for low-temperature experiments can \nly-be produced by nuclear reactions such as t,,

3ri + åtr* trg+|ot

(1.la)

and the subsequent decay of the tritium

1H-- lHe + e-

(1.lb)

Clearly, this was not possible until after the end of the Second World V/ar. The condensation of 3He gas and the first experimental work on the new liquid \ryere achieved by Sydoriak et al. (I949a,b). Somewhat larger quantities of 3He became available only in the late 1950s. Since then, maõroscopic samples of 3He have been investigated intensively and at lower and lower temperatures.

1. Introduction By the time experimental 3He physics eventually started, the theoretical and experimental investigations of oHe had been going on for such a long time and had yielded so many novel results that "helium" was used synonymously for oHe. White this tradition can be found even in today's scientific literature, one may expect the situation to change in the future. After all, the low-temperature phases (normal, superfluid, solid) of 3He have led to a unique multitude of fundamentally new concepts and have thereby enlarged our knowledge of the possible states of condensed matter more than any other single-component system previously studied. From a microscopic point of view, helium atoms are structureless spherical particles interacting via a two-body potential that is well understood (see Chapter 2). The attractive part of the potential, arising from weak van der Waals-type dipole (and higher multipole) forces, causes helium gas to condense into a liquid state at temperatures of 3.2K and 4.2K for 3He and aHe respectively, at normal pressure. The pressure versus temperature phase diagrams of 3He and aHe are shown in Figs. L.l, and 1..2. When the temperature is decreased even further one finds that the helium liquids, unlike all other known liquids, do not solidify unless a pressure of

around 30 bar is applied. This is the first remarkable indication of macroscopic quantum effects in these systems. The origin of this unusual behaviour lies in the quantum-mechanical uncertainty principle, which requires that a quantum particle can never be completely at rest at given position, but rather performs a zero-point motion about the average position. The smaller the mass of the particle and the weaker the binding force, the stronger these oscillations are. In most solids the zero-point motion is confined to a small volume of only a fraction of the lattice-cetl volume. In the case of helium, however, two features combine to prévent the formation of a crystalline solid with a rigid lattice structure: (i) the strong zero-point motion arising from the small atomic mass (helium is the second-lightest element in the periodic table); and (ii) the weakness of the attractive interaction due to the high symmetry of these simple atoms. It is this very property of helium- T.. A well-known example of such a transition is that from a paramagnetic to a ferromagnetic state of a metal when the system is cooled below the Curie temperature. In the paramag-

10

1- Introd.uction

netic regime the spins of the particles are disordered such that the average magnetization (M) of the system is zero. By contrast, in the ferromagnetic phase the spins are more or less aligned and (M) is thus finite. In this case the system exhibits long-range order of the spins. The degree of ordering is quantified bV Kru) l, the magnitude of the magnetization. Hence M is called the "order parameter" of the ferromagnetic state. Clearly, the existence of a preferred direction M of the spins implies that the symmetry of the ferromagnet under spin rotations is reduced ("broken") when compared with the paramagnet: the directions of the spins are no longer isotropically distributed, and the system will therefore no longer be invariant under a spin rotation. This phenomenon is called "spontaneously broken symmetry"; it is of fundamental importance in the theory of phase transitions. It describes the property of a macroscopic system (i.e. a system in the thermodynamic limit) that is in a state that does not have the full symmetry of the microscopic dynamics. The concept of spontaneously broken symmetry also applies to superconductivity and superfluid 3He. In this case the order parameter measures the existence of Cooper pairs and is given by the probability amplitude for a pair to exist at a given temperature. It follows from the discussion of the possible structure of a Cooper pair in superfluid 3He that the associated order parameter will reflect this structure and the allowed internal degrees of freedom. What then are the spontaneously broken symmetries in superfluid 3He? As already mentioned, the interparticle forces between the 3He atoms are rotationally invariant in spin and orbital space and, of course, conserve particle number. The latter symmetry gives rise to a somewhat abstract symmetry called "gauge symmetry". Nevertheless, gauge symmetry is spontaneously broken in any superfluid or superconductor (see Chapter 3). In addition, in an odd-parity pairing superfluid as in the case of 3He, where l: I, the pairs are necessarily in a spin-triplet state, implying that rotational symmetry in spin space is broken, just as in a magnet. At the same time, the anisotropy of the Cooper-pair wave function in orbital space calls for a spontaneous breakdown of orbital rotation symmetry, as in liquid crystals. All three symmetries are therefore simultaneously broken in superfluid 3He. This implies that the A phase, for example, may be considered as a "superfluid nematic liquid crystal with (anti)ferromagnetic character". One might think that a study of the abovementioned broken symmetries could be performed much more easily by investigating them separately, i.e. within the isotropic superfluid, the magnet, the tiquid crystal etc. itself. Flowever, the combination of several'simultaneously broken continuous symmetries is more than just the simple sum of the properties of all these known systems. Some of the symmetries broken in ¡$[çerfluid 3He are "relative" symmetries, such as spin-orbit rotation ty-ffi or gauge-orbit symmetry (Leggett L972, L973b, Liu and Cross L978).'.Because of this, a rigid connection is established between the corresponding degrees of freedom of the condens-

1.3

3He Elementary Discwsion of Superfluid

11

ate, leading to long-range order only in the combined (and not in the individual) degrees of freedom. This particular kind of broken symmetry, for example the so-called "spontaneously broken spin-orbit symmetry", gives rise to very unusual behaviour, as will be discussed later. The whole 3He and its consequences will be concept of broken symmetries in superfluid discussed in detail in ChaPter 6. It is clear that in principle the internal degrees of freedom of a spin-triplet p-wave state allow for many different Cooper-pair states and hence superfluid phases. (This is again different from ordinary superconductivity with S:0, l:0 pairing, where only a single phase is possible.) Of these different states, the one with the lowest eiergy for given external parameters will be realized.In fact, Balian and Werthamer (1963) showed, that, within a conventional "weak-coupling" approach, of all possible states there is precisely one state (the nW state) that has the lowest energy at all temperatures. This state is the one that describes the B phase of superfluid 3He. The state originally discussed by these authors is one in which the orbital angular momentum I and spin S of a Cooper pair couple to a total angular momentum J: , + S :0. This 3Po state is, however, only a special case of a more general one with the same energy (in the absence of spin-orbit interaction), obtained by an arbitrary rotation of the spin axes relative to the orbital axes of the Cooper-pair wave function. Such a rotation may be described mathematically by specifying a rotation axis fi and a rotation angle 0. In the BW state all three spin substates in (1.2) occur with equal measure. This state has a rather surprising property: in spite of the intrinsic anisotropy, the state has an ßotropic energy gap (see Fig. 3.4a). (The energy gap is the amount by which the system lowers its energy in the condensation process, i.e. it is the minimum energy required for the excitation of a single particle out of the condensate.) Therefore the BW phase resembles ordinary superconductors in several ways. On the other hand, even though the energy gap is isotropic, the BW state ¿s intrinsically anisotropic. This is clearly seen in dynamic experiments in which the Cooper-pair structure is distorted. For this reason the BW state is sometimes referred to as "pseudo-isotropic". Owing to the quantum coherence of the superfluid state, the rotation axis û and angle 0 characterizing a Cooper pair in the BW state are macroscopically defined degrees of freedom, whose variation is physically measurable. Since in weak-coupling theory the BW state always has the lowest energy, an explanation of the existence of the A phase of superfluid 3He obviously requires one to go beyond such an approach and to include "strong-coupling effects", as will be discussed in Chapter 5. In view of the fact that at present microscopic theories are not capable of computing transition temperatures for 3He, it is helpful to single out a particqlar effect that can explain the stabilization of the A phase over the B pha$ê. As shown by Anderson and Brinkman (1973), there is such a conceptually simpte effect, which is based on a feedback mechanism: the pair correlations in the condensed state

L

T2

lntroduction

change the pairing interaction between the 3He quasiparticles, the modification depending on the actual state itself. As a specific mechanism, these

authors considered the role of spin fluctuations and showed that a stabilization of the state first considered by Anderson and Morel (1960, 1961) is indeed possible. This only happens at somewhat elevated pressures, since spin fluctuations become more pronounced only at higher pressures. This state, which is now referred to as the "ABM state" (from the initials of these three authors), does indeed describe the A phase. It has the property that, in contrast with 3He-B, its magnetic susceptibility is essentially the same as that of the normal liquid. This is a clear indication that in this phase the spin substate with S, :0, which is the only one that can be reduced appreciably by an external magnetic field, is absent. Therefore 3He-A is composed only of 111) and lJ{ ) cooper pairs. This implies that the anisotropy axis of the spin part of the Cooper-pair wave function, called d, has the same, fixed, direction in every pair. (More precisely, â is the direction along which the total spin of the òooper pair vanishes: d'S:0.) Likewise, the direction of the relative orbital angular momentum Î is the same for all Cooper pairs. Therefore in the A phase the anisotropy axes â and Î of the Cooþer-þair wave function are long-rangeordered, i.e. are preferred directions in the whole macroscopic sample. This implies a pronounced anisotropy of this phase in all its properties. In particular, the value of the energy gap now explicitly depends on the direction in É space on the Fermi sphere and takes the form A¿Q): Áo(T)[r - (k . î'¡z7rrz (r.4) Hence the gap vanishes at two points on the Fermi sphere, riamely along tÍ (see Fig. 3.4b). Because of the existence of an axis i the ABM state is also called the "axial state". The existence of nodes implies that in general quasiparticle excitations may take place at arbitrarily low temperatures. Therefore, in contrast with 3He-B or ordinary superconductors, there is a finite density of states for excitations with energies below the average gap energy, leading for example to a specific hêat proportional to T3 at low temperatures (see Chapters 3 and 7). The third experimentally observable superfluid phase of 'FI", the A1 phase, is only stable in the presence of an external magnetic field. In this phase cooper pairs are all in a single spin substate, the 111 ) state, corresponding to S, : +L; the components with llJ ) + ltl ) and ll"[ ) states are missing. It is therefore a magnetic superfluid, the first ever observed in

nature.

..

1.3.3 Orientational effects

I ,W,

.g

I

I

For a pair-correlated superfluidi the pairing interaction is the most important interaction, since it is responsible for the formation of the

l.j

iHe Elementary Discussion of Superfluid

13

condensate itself. Nevertheless, there also exist other, much weaker, interactions, which may not be important for the actual transition to the pair-condensed state, but which do become important if their symmetry differs from the aforementioned. In particular, they may be able to break remaining degeneracies. The

dipole-dipole interaction and other weak effects

The dipole-dipole interaction between the nuclear spins of the 3He atoms leads to a very weak, spatially strongly anisotropic, coupling. The relevant coupling constant

S"(f)

is given by

Bo(r):#(X)'"

!

I

I

Et

tl {

(1.s)

Here #o is the nuclear magnetic moment, such that ¡Êsla3 is the average dipole energy of two particles at relative distance a (the average atomic distance), while the second factor measures the probability for these two particles to form a Cooper pair, and n is the overall particle density. Since þt3la3 corresponds to about 1-0-7 K, this energy is extremely small and the resulting interaction of quasiparticles at temperatures of the order of 10-3 K might be expected to be completely swamped by thermal fluctuations. This is indeed true in a normal system. However, the dipole-dipole interaction implies a spin-orbit coupling and thereby has a symmetry different from that of the pairing interaction. In the condensate the symmetries with respect to a rotation in spin and orbital spa^ce are spontaneously broken, leading to long-range order (for example of ãand iin ttre case of 3He-A¡. Nevertheless, the pairing interaction does not fix the relatiue orientation of these preferred directions, leaving a continuous degeneracy. As pointed out by Leggett (I973a,b, L974a), in this situation the tiny dipole interaction is able to lift the degeneracy, namely by choosing that particular relative orientation for which the dipolar energy is minimal. Thereby this interaction becomes of møcroscopic importance. One may also view this effect as a macroscopic amplification of a microscopic interaction via the quantum coherence of the pair condensate. This correlation is equivalent to a permanenr local magnetic field of about 30 G at any point in the superfluid (in a liquid!). Jn 'H: the dipolar interaction is minimized by a parallel orientation of d and l; for details see Section 6.3. Besides the dipole interaction, there are also other, even weaker, interactions, which-although completely unimportant in a normal systemmay be amplified to macroscopic size by the long-range ordèr in the superfluid (Leggett L977a,b,1978a). For example, the electromagnetic interaction of the two 3He atoms in a Cooper pair leads to a minute distortion of the electronic shell of the 3He atom, thereby shifting the centres of the positive and negative charges relative to each other. The orbital motion within the Cooper pair then leads to a magnetic moment

t4

L

lntroduction

along i in every Cooper pair. Because of the long-range order of i these magnetic moments add up to an equivalent magnetic fleld of about 0.02 G. Although this is very small, it makes the system behave differently upon reyersal of the direction of an external magnetic field. Such a change has indeed been detected (Paulson and Wheatley 1978a). Therefore 3He-A is a liquid orbital ferromagnet ! Leggett (L977a) also predicted that the macroscopic quantum coherence in superfluid 3He will raise weak-interaction effects in elementary particle physics to a macroscopic level. For example, the interaction between electrons and protons within a 3He atom has a parity-violating part given by the exchange of a neutral Z0 boson. This effect is expected to induce a finite electric dipole moment along the preferred direction fi in 3He-8, with an overall magnitude of about l0-L2e cm-', where e is the electronic charge. This is still a very small effect, but it does not seem to be hopelessly outside experimental reach. If an experiment succeeded in measuring this effect, it would be the first detection of parity violation on a macroscopic scale. The amplification of these and other effects due to the specific long-range order in superfluid 3He is discussed in Chapter 12. Clearly, of all those interactions the nuclear dipole interaction is by far the most important. Its effect on the relative orientation of the spontaneously preferred directions in spin and orbital space has to be included in any description of low-energy phenomena in superfluid 3He (i.e. where the order-parameter structure itself remains unchanged). Effect of a magnetic field

An external magnetic field acts on the nuclear spins and thereby leads to an orientation of the preferred direction in spin space. In the case of 3He-A the orientation energy is^ minimal l1 d is perpendicular to the freld H, since (taking. into account d. S - 0) this orientation guarantees ^S ll ^tl. Walls Every experiment is performed in a volume of finite size. Clearly, the walls will have some effect on the liquid inside. In superfluid 3He this effect may readily be understood by using a simple picture. Let us view the Cooper pair as a kind of giant "molecule" of two 3He quasiparticles orbiting around each other. For a pair not to bump into a wall, this rotation will have to take place in a plane parallel to the wall. In the case of 3He-4, where the orbital angular momentum I has the same direction in all Cooper pairs (standing perpendicular on the plane of rotation), this means that I has to be oriented perpendicular to the wall. So there exists a strict orientation of i caused by the walls (Ambegaokæ et aI. 1974). In the B phase, with its (pseudo)isotropic order parameter, the orientational effect is not as pronounced, but there are qualitatively similar boundary conditions.

1.3

3He Elementary Discussion of Superfluid

15

1.3.4 Textures From the above discussion, it is clear that the preferred directions i and ä in 3He-A are in general subject to different, often competing, orientational 3He-A). At the same effects (for simplicity, we shall limit our description to time, the condensate will oppose any spatial variation of its long-range order. Any "bending" of the order-parameter field will therefore increase the energy, thus giving an internal s^tiffness or rigidity to the system. While the orientational effects might want d and I to adjust on the smallest possible lengthscale, the bending energy wants to keep the configuration as uniform as possible. Altogether, the competition between these two opposing effects will lead to a smooth spatial variation of â and i througho,ti the sample, called a "texture". This nomenclature is borrowed from the physics of liquid crystals, where similar orientational effects of the preferred directions occur (de Gennes 1974). The largest part of Chapter 7, which itself is by far the longest chapter in this book, is devoted to the discussion of textures in superfluid

3He.

The bending energy and all quantitatively important orientational

energies areînvariant under the replacement â--ã, î--î. A state parallel are therefore and I has the same energy as one where ã and where ä This leads antiparallel. to two different, ate degenerate, ground states. î There is then the possibility that in one part of the sample the system is in one ground state and in the other in a different ground state. Where the two configurations meet they form a planar "defect" in the texture, called a "domain wall" (Maki L977a) (see Fig. 7.22). This is in close analogy to the situation in a ferromagnet composed of domains with a different orientations of the magnetization. Domain walls are spatially localized and are

quite stable against external perturbations. In fact, their stability is guaranteed by the specific nature of the order-parameter structure of 3He-4. Mathematically, this structure may be analysed according to its topological properties. The stability of a domain wall can then be traced back to the existence of a conserved "topological charge". Using the same mathematical approach, one can show that the order-parameter fields of the superfluid phases of 3He not only allow for planar defects but also for point and line defects, called "monopoles" and "vortices" respectively. Defects 'can be "nonsingular" or "singular", depending on whether the core of the defect remains superfluid or whether it is forced to become normal liquid. The concept of vortices is of course well known from superfluid aHe. However, since the order-parameter structure of superfluid 3He is so much

richer than that of superfluid aHe, there exist a wide variety of different vortices in these phases. Their detailed structure has been the subject of intensive investigation, in particular in the context of experiments on rotating superfluid 'H", where they play a central role (Hakonen and Lounasmaa 1987); see Chapter 7.

L

L6

1.3.5 Superfluid mass currents in

Introduction

3He-A

to their ability to flow through tiny pores and narrow slabs, apparently without friction. This extraordinary property, Superfluids owe their name

which is in sharp contrast with our usual experience with liquids, is certainly the most impressive manifestation of the macroscopic quantum coherence of the pair condensate in a Fermi system. Mathematically, superfluidity finds its expression in a complex-valued order parameter, i.e. the existence of møcroscopic quantum-mechanical phase variables. To understand the flow properties of a superfluid, it is helpful to consider it as a system of two independent, interpenetrating components: a "superfluid" and a "normal" component, with densities p, and p,, respectively, such that p,* pn: p, with p the total density. At T > T. one has p,: 0 and pn: p, while at T :0 one has p.: p and p,,:0. This "two-fluid model" is useful in many ways (but, of course, it is only a model, since in reality the liquid is not composed of distinct superfluid and normal particles). In this model a mass current g is the sum of a superfluid and a normal part g, : p"u" and gr,: pnun respectively. Here u" and un ate the velocities of the two components. In a superfluid with isotropic order parameteÍ, W: tpoeiâ, as in aHe, the superfluid velocity u. is simply given by the gradient of the macroscopic, i.e. long-range-ordered, phase of the order parameter:

o,:

!vþ,

(1.6)

lfita

where ma is the mass of a aHe atom. This simple dependence implies Vx

u":0,

(1.7)

i.e. the flow is "curl-free"-1þs¡e is no rotational motion

(*. do not

consider the rotation of the system as a whole). For any closed contour ? in the liquid, lve may define the "circulation" r as the line integral over the velocity:

r:gJßff ds.q:

I

Jarea

d/.

(V x u,).

(1.8)

Using (1.7), it follows that r:0. On the other hand, in a container with a hole (e.g. a vessel with the shape of a doughnut-a "torus") the situation is quite different. The circulation r along a contour € enclosing the hole will clearly be finite as long as u" * 0. Since the phase of the order parameter is a macroscopic quantity, it is well defined along % and can only change by multiplies of 2n when ß is trdversed. In this case K is quøntized and (1.S) takes the well-kno\ryn form of the Bohr-Sommerfeld quantization condition for electronic orbits in an atom:

*-L*. lll4

(1.e)

ry 1.3

3He Elementary Discussion of Superfluid

N:0,

t1,,

t7

12,...,

and hlm¿ is the so-called "flux quantum". This means that supercurrents in a toroidal geometry are quantized and hence cannot decay continuously: they are (meta)stable, their lifetime being of the order of cosmological times. In an anisotropic superfluid such as 3He-A the situation is very different. Here the order parameter not only has a phase but also has an orientation, namely the preferred direction i lttre spin structure is unimportant in this discussion). The anisotropy implies that the system can now distinguish directions relative to L In this case the superfluid velocity will depend not

Here

only on the spatial change Parametrizíng form

of the phase A but also on that of

î.

I by azimuthal (B) and polar (e) angles, us now takes the

,":#r(vø-cosBvø)

(1.10)

(note that the mass of a Cooper pair, 2mr, appears in (1.10)). The fact that u, is no longer given by a phase gradient has an immediate, drastic, consequence: V x u"* 0. This means that V x u", and thereby r in (1.7), can in general take any value, depending on how Í changes. There is no longer any flux quantization! Consequently, a superfluid mass current in 3He-A is unstable (Mermin and Ho 1976). Therefore 3He-A does not seem to be a "superfluid" at all. However, \rye have so far neglected the surface of the coniainer. Since Î has to be perpendicular to thè wail, i.e. it has a fixed orientation there, the circulation at the wall is quantized and the stability of a supercurrent is guaranteed at least very near to the surface. These and related questions are discussed in detail in Chapter 7.

1.3.6 Dynamic properties From the discussion presented so far, we have already seen that the static properties of an anisotropic superfluid are very unusual. Clearly, the dynamic properties can be expected to be at least as new and diverse. Indeed, the fact that in superfluid 3He Cooper pairs have an internal structure can only be investigated in detail by studying the dynamics, i.e. the frequency and momenturn dependence, of the condensate. One may roughly distinguish between magnetic and nonmagnetic dynamic properties, depending on whether the magnetization of the system is probed or whether properties such as mass transport (see Chapter 10) or the propagation of sound are studied. For the investigation of dynamical effects, it is instructive to have an idea of the typical frequencies inherent to the superfluid condensate. For this, we again employ the two-fluid model. Both the normal and the superfluid components are essentially characterized by a single timescale each: for the normal component this is the quasiparticle lifetime t, and for the superfluid

18

L

Introduction

component it is klA(T), where AQ) is the average of the temperaturedependent energy gap. The orders of magnitude of the equivalent frequencies are given by {':L0MHz and A(T)ln-103(1 -TITL)t'' MHz, i.e. usually one has ø-1 Y,€knn, . Y *n*n,þ: I{n*np}. u'

(2.2g)

u'

The left-hand side of (2.29) accounts for the continuous change of the distribution due to the streaming of quasiparticles in phase space. The collision integral I describes discontinuous changes brought about by collisions. It is now necessary to express the hydrodynamic densities and currents in terms of the quasiparticle distribution function. This has been done in part in (2.8)-(2.L0). The remaining quantities are obtained as follows (Nozières 1964). Multiplying the kinetic equation (2.29) by hk and summing over /c, one generates the momentum conservation law (2.22). Since momentum is conserved in each quasiparticle collision, the collision integral drops out. By comparison with (2.22), the stress tensor is identified as

il,j:Z

kuþ

hk¡u¡r¡nBn¡rpo,

(2.30)

E r,l i

2

34

Theory of Normal Fermi Liquid"s

where we have introduced the quasiparticle velocity

(2.31)

uknþ:Y *ê*oþ.

for the energy current i.(r, t) for the

Similarly, the expression

spinsymmetric case is derived, using the definition of the energy-density change

ôe

:2

(2.32)

,onorrooo,

kø

by multiplying with e¡,oo and summing over k, a. The result is

j.:2

(2.33)

€kn&knofl*on' ku

Finally, the spin current

j,

is obtained as

ji::L

(o*) kuu'þ'

nn,u *n,F,flkp,n.

(2.34)

2.3.3 Ouasiparticle lifetime Before we turn to a detailed discussion of the collision integral, let us first consider the lifetime of a quasiparticle, rkn. At low temperatures, when binary collisions dominate, this is given by the Golden-Rule expression

:'+2 Tkrn,

x

@G,

z;3,

õu1+a2,u3*.0

4)12 nz(L

-

nr)(r

-

n+)ô*,*k2,k3+ka

ô(e, * ez- et- ec).

Q.35)

Here the delta functions express conservation of momentum, spin and energy in the scattering process, and the Fermi factors represent the probability for state 2(:kr, ut) to be occupied and 3,4to be empty. The transition probability is given by the square of the quasiparticle scattering amptitude a('1.,2;3, 4) lor the binary collision process (I,2)+ (3, 4). Spin-rotation invariance implies that there are only two independent scattering amplitudes because both the initial and final states have to be either spin-singlet or spin-triplet. Also, in the limit of low temperatures (f ^:

#,*rr, rr, /'o{"o follows from symmetry that t¡n is isotropic and

";øf

such that (1)"-1. It independent of spin. The quasiparticle relaxation rate is seen to be proportional to Tz or (.0 - ¡t)2, whichever is larger, and hence becomes much smaller than the excitation energy itself for sufficiently low temperatures. This property

F= H

2

36

Ë

Theory of Normal Fermi Liquids

r

t !

forms the basis of the quasiparticle picture, because it guarantees that the quasiparticles are sufficiently long-lived to be treated as well-defined Note that this is a consequence of the Fermi statistics and of energy "ìtitièr. conservation, leading to the restriction €¿ - þ l ktT in the two independent energy integrals in (2.35). It is difficult to measure the quasiparticle lifetime 2fl, in the normal state. However, one can determine rfl, from experiments probing relaxation effects in the superfluid state close to the transition. In this way one frnds a value T,t?r:0.3 ¡rs (mK)2 (Wheatley 1978) at intermediate pressure. Using this number, we may estimate the temperature Tq at which the quasiparticle energy width nldu is equal to the thermal energy ksT as approximately 3He is only a0 mli. Strictly spèaking, the quasiparticle description of liquid valid in the temperature regime below TqHaving calculãted the quasiparticle lifetime, we are already in a position to derive the low-temperature limiting behaviour of the transport coefficients (Abrikosov and Khalatnikov 1959). All we need for a qualitative discussion is to recalt the results of the kinetic theory of gases as applied to the quasiparticle gas. The gas-kinetic expressions for the shear viscosity 46, the thermal condùctivity ro and the (spin) diffusion coefñcient Doo tead

qo: tpulz$ *

ro:

T-', I

jCpu'."$ *

t-t,

I Doo: år'."R, n T-2. )

(2.40)

Here we have made use of the fact that the quasiparticle velocity is given by the Fermi velocity up and that the collision time is of the order of 2fl,. Rtt the transport coefficients diverge as Z+0 because the mean free path /o: urz$-ot ttt" excitations becomes infinitely long. While r¡ and Dohave a f-z ãeþendence, which is due solely to z$, the thermal conductivity diverges only as T-1 owing to the linear T dependence of the specific heat C¡. This behaviour has been observed experimentally, although only at temperatures below about 30 mK, i.e. far below the Fermi temperature (Wheatley 1966, 1970, 1975, Greywall 1984). The reason for the substantial àeviations from Landau theory at intermediate temperatures (see e.g- FigZ.'1. for the specifrc heat) lies in the excitation of low-lying collective modes, in particulai spin fluctuations (Berk and Schrieffer 1966, Doniach and Engelsberg L966, Brenig et al. L967, Brinkman and Engelsberg 1968, Riedel 196¡, Amit et al. 1968a,b, Baym and Pethick L978, Mishra and Ramakrishnan 1985). These effects are, strictly speaking, beyond the regirne of applicability of Fermi-liquid theory. The temperature dependence of the thêrmodynamic quantities at finite temperatures, i.e. above the lowtemperaiure regime where Landaulp-theory is valid, has been considered in several model calculations. These include (i) a systematic extension of Landau theory (Pethick and Carneíro 1973, Carneiro and Pethick 1977,

i i

I

2.3

37

Trønsport ProPerties

Coffey and Pethick 1988), (ii) the spin-fluctuation model (Levin and Valls 1.983), and (iii) the Gutzwiller-Hubbard lattice-gas description, discussed below in more detail (Seiler et al. L986), While the simple expressions given in (2.40) certainly give the correct temperature dependence, their prefactor only gives the correct order of magnitude. In the following we shall show that (and how) these prefactors b" calculated exactly. This, however, requires a more widely ranging "un treatment.

2.3.4 Local equilibrium The collision integral in the kinetic equation (2.29) is the rate of change of the distribution function for a nonequilibrium state by collisions- It is precisely given by the relaxation rate (2.35) of a nonequilibrium state times the distribution function n¡,nr, i.e. Io,n,{non)

: -'+2 x

ø>1) one

finds

,o-,,{r

.i#(1

+

+Få)(î)'.

"[(i)J],

(277a)

"":lÇ+#(i)'{'.4(i)1}

(277b)

As mentioned above, the deviation of the zero-sound velocity ca from the hydrodynamic sound velocity c1 is only of order (urlrr)í

sära[. The

2.4

Collectiue Modes

45

attenuatioa uo, however, is a factor of (uplc)3 smaler than what one might expect on dimensional grounds. The transition from first to zero sound may be achieved by working at fixed frequency and lowering the temperature, whereby z n T-z increãses (Keen et al. L963; Abel et al. 1965). One finds an increase in sound velocity by a fraction of a per cent and a maximum in the sound attenuation ai a)r - L, in agreement with (2.77a,b) (see Fig. 2.3). There are other forms of zero sound that have been found experimentally, too, although they are much less well defined, e.g. transuerse zero sound (Roach urrã K"tterson 1976a,b, Lea et al. 1977) and spin zero sound (Masuhata et al. L9g4). The coupling of density and spin zero sounds induced by a small particte-hole asymmetry has also been discussed (Ketterson 1,983, Fishman and Sauls 1e8s).

The density response of liquid 3He at frequencies several times the Fermi frequency has been studied by means of neutron scattering, (Sköl d et al. L976, Sköld and Pelizzari L976, 1980, Scherm et aI. 1974, rgg7, Stirling

cîF

oo ç

3l-1.

29.3 bqr 20.2t, M Hz

e-¿ ola --¿

?o m

3 C

.o d coJ ú l0 -o

ca o

(,/)

I

2

/. 6 810 ïemperoture (mK)

Figare 2.3

20

t0

60

Sound attenuation and velocity change in liquid tFle versus temperature as measured by Ketterson eú al. (1975). The solid lines are the result of a fit to (2.76). The onset of the superfluid state is signalled by the sharp features at the low-remperature end. (After Wölfle (tfSa¡.¡

2

46


et al. 1976, Stirling 1978). The scattering is dominated by excitation of particle-hole pairs generating a broad continuum for energy transfer hu and momentum transfer q satisfyíng lka - qzlLm*l< qu¡. Sköld and Pelizzari (1976, 1980) observed what seems to be a continuation of the zero-sound mode into this high-frequency regime. The existence of such a mode was anticipated by Pines (1966) and Aldrich and Pines (1978) on the basis of the so-called "polarization potential theory" to be discussed below (see also Aldrich et aI. t976).

2.5

THEORIES BEYOND THE FERMI-LIOUID MODEL

Landau's theory of Fermi tiquids is phenomenological in the sense that the observable properties are expressed in terms of an effective interaction (something iitt a set of pseudopotentials), namely the interaction function Ao'1(9,0). It is then io,o,U or, nror" generally, the scattering amplitudes ãrËuã¿ that these pseudopotentials are well described by a small number of parameters, which depend on pressure but not on temperature and magnetic field provided that T3 at all (Herring L966, Stamp 1985). The paramagnon theory assumes that longùved and long-ranged spin fluctuations ("paramagnons") determine the 3He. Within that particular theory the low-temperature properties of liquid strong enhancement of the spin susceptibility is interpreted as a consequettð" of an incipient ferromagnetic instability of the liquid. This is why liquid 3He has long been called "almost-ferromagnetic". A variational solution for the lattice model (2.56) was proposed by Gutzwiller (1963,1965). He constructed a variational wave function that restricts local density fluctuations in lÚo), the wave function of the noninteracting system, i.e. one that reduces the weight of those spin configurations in lúo) that allow for too many on-site interactions. The appróach is therefore similar in spirit to that of the correlated basis function tC-ge) approach (Feenberg1969, Woo t976). To calculate the expectation value of (2.86), Gutzwiller used an approximation that is equivalent to calculating the classical statistical weights of different configurations of particles (for a discussion see Vollhardt 1.984a) and that becomes exact in d--* dimensions (Metzner and Vollhardt 1988, 1989). If an equal number of particles and lattice sites is assumed (half-filled band) then the results of thii approximation describe a transition to a locallz;ed state with single occupancy of lattice sites at a finite interaction strength U : U" (Brinkman and Rice 1970). Although the transition itself is an artefact of the approximation, the results for U < (1" for the effective mass' the spin susceptibility and the compressibility closely resemble the experimentally 3He at T 2ke.Similar forms have been deduced from real-space models of the (1978) and from þolanzation potentials employed by Aldrich and Pines tni..or.opic CBF calculations by Krotscheck and collaborators (Krotschek 1e83).

2.5.6 Pair interact¡on Finalty, we consider the pair interaction, i.e. the interaction responsible for the superfluid phase transition in liquid'He. As will be discussed in detail in

Table2.2 Parameters a* characterizing the quasiparticle scattering amplitude (2.96) and pair-interaction coupling constants Â¡ according to (2.98).

Pressure

a6

Au

An

Ao

h2

(bar)

0 L2 2t 34

4.18 2.39 -0.r2 t.62 1.35 5.30 3.50 -0.07 1..74 L.54 5.54 3.62 -0.09 1.82 1.65 0.34 2.L0 L.75 5.13 3.23

-o.22 0.15 -0.33 0.31 0.14 0.06 -0.28 0.34 0.14 0.07 -0.30 0.35 -0.02 0.29 -0.36 0.40

2

Figure

2.5


Momentum-dependent Landau parameters as functions of the reduced particle-hole momentum Q : q lkr. (After Pfitzner and V/ölfle (1987).)

Chapter 3, this is the interaction between two quasiparticles with momenta (k, -k).The matrix element V*-*,of this interaction describes the process where rwo quasiparticles with momenta (k, -ft) are scattered into states (k', -k'). There are arguments (Patton and Zaringhalam 1975) that the pair interaction is given to a good approximation by the scattering amplitude

A(0, O)

for 0 : Ir, and cos @ : ü' k'. More precisely, the exchange-

symmetrizedinteraction V is given by a(0, @) averaged over a small region around 9, such that the weak logarithmic singularity in A(0, @) for 0 : n is removed. The pair-interaction constants V,, obtained from a Legendre expansion of V¡,-¡,(see (3.5a)), tttuy then be calculated from "1.f1

Âr

| d(cos = N(O)V.: 1] J-t

@) P¿(cos

þ)A'(n,

(2.e7)

Q),

:0

or L for / even or odd. Substituting the expression (2.96) and using the property of the Jacobi polynomiah ff;'ol1-1) : (-l), one obtains with

s

t, - l(2t * !)-rrz ) tr + 1)t"(k:t

For the values

of. a¡¡,

(2.e8)

r)oo,o.

given in Table 2.2, one finds )ro, )rr, )r2as shown in the

same table.

As can be seen from Table 2.2, the pair-interaction constants positive (reputsive interaction) for

l:0,

2 and negative for the / :

Â¿ are

L channel

Further Reading

59

(see also Bedetl and Pines 1980b). An attractive interaction in the p-wave channel mediated by spin fluctuations was first proposed by Layzer and Fay (1968, L97L); see also Østgaard (1973,1974,1975).In the next chapter it will be shown that the attractive p-wave attraction leads to a phase transition characterized by the formation of pairs in a relative p-wave state. 3He. We This concludes our review of the properties of normal liquid have focused mainly on the limiting low-temperature behaviour, which provides the starting point for the theory of the superfluid phases, our main subject. It appears that the semiphenomenological Landau theory of Fermi liquids provides an accurate description of the low-temperature^properties

3He is still of nor*ul liquid 3He. A reliable microscopic theory of liquid

lacking, although important progress has been made in recent years. 3He at Much interest has recently been devoted to the behaviour of liquid intermediate temperatures up to the Fermi temperature, where precise measurements (Greywall 1983) for example of the specific heat show distinct structure. This behaviour appears to reflect a single-particle spectrum with a rapidly decreasing effective mass, as one moves away from the Fermi edge, both in momentum and energy (Brown et aI. 1982, Krotscheck 1983). Naturally, the structure of the excitation spectrum for higher energies and momenta also enters a microscopic calculation at zero 3He must account for the temperatut". n complete theory of normal liquid quasiparticle energy and interaction in a consistent way.

FURTHER READING Abrikosov A A and Khalatnikov I M 1959 Rep. Prog. Phys. ?2 329 Physics of Liquid and Solid Helium, Part II, Baym - G and Pethick C J 1978 inB The ed. K H Benneniann and J Ketterson (Wiley, New York), p. L Campbell C E 1978 in Progress in Liquid Physics, ed. C A Croxton (Wiley, New York), p.2t3 Feenberg'E 1969 Theory of Quantum Fluids (Academic Press, New York) Keller W ¡, t969, Helium-3 and Helium-4 (Plenum, New York) Landau L D 1956 Zh. Elcsp. Teor. Fiz.30 1058 lSou. Phys. IETP3920 (195J)l Landau LD lg57 Zh. Elcsp. Teor. Fiz. 32 59lSou. Phys. JETP 5 101 (1957)l Landau L D 1958 Zh. El<sp. Teor. Fiz. 35 97 lSou. Phys. JETP S 70 (1959)l Leggett A J 1975 Reu. Mod. Phys. 47 331 Levin K and Valls O T 1983 Phys. Rep. 981 Nozières P 1974 Theory of Interacting Fermi Systems (Benjamin, New York) Pines D 1985 in Highlights of Condensed Matter Theory; Course 89 of the Varenna Surnmer School (Societa Italiana di Fisica, Bologna, Italy), p. 580 Pines D and Nozieres P 1966 The Theory of Quantum Liquids (Benjamin, New York) Vollhardt D 1984 Reu. Mod. Phys. 56 99 rWheatley J C Lg70 in Progress in Low Temperature Physics, Vol. VI, ed. J C Gorter (North-Holland, Amsterdam), P. 77 rwilkò J 1967 The Pioperties of tiqiia ønd Solid Helium (Clarendon Press, Oxford) "/ Wölfle P 1979 Rep. Prog. Phys. 42269

I 1i

li

Pair Gorrelations in the Weak-Goupling Limit

The concept of pair correlations in interacting Fermi systems was frrst introduced in 1957 by Bardeen, Cooper and Schrieffer (1957) (BCS) to provide a microscopic understanding of superconductivity. This phenomenon had resisted theoretical explanation for more than 40 years. According to modern interpretation, the salient feature of this theory is the concept of spontaneously broken gauge symmetry, or, in more elementary language, tÈe fact that the pair-correlated state is described by a complex rather than a real-valued order parameter. This fact is usually related to the appearance of a gap in the single-particle excitation spectrum. The phase rigidity of the \ryave fúnction (or order parameter) of the pair-correlated state is respon-

superfluidity or superconductivity, which is probably the most spectacular property of these systems. We shall see later 3He deserve the name superfluid. In this io what extent the new phases of chapter we present the so-called weak-coupling theory of pair correlations in Fermi systems. As the interaction giving rise to pair correlations becomes weaker, we should expect both the transition temperature I to the paircorrelated state to decrease and the characteristic length Eo of pair correlations to increase. The small parameter of weak-coupling theory is the ratio T.lh (or equivalently (k.fo)-t). For tiquid'FI", this ratio is of order 10-', suggesting that weak-coupling theory, which becomes exact in the limit T.lTp-¡'O, should be a reasonable starting point. Why weak-coupling theory is not quite as good as one would expect from the smallness of the parameter f"lTF will be discussed in Chapter 5-

sible

for the property of

3.1

GOOPER INSTABILIW

Let us flrst consider a simple model problem introduced by Cooper (1956) in the preliminary stages of the development of BCS pairing theory, which itluminates best the basic idea of pairing theory. Consider a system of N

3

62

Pair Conelations in the Weak-Coupling Limit

non-interacting identical fermions of mass rn and spin I in the ground state (the "Fermi sea"). If we add two more particles, the ground state of the N + 2 particle system is of course obtained by putting the two particles in the lowest available states at the Fermi energy. Flowever, what happens if \rye now switch on an attractive interaction V(r1- a) just between the two added particles? Quite generally, the wave function of our model system is given by the antisymmetrized product of a correlated pair wave function Q(rr, rzi u, þ) and an N-particle Slater determinant describing the Fermi sea. The pair wave function in turn is a product of the centre-of-mass plane wave, the wave function describing the relative motion !t(rt-rr), and the spin functions X@, þ): (3.1) Q(rr, rzi &, É) : e*p Í*¡p . (r1+ r)l tþ(rrrr)X(u, þ). Since the particles are identical by assumption, the pair wave function must be antisymmetric upon interchanging particles L and 2, implying that either r/(r) is odd and X@, þ) is even (spin-triplet state) or vice versa (singlet state). The important effect of the particles in the Fermi sea is to block the single-particle states below the Fermi energy. This is most simply taken into

account by working

in the momentum representation. Deflning

Fourier

components by

ú(r) :2

o'

Vo:

.to''rl,o: (zn)-s I

\ OtO

"'o''rl,o,

r^dtr e-'*''V(r),

(3.2a) (3.2b)

J

the pair Schrödinger equation takes the form (Eo**,t

*

E*-r,,

- E),þ o : -(zù-t Ir,_od k

v*-*.1þ *,.

(3.3)

Here the blocking effect must be taken into account in the sum over intermediate states ft'. For the following, it is useful to measure singleparticle energies from the Fermi level, deñning the quasiparticle energy

ç h2k2 5*: * - lt,

(3.4)

where ¡r is the chemical potential (equal to the Fermi energy e, at zero temperature). It is clear from (3.3) that the lowest energy E is obtained when the two particles have equal and oppositely directed momenta. rWe put P:0 in the following, because finite P corresponds to a trivial centre-of-mass motion of the pair with momentum kP. Equation (3.3) may be separated into angular-momentum components owing to the assumed spherical symmetry of the interaction potential. By in terms of Legendre polynomials Pt(k' /c') and expanding Vo-o, and

3.1. Cooper Instability sphericat harmonics

63

Y,^(î), one has

v*-*,:Ë e, + L)U(k, k,)n(îc, . îr,¡, t:o 1þ

*

: 2 o,^rlt,(k)Y,^(Ê). lm

(3.5a)

(3.sb)

In order to allow for an explicit analytic solution of (3.3), we now introduce a further simplifying assumption, by putting V{k, k') equal to a constant within a thin shell around the Fermi surface and zero elsewhere:

U(k,

(V, (l6ol, l5o'l < e. (( ep), k'): to' otherwise.

(3.6)

Equation (3.3) then reduces to the separable integral equation (zEo

- E)q'ík): - Y,N(O) ["

,1,,{O') d|o,,

(3.7)

where the density of states (for one spin species) N(0) has been taken out of the integral, N(f) being a slowly varying function on the scale of e.. It is worth noting that (3.7) has essentially the same form as the Schrödinger equation for just two particles in two spatial dimensions. This is due to the fact that the effective density of states in the Cooper problem is finite (and constant) above the Fermi energy and zero below, exactly as in the case of the scattering of two particles in two dimensions. By inspection of (3.7), one can see immediately that for attractive interaction (V, < 0) the energy eigenvalue E is necessarily negative. So, in the presence of the Fermi sea, the two test particles form a bound state for

arbitrarily weak attractive interaction. The bound state is formed with relative orbital angular momentum L corresponding to the strongest attractive interaction parameter V2. The correlated pair is called a "Cooper pair". Cooper's problem is in marked contrast with the two-particle problem without blocking effect, where a bound state is formed only if the potential is sufficiently attractive. An accurate necessary condition for the existence of a bound state in three dimensions is (Rosen 1982)

Iv for electrons in superconductors that is a properly antisymmetnzed product of pair wave functions.

3.2

65

Generalized Pairing Theory

3.2.1 Generalized

BCS wave funstion

Phrased in the language of second quantization, this ansatz, appropriately generalized to anisotropic and spin-dependent pairing, takes the form

l,/)

:

T' T

(uo*,*à

ro.uo;.o:ou) P¡,

(3.11)

where the product over /< is restricted to a half-space, say k ' î > 0, to avoid double-counting, and s: 1, J. The wave function lrp) reduces to that for the normalstateif we putu¡,nn:0(k -k.) andu¡on:0(kç-k), ukn-n:0' The anticommutation relations of the field operators imply u-knn: tt¡,aq ãlrLd n-*sþ: -ukpn. The operator af¿at¿B acting on.the vacuum state l0) (the state without particles) creates a pair of (quasi)particles in single-particle states (ku, -kB). The probability amplitudes for the pair state to be occupied or empty, uknB and u¡nn respectively, are considered as variational parameters. They may be determined by minimizing the energy expectation value fornted with lV) or by diagona\izing the Hamiltonian directly, taking advantage of the particularly simple type of correlation admitted by (3.11). We shall follow the latter procedure. Before doing so, let us remark on an unusual property of the wave function (3.11). By expanding the product, it is seen that lr/) consists of a superposition of \ryave functions describing O-,2-,4',. .. ,N-particle systems. In other words, the state lV) is not an eigenstate of the particle number operator N. On the other hand, in an isolated system the total number of pãrticles is fixed. In this case the system has a particular symmetry, called ;'g",rg" invariance". This means that the Hamiltonian of the system is inva.iant under the global gauge transformation aln+ Q'"oln, where 0 is a fixed, bat arbitrøry, phase variable. Clearly, the state.lrl), which may be characterized by the set of coefficients {uonn,u*.,p}, is in general not gauge-invariant since it transforms into a state l!,Q9)> specified by \uo,,, eneu*np]r.In the thermodynamic limit the states lrl(Ô)) with different A are mutually orthogonal, i.e. they represent inequiuøIent quantummechanical states. (By contrast, the gauge-transformed \ilave frunctions for the normal state differ only by an overall phase factor and are therefore unitarily equivalent, i.e. the ground state is unique.) Despite the inequivalence of the states lqt@)), they are still energetically degenerate since the energy does not depend on O at all. This implies that the ground state is not unique but has a continuous degeneracy. Some of the most intriguing properties of pair-condensed states are a consequence of this fact, as will be shown later. Any one of the pair-correlated states lrl(0)) is a state with broken gauge symmetry, because it no.longer possesses the gauge symmetry of the Hamiltonian. Since an invariance under a phase transformation is mathematically equivalent to a U(1) symmetry, the broken gauge symmetry corresponds to a broken U(1) symmetry (see Section 6.2). The same

3

66

Pair Coruelations in the Weak-Coupling Limit

concept holds in the case of a Bose-Einstein condensate, i.e. superfluidity in a Bose system; see e.g. the discussion by Huang (1987). Because of the long-range order associated with the broken gauge symmetry, the phase e of the pair-correlated state lú(ø)) is defrned throughout the system. It will become clear later (see Chapter 7) that the most important property of this macroscopic phase variable is its "stiffness" or "rigidity", i.e. the fact that it costs energy to change it in space or time, whereas its absolute value does not lead to observable consequences (as usual in quantum mechanics). An elementary but lucid discussion of phase rigidity ih a Bose-condensed state (where again the same notion applies) and of the concept of rigidity as such, has been given by Anderson (1984). The fact that the absolute value of the phase is not an observable may be used to construct a pair-condensed state with definite particle number by forming a phase-coherent linear superposition as

lrlr.)

:

f"

dE

sioNorz

lq)@)>,

(3.r2)

where No is the particle number (Anderson 1958a,b, 1966). This relation is analogous to a Fourier transformation relating position and momentum operators respectively. Since the latter operators are canonically conjugate, there exists an uncertainty relation between them. The same is true for the phase and the particle number, which obey the commutation relation I+N, Ql: i, with N:2iAlAA in the @ representation. It is important to bear in mind that the integration over the phase @ in (3.12) only removes the ouerall phase but leaves the phase rigidity intact! It is the rigidity, not the existence of an overall phase, that is responsible for the phenomenon of superfluidity, as will be discussed in detail in Sections 7.1 and7.2. The above discussion is of course not limited to systems with broken gauge symmetry, but applies to all systems with a broken symmetry. Let us therefore analyse the situation in the case of a crystal, where things are more easily visualized. In the crystalline phase, with its strict periodicity, translational invariance is spontaneously broken. This implies that the conjugate variable to a displacement, the momentum, is no longer conserved. A state describing the system may be written as lrl(Rr)), where R¡ are the lattice vectors. In the thermodynamic limit the states l?(Rr +¡)) with different r, where ¡ is some displacement, are inequivalent, i.e. mutually orthogonal. Superposition of such states by integrating e*'' lú(R,+¡)) over .r leads to a state with definite momentum P. This means that the overall position of the crystal (given by, say, the centre-ofmass coordinates) is now completely uncertain. But, at the same time, the periodicity of the crystal with its rigidity against displacements of the atoms from their positions R¿, i.e. its long-range order, is of course still intact. The properties of the BCS wave function (3.11) discussed above seem to require one to abandon the concept of a fixed phase variable. However, in a macroscopic system the fluctuations ÂN about the average particle number

3.2

67


are proportional to Nuz, so that fluctuations in the phase are proportional to y-t]z and hence are negligibly small. Therefore in the thermodynamic limit the phase ¿s fixed. It is then simply more convenient to work with (3.11) rather than (3.12).

g.2.2 Diagonalization of the mean-field Hamiltonian Let us now embark on the calculation of the parameters u¡,no ãîd

U*uþ-

euite generally, âtry expectation value of a product of creation (destruction) operatórs for fermion quasiparticles al (a*), taken in terms of the wave function (3.11), may be factorized into products of expectation values of bilinear combinations of freld operators. These expectation values are found AS

(Wl oi"o o, B lW) :

õ ¡,¡,, ô

(Vl a-¡,Bao',lW) :

(3.13a)

oBn¡,n,

(3.13b)

õ*r,F*op,

where

fikn:2

(3.laa)

u*npuloB,

p

: - F-*po . (3.14b) The correlation function nkn describes the distribution of particles in F*op

:

U

*oBuTpp

momentum Space, while F¡,.,Bis referred to as the "pair amplitude"The Hamiltonian of the interacting quasiparticle system is given by

H

-

pN

:2 ku

++> kk,q

E*,al.oo,

uþu'þ, (3. 15)

where Eoo:E*- aþoH for a flnite magnetic freld ^F/ and a: *1- or -L for by the pair 1 ot I respectively. In the subspace of Hilbert space spanned *uu"-funciions (3.11) only pairs of particles with equal and opposite momenta are correlated. The fact that pair correlations are of dominant importance can be exploited to simplify the product of field operators in the interaction term of (3.15). This is achieved by replacing one pair of operators, alal,ot a¡,a¡,,, by its expectation value. Thus, for a given pair, the other pairs act effectively as a kind of mean field. The resulting mean-field Hamiltonian may be written as

H*,

-

ltN

:2

Eo*aip*,

ku

1

-2

* ä Z (ni,uo-*fl*o kuþ

Al*pF*np. kuþ

+ af¿aleBAop) (3.16)

3

Pair Correlations in the Weak-Coupling Limit

Here \ile have defined the "off-diagonal energy" or "off-diagonal mean

field"

AonpbY

A*nþ:

k'a'þ'

with the pair amplitude F¡np given by (3.13b). The mean-field Hamiltonian (3.16) is a bilinear form in the quasiparticle field operators and may be diagonalized by a canonical transformation (Bogoliubov L958, Valatin 1958), i.e. the off-diagonal terms a-¡pØ¡.o rn'd ai,nloe may be eliminated. The new field operators bon and þ[, are defined by

boo:2 p

bï,

@o,ñ*p

-

: 2 @i,uolp F

u¡oBalop),

(3.18a)

uf,,Ba

(3.18b)

-

kp),

where u¡np ãrrd u¡nB îra again variational parameters (in fact the same as those in (3.11)). By means of the convention (3.1e) It-ka7: Ukpo, 'U -knþ: -U kþo , it is easily shown that the anticommutation relations of the new operators are indeed canonical, i.e. (3-20a) {bon, bn,p} : {bI* ,bl,p} : O,

ô**,6nþ,

{boo, bl,p: if the 2x2 matrices u¿ and v¿ (defrned respectively) satisfy the conditions

-

V*U*

u¿uf, *

v¿vf,

V*Y*

:

by their elements u¡nB a;îd 0,

: )(voufv lt),pb ou * 2, pp

u¡,nB

(3.2t)

:1. )]

With the help of these relations, the inverse transformation is found akn

(3.20b)

o,pb!*p .

as

(3.22)

Equation (3.22) may be simplified for Hermitian matrices u¡, using (3.2I), as

akn

:

Z(uo,pb op * u e,pb!

¡B).

(3.23)

þ

3He are such that As will be shown later, the pair correlations in superfluid

(3.23) applies. Substituting the field operators a¡ and af in the Hamiltonian (3.16) with the aid of (3.23), one is led to a bilinear expression in the new fermionic field operators b¡, and bf. For (3.20a) to be fulfilled the coefficient of the operator product bib:k (and therefore, by Hermiticity, of. b¡,b-¡,) must vanish by suitable choice of u¿ and v¿, yielding the condition (3.24) 2Ç,1ra¡,v ¡, - v eL[v ¡, * u¿A¿u¿ : 0,

3.2


where A¿ (etc.) is the 2xZ matrix defined by the elements A¡*B (etc.). Equation (3.24), together with the normalization condition (3.2I), may be easily solved for two cases that will later be found to be of special interest: (i) unitary v¿ and spin-independent ri¿; and (ii) diagonal v¿ and u¿. Writing u¿ and Y¿ âs

ttknp: ô'BxenlD¡',\ t)kny: - AooplDo*, )

(3.2s)

with x¿, and Don to be determined, (3.24) reduces to a quadratic equation fot x¡o, with the (positive) solution (3.26) Xkn: Eon I Eon . Here we have defined

Eko:fE"o,*

(Lo\l),*l'''.

(3.27)

assumption, AoÂå is a diagonal matrix. From condition (3.2L), one finds immediately

By

Dko:

fLEo*(Eo*

the normalization

+ Eon)lt''.

(3.28)

After substitution of u¿ and v¿ into the Hamiltonian, rearranging operator products to normal order and collecting terms, one finally obtains H"",-

t.tN:)

[+to-o

Ð""(+-#r;)

As the operator b*b is positive, the ground state condition b¡,* lw) :0.

lV)

* nr,øi**,]-

Q-zs)

is characterized by the

It is easily checked that the wave function given by (3.11)

(3.30) has this property-

3.2.3 Single-particte excitations Excited states are obtained by operating with by

b[* on lW). The state defined

(3.31) lW; kal : bi*l|Ø) describes a single elementary excitation, a so-called Bogoliubov quasiparticle. It is a momentum eigenstate with eigenvalue kk. The single-particle excitation energy is found from (3.29) to be

Ekn:U#-

p)'

*4

oo*uoi,,rf'

(3.32)

At the Fermi surface, hzkzl2m*: ¡t, the excitation energy does not tend to zero as in the normal state, but rather remains finite: the energy spectrum

3

70

Pair Correlations in the Weøk'Coupling Limit

E¡

EF

1o--

0

a

a

a

¿

kFk Figure

3.2

itt momentum space (see The otr-diagoãal mean freld A¡*B is therefore referred to as the

has a gap depending

eis.

l2i.

Energy spectrum of Bogoliubov quasiparticles.

in general on the direction

Ê

"gap parameter"-

3.2.4 Singlet versus triplet pairing materials the pairing of conduction electrons F¡nB and the gap takes place itt ttt" spin-singlet state, i.e. the pair amplitude p"r"."t"¡ A*nF utã both antisymmefüc 2X2 mattices, leaving only one independent element. One may express this fact as

In usual superconducting

A*: (-i î)

:

aio2,

(3.33)

where o2 is the second Pauli matrix'

3He unambiguously contrast, experimental evidence for superfluid imptie, spin-tripleì pairing- In this case F*nþ and A¡,oB are, symmetric of matrices, determine¿^ UV tñree independent elements. This multiplicity, :0, tL of a course, corresponds to the three possible projections of S' and Fk spin-s:l system. Rather than taking the amplitudes of ak to the three S' projections, it is more convenient to combine is achieved "ärr"rponding the thiee spiñ components intô a vector d(k) in spin space. This p:1,2,3, by ,"punding A¿ itt ttt" three symmetri" 4*2 maffices io,,o2, where atrare the Pauli matrices, as

In

:

A*nþ:2 It

¿-@)(orio2)oB

: (-o';'o' o,f,o,).

Q.3aa) \

3.3

7t

Pairing Theory at Finite Temperaure

Conversely, this yields

d*(k)

:

(3.34b)

-+tro (A¿o2ior), ld.(k)l' : Ltr.- (^fAo)

(3.3ac)

tr, (...) it the trace over the 2x2

spin matrices. It is easily verified from (3.34) that d(/c) transforms like a vector under rotations in spin space, since

where

u(ß)Aeur(R)

:2

(3.3s)

¿Lrn> - snþo,þ, a*n,p,:0, 2 ø'þ'

(3.36)

where (S*p,,p,)p: (a*)oo,ôpp'* õ*,,(ou)pB' is the total spin operator of the Cooper pairs and (o*),B is the u, þ component of the Pauli matrix o*. In terms of. d(k), the square of the energy gap is given by

(Âoaf),B -- d(k) - d*(k)õ*p

* tì

Íd.(k) x

d*(k)lu(o*),8.

Q.37)

oc 1 are "unitary". It follows that for with the property "urr"o ^^+ :0. The quantity d x d* represents a kind of spin unitary states d x d* angular momentum of the Cooper pairs, at least locally on the Fermi

States

surface (the average over the Fermi surface may still be zero). Note that the in the general case are given by (Takagi 1973) eigenvalues of ^^+ (3.38) laol',.,r: d(k)' d"(k) {ld(k)' d" (k)l' - Êg¡a*'(k)}'"

r

-

The self-consistent theory of a pair-correlated Fermi system described above may now be used to calculate physical quantities at zero temperature. However, it is more useful to extend the theory to finite temperatures first. The zero-temperature properties are then easily derived by taking the limit T-+ 0.

3.3

PAIRING THEORY AT FINITE TEMPERATURE

The preceding treatment of the many-body problem is readily generalized to finite temperature T. For T > 0 a certain fraction of pairs will be broken into single-particle excitations, such that the probability for pair formation


3

will be reduced. Above a critical temperature in equilibrium.

I

Cooper pairs cannot

exist

3.3.1 Effective Hamiltonian The temperature-dependent probability amplitude for the existence of Cooper pãirs is given by the generalization of (3'13b):

(a-*pa*,n):

ô*,Fo'p-

(3'39)

Here the angular brackets denote the thermal average, defined by

(A)

H.'rF-PN\I ( .l / Hr,rc-¡¿N\ll / I ((-ä/l :t'{al"*n (-ff )lltr lexn

(3.40)

The full Hamiltonian (3.L5) is approximated by a temperature-dependent mean-field Hamiltonian of the form (3.16), with A¡,'B given by (3'17) in terms of the temperature-dependent FpnB defined above. Passing through steps analogous tõ (f .fa)-Q.29), one arrives at the diagonalized effective Hamiltonian

H\/rr,-

þtN

:> lkaL

ä(Eo, +

- +> p

Eo)-'(LoLl),'

al,pF*,p+

Ek,bi;bk,f..

(3.41)

The single-particle excitation energy E¡"n and the Bogoliubov quasiparticle operatois 6oo, bfo are given by (3.32) and (3.18), with, however, the tËmperatut"-ã"p"n¿ent gap parameter A*oþ.For a detailed derivation of ql.[t¡ in rhe case of s-wave pairing see Eilenberger (1965). It iollows from the form of the effective Hamiltonian (3.41) that the Bogoliubov quasiparticles are independent fermions. At finite temperature these states ute oð"opied according to Fermi statistics, which implies

(bi,bo,el: ô**,6nþf*n, where

(?")* r] ', (bo,bo,p> : > A, the f integral on the right-hand side of (3.46b) is dominated by the f-l dependence of the integrand, leading to a logarithmic dependence on €c. The gap equation (3.46b) may also be written in an integrated forl !¡ performing the trace and averaging over k. lf-, multiplying both sides by ^å, is given by an arbitrary linear combination furthôrmoie, the gap parameter fo spherical harmonics Y¡^(d}¡) of a definite order l, i.e' Af!>'

: e

I

ì-/*,

o

þY'*(dZ a)'

then, using the ProPertY (2L + L)(P'(!,

'

îc"¡dg)'u) n,

-

ô,t&Íllp,

Q.a6c)

the integrated gap equation has the form

) (rur ù'-lt- lv¿lN(o,f"ot

tanh (E¡nlZkBT) E*o

l), - o.

(3.46d)

Employing (3.a6d), the average value of the gap parameter in the limits T + T. and T + 0 may easily be calculated (see below). In conventional superconductors, the pair interaction is strongest in a

3.i

Pairing Theory at Finite Temperature

relative s-wave state. The gap parameter A*op is then completely specified by a single />'1,, the

Jt]'-"

Pair Correlations in the Weakûoupling Limit

'-'

f integral in (3.47c) is calculated as

orw::l'"!*,."(+\ \€./ Jt z z :rn (c, ¡;f),

where

fl'dz--l tanhz f- dz- I-tanhz c:l JozJlz

:ln1+y Jf

and

C6:äe'_

(Zln)er. Here y:0.5772 is Euler's constant, yielding

1,.L34.

Equation (3.47b) yietds a relation between the critical temperature cutoff energy e. and the pair interaction constant V¡:

r":co,fi"*o

t-treh]

Co:

4,

the

(3.48)

as appropriate at melting pressure, and a cutoff energy of (O.O+-t.O)k"T, (see below), we find from (3.48) that the coupling constant ¡¿tol lyrl vaiies Ùetween 0.3 and 0.2. This is within the range of values for from the normal-state quasiparticle scatte¡ing amplitude (see

For

?"

:2.6mK,

Âr'áeduced Section 2.5).

3.3.4 Free energy To a very good approximation, the entropy of a pair-correlated Fermi system is cãrried onty by the singte-particle excitations. The collective excitations (see Section 9.4) contribute only to higher order in T"IT, owing to phase-space restrictions. Hence the entropy is that of a system of indåpendent fermions-the Bogoliubov quasiparticles-and is given by the familiar form ,S

:

+ (r - fo') ln (1 - fo)l' -ke 2Ífo*lnfo.kæ

(3.4e)

Substituting Eon from (3.32) and performing partial integrations with respect to Eo, this maY be written as ðfo* ç--1s tz t* u* ðEoo' T

e

(3.s0)

The internal energy is found from (3.41) by substituting the pair amplitude

3.3

77

Pairing Theory at Finite Temperature

from (3.43b) and performing the thermal average

(H -pN) :

as

)> ro'or)""1æ - Ç+t] . à Eoor*o.

(3.s1)

The free energy is obtained by combining (3.50) and (3-51)- Partially integrating the terms involving /¿' yields

p=(H-pN)-rS ø

E> 1), with x : lA*llknT, where we have used (3.56b) and (3.80) respectively- The function Yo(Ê; T) is a measure of the density of thermal excitations at the point É on the Fermi surface. Note that (3.87a) does not reduce to pôa for T-7.Lf. m"lm*1,. This is because Fermi-liquid corrections have not yet been taken into account. They are necessary, however, to ensure Galilean invariance. As discussed in Section 2.2, the bare velocity field u,, is screened by backflow effects involving the Landau parameter Fì, and must be

86

i

Pair Correlations in the Weak-Coupting Limit

replaced by the effective field

un,en:

"^-+#o*^.

(3.e0)

In writing (3.90), we have assumed that the Landau parameters are not

affected by the pair correlations to order T"lTi. This may be understood by observing that the Fermi-liquid interaction involves poiarization contributions from intermediate states with momenta all ovèr the Fermi sphere. However, only a small poftion of these states near the Fermi energy are modiñed by the pair correlations. Substitutitrg u,, by u,,,.r in (3.g6), the Fermi-liquid-corrected normal-fluid density is obtained as (ieggeìt 1965a,b, Combescot 1975e) Ptt:

{m (',+ " -p/ \ åFìd)-'og,

(3.e1)

wùere in general pfl and p,, àre tensors and (. . .)-t is the matrix inverse for pi considered as a matrix. The superfluià density p" is then given by p": p1 - p,, (see (7.15b)). In the BW state the Yosida function yo@;T)-yo(T), see (3.101c), is independent of É and hence p,, is isotropic:

Pn¡:

ô¿¡P

(1 + åFì)% 1

+ åFìyo

(3.e2)

pn decreases linearly for T < T; and vanishes exponentially fast for T+ 0. The Yosida function Ye(T) is shown in Fig. 3.5. In the ABM state p,, is an axially symmetric tensor, oriented along i. rne components of pn near T" are obtained by substituting the expansion 1:.SO¡

Yo(T)

0.8

0

Figure

3.5

0.4

0.8 r tTc Yosida- function Yo(T) for the BW state versus reduced temperarure f lf..In the evaluation the weak-coupling BCS gap function /1f¡ nut been used.

3.4

Thermodynamic Properties of Model States

87

into (3.87), with the result

p't- p -z(pû- p): -g E@# r(#)'

(3.e3)

P,, in the ABM state is even stronger at lower temperatures. As argued before, for AslksT >> L, quasiparticles are excited ãuinfy near the "po.-l" caps" of the Fermi sphere. The weight functisns Æ? appearing in the angular average in (3.87b) are of order L for components päiallel io i, Uut of order (keT I Aò2 for the perpendicular components. i{rn." for T + 0 the two independent components of p,., tend to zero as

The anisotropy

of

P"tr

., ffi* lktT\t :fr-P m \ Áo /, 7na m* /kBT\4

(3.e4)

Pnt:15P^\^),

rather than vanishing exponentially. Accordingly, the anisotropy of the superfluid density P": P1- p,, vanishes at T :0. For a discussion of p,' in other superfluid model states see Takagi (L97ac)-

3.4.5 Spin suscePtibilitY One expects the spin susceptibitity of a general triplet staie to be reduced from the Pauli value owing to the presence of Cooper pairs with magnetic quantum number ,S, :0, which do not couple to a magnetic field. Since the orientation of the Cooper-pair spins is described by the preferred direction in spin space, d, the susceptibility 1 is a tensor in spin space, with axes determine dby d. The situation is slightly complicated by the fact that d is in general a function of position on the Fermi sphere. Let us therefore consider frrst the simpler case of the ABM state, where the preferred direction d in spin space is independent of lc (see (3-68)). . When a weak uniform static magnetic field ^[I acts on the system the magnetization M is generally determined by

M

: XH,

(3.95a)

where in the following 1 is a diagonal tensor with components 7t and 2¡,,. The component of. M along .ãf is given by the number of up-spin particles minus the number of down-spin particles, multiplied by the magnetic moment per particle, ¡ts:

M

-

po} k

(þipr) - þipotl).

(3.esb)

Let us first consider the case dl-.ÍL As can be seen from (3.34), ArnBis diagonal and we may use the results derived in Section 3.3, with Eo replaced by Eo,: E* - aHlto. In particular, the number densities of up- and

3


down-spin particles are given by (3.43a), and hence the magnetization is

M:-þo?"ær"n(yo).

(3.e6)

Neglecting Fermi-liquid corrections, the susceptibility is obtained from (3.96) by differentiating with respecr to .FI and purting Ë1:0: xot

: zpu@ I or

+ (fio"n

*")-

.rR

.

(3.e7)

Since in this case the occupation numbers (a[¿a¡,.) depend on Ii only via $on, the result is the same as in the normal state. Physically speaking, ih"r" are no cooper pairs with s, :0 if d r H, i.e. all pairs couple to the magnetic ñeld. In (3.97) the superscript zero indicates the absence of

Fermi-liquid corrections, which will be included later. If particle-holeasymmetric terms (generated, for example, by the energy dependence of the density of states) are taken into account, one finds a small enhancement of the spin susceptibility of order TLITï over its value in the normal state

(Takagi 1974a). The result is different for dllH, since then dr-d2:0 in (3.34a), i.e. A¡nB is purely off-diagonal. This case is not covered by the discussion in Section 3.3. Solving (3.21) and (3.24), one finds the coefficients of the Bogoliubov-Valatin transformation as

E* ukno: E**

Dk'

uknþ:+,

(3.e8)

with D*

:

[zEk(Ek

* Eo)l''', E* :

[87

+ l&olrlr,r.

Thus for psH (( Á there is no explicit dependence on I{ ! As before, the single-particle excitation energy can be read off from the transformed Hamiltonian as

Eon: Ek - upoH.

(3.ee)

The occupation numbers are readily found by substituting the abov e ukno and u¡nn factors into (9*l4a): ?

3.+Jt

!' (, 5¿\ , (ol,on*> :!(t* *\' ¿\ ,;)fr'*;(t -ä)o-fo-)' Ü

L

(3'1oo)

The magnetization and the susceptibility without Fermi-liquid corrections

are obtained

as

M:tro)(,f*,-foù, k

xît:2pz?(-

#):

x,?,Yo,

(3.101a)

(3.101b)

3.4

Thermodynømic Properties of Model States

89

where the Yosida function YoQ) is given by

%(r): (%(¡e; r)l¡,

(3.101c)

with Yo(ie; T) defined by (3.8S). Note that for spin-singlet pairing AnB is purely off-diagonal (see (3.32)) and the preceding discussion applies, i.e. 7¿o(T):X$yo(T), a result due to Yosida (1958) for s-wave pairing. We see that the spin susceptibility of the ABM state depends strongly on the relative orientation of d and H. In its general form the susceptibility tensor reads

)(o*n:tR,{ôu"

- ârâ"¡t- %(r)l}.

(3.102)

We are now ready to tackle the conceptually more complicated case of the BW state. The clue is to first consider the contribution to the susceptibility tensor from a given point É on the Fermi sphere. This may be calculated in the local frame of reference parallel to d.(k), the local preferred direction in spin space. For the BW state , âr(Ò: E¡ R*¡Ê,¡, where Rr¡ is the rotation matrix characterizing the BW order parameter. The contribution to Xo¡"" from each point É hur the form of (3.102), with, however, ã replaced by ã&). The total spin susceptibility Xo*n is arrived at by summing over ft: (3.103) xo*": xR( ôr" - â*1*¡â"(k)[1 - yo(Ê; r)])¿. Here Yo(Ê; f) is defined by (3.88). For nonunitary states, d.(k) may be a complex vector, in which case â*â" should be replaced by ne {áuá}¡ in (3.103).

It is now a simple task to evaluate (3.103) for the BW

state. Observing

that Yç(i,; Z) is isotropic, one flnds

xo*":

ôr"1R,[1

- å(1 -

%)],

(3.104)

a result that is easily interpreted as follows: the BW state is an equal

mixture of S, :0, tL pairs, of which the S, : 0 species does not contribute to the magnetization. Hence the susceptibility is given by its normal-state value minus one-third of the superfluid contribution 1 - Y". The above results are again subject to Fermi-liquid corrections. As discussed in Chapter 2, an applied magnetic field is screened by a polanzation of the medium described by the Landau parameter Ffi, resulting in the effective magnetic field

H"n: H -

FW

(3.10s)

,rR,

In the presence of Fermi-liquid interactions the magnetuation M induced by a magnetic field is equal to that of .the noninteracting system in a magnetic field H.n, or M : xoH"n. Solving for M yields the renormalized spin

3

90

Pair Cotelations in the Weak4oupltng Limit

susceptibility (Leggetr 1965a,b)

,:(t.#)

'ro,

(3.106)

where yandTo are tensors in spin space. For the BW state, the final result is

xe)-r*ffit.

(3.107)

For Z->0 and Fi:-1, Q.107) rends ro about tN,*, i.e. the reducrion is twice the value of the unrenormalized result.

rn general, there are also corrections to 2¿ from higher Landau parameters. For the BW state, owing to its high symmetry, onry F) enters (Czerwonko 1967) (see (10.56)).

3.5 EXPECTATION VALUE OF TWO.PARTICLE

OUANTITIES

Following Leggett (1975a), we finally want to determine how the twoparticle correlations involved in côoper pairing influence one_ and two-particle quantities in the superfluid. It tu*r out that in most cases the expectation valueôf a one-particle quantity acquires only a small correction of order (T/TÐt - 10-6 caused by the- sligirt energy variation of the one-particle density of states (namely if this quantity is ân even function of the energy Eò. Examples are the total num-be, of particles and the total momentum. An important exception is the kinetic_energy part of the Hamiltonian (3..15), E*E*nu The expectation value of this ãne-particle operator, Eo Eoþù, is indeed substantially modified by the formation of the condensate owing to the linear dependence on Ç¿. In the case of two-particle quantities (those involving an interaction . between two particles) we should, on the other hand, añeaoy intuitively expect that they are particularly sensitive to the pair correlaiion. Let us write such a quantity

QØ

: ä2

n*m

as

e(rn, rî pn, på ofP,

oy\,

þ, v :

I,2,3),

(3.10S)

rhere

we have specified that it generally depends on the position, nomentum and spin of any two particles in the system. We denote its :xpectation value by (QØ). The most obvious example is the interaction part V of the Hamiltonian 3.15) itself. By means of the pair amplitude F*nB, one finds from (3.15)

(v)

: 12 Ai*eF*nþ. kuþ

(3.109)

Ising the gap equation (3.L7) and the simplified spin-independent interac_ ion (3.45), this gives

(v): 2 vo_o,F*npFl,,p. kuk'B lhis result is not surprising-it merely expresses the fact that

(3.110)

(Z)

is the

3.5

Expectation Value

of Two-Particle Quantities

9t

sum over all momenta of the interaction weighted by the probability amplitudes F*oF, Fl'*p for the scattering from a pair state ft to one with fr'. More generally, we may assume Q to be a spin-dependent operator of the form Q

:>2 nm

- r^)of)of,),

Q*"(r,

p,v

(3.111)

involving particles fl, ffi; the indices ¡r, v specify the components of the spin, and we assume Q*u(rn - î*): Q¡"u(r^ - r,): Quu(rn - r^). This is not the most general form that Q can have, but it is sufficient for our purposes. We now insert (3.111) into (3.108), multiply by Jd'" I d'r' ô(r- r,)ô(r'r^) and write the sum oveÍ n, m as En+^:En,^-8,. Next, the spin density or(r) is introduced as

o*(r):4

"Í:,

ô(r

- r,)

(3.1r2)

: å > (o),prþI!)rpe!),

(3.113)

uþ

where we have written

o*(r) in second-quantization form, with

q,,t) -) This yields

). o(')::'eå à x

aoo eik''.

(3.

U4)

t ). [ t'r ¿" d'r' Q*"(r - r')(o*)*p(o")ro J Èr"J

tpit)lti!')r¿oþ')rlteþ).

(3.11s)

The effect of the pair correlation on the expectation value of QØ involves only the anornalous averages (rprp) etc., and we therefore find

:, (eQ)>:+

) t Ia3rg,"1r)(o,\*6(o" o"' Q*"Q)(o*)*B(o")"aFl'(r)F5BQ), à",oà J

(3.116)

where

F*eQ)

-) k

Fonu eik'',

(3.117)

To simplify the notation in the present case of triplet pairing, we proceed in (3.3a) and introduce auector F¡in spin space via

F*oþ:) @rior)nþF*t"

as

(3.118a)

It

such that F*p

: -+> (o2ia*) nB FpnB, uþ

(3.118b)

in exact analogy with Aonu and d(k); 11* is the Fourier transform of ^F(r) in position space. Inserting (3.7L7) and (3.118a) into (3.116) and performing the spin summation (with )tro (o*auc¡,op): ô*uô4 - õ¡,¡õnp + ôt pô"^), we find

(QQ>>: å

)

o', Q*"Q){lF(r)12

ôr"- 2 Re [{,(r)ri(r)]},

(3.11ea)

3

92

Pair Coruelations in the Weak-Coupling Limit

which is identical with

(eØ): å > 28*"@- k'){Fn.Fl,õt""*2Re [Fo*Fi,"l], kk' ¡tv

(3.119b)

where 0r"@) is the Fourier transform of Q*"(r). Note that the simpler spin-singlet case, which is spherically symmetric, can be obtained from the above expressions by making the replacement i Q*"+ Qô¡"u and Fe,*- F*, where F*: (a-*g*¡\. In terms of F(r): Xo F*e*'', one finds (O(')).,.* r:

I Êr Q¡)lr(t)l'.

(3.120)

The basic structure of (3.119a) and (3.L20) is essentially the same, the additional complications in (3.119a) are only due to the spin-triplet pairing. Now lF(r)l'is just the probability density for a Cooper pair with separation r, giving a very plausible interpretation of the result in (3.120) for the the operator Q in a pair-correlated state. In this expectation value (Q)

"fsimilarity between a Cooper pair and a very large respect there is a certain "diatomic molecule" (Leggett 1980a,b), where the expectation value of an

operator is given by an expression identical with (3.120) but with F(r) replaced by the pair wave function of the molecule. We shall later discuss this molecule picture in more detail (Chapter 12).If used with proper care, it proves to be helpful in calculating macroscopic properties of superfluid 3He that involve the internal structure of the 3He Cooper pair. FURTHER READING

A selec'tion of books on BCS theory Abrikosov A A, Gorkov L P and Dzyaloshinski I E 1963 Methods of Quantum Field Theory in Statistical Physics (Prentice-Hall, Englewood Cliffs, New Jersey; reprinted L975 Dover, New York) Fetter A L and Walecka J D t971, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York) de Gennes P G 1966 Superconductiuity of Metals and Alloys (Benjamin, New York) Lifshifz E M and Pitaevskii L P 1980 Statßtical Physics, Part 2 (Vol. 9 of the Landau-Lifshitz Course of Theoretical Physics) (Pergamon, Oxford) Rickayzen G 1965 Theory of Superconductiuity (Wiley, New York) Schrieffer J R 1964 Theory of Superconductiuity (Benjamin, New York) Tinkham M L975 Introduction to Superconductiuiry (McGraw-Hill, New York) Review articles on weak-coupling pairing theory of liquid 3He Anderson P W and Brinkman W F 1978 in The Physics of Liquid ønd Solid Helium, Part II, ed. K H Bennemann and J B Ketterson (Wiley, New York), p. I77 Lee D M and Richardson R C 1978 in The Physics of Liquid and Solid Helium, Part II, ed. K H Benneman and J B Ketterson (Wiley, New York), p.287 I-eggett A J 1975 Reu. Mod. Phys. 47 33L Wheatley J C 1975 Reu. Mod. Phys. 4il 4L5

Basic Experimental Propert¡es 4.1

EXPERIMENTAL TEGHNIOUES FOR ATTAINING ULTRALOW TEMPERATURES

The experimental investigation of liquid and solid 3He

in the low-

temperature, quantum regime could not have taken place without several important improvements in cooling techniques that took place in the 1.950s 3He-oHe dilution refrigerator, based on a cooling cycle and 1.960s. First the 3He in liquid oHe as proposed by London involving the heat of solution of (1951), had been developed (London et al. 1962, Das er al. 1965, Neganov e/ ø1. L966, Hlall et al. 1966, Wheatley et al. I97I). It provides a means for removing relatively large amounts of heat down to temperatures of several millikelvin. In order to go below these temperatures, one has to work with systems that order at even lower temperatures. These are spin systems of various types whose ordering temperature is below the temperature one wants to reach and that at the same time possess a large heat capacity, good thermal 3He. A single-cycle conductivity and tow Kapitza boundary resistance to cooling process is possible owing to the fact that, while the total entropy is conserved in an adiabatic process, the entropy of the spin system (i.e. the coolant) placed in a magnetic field increases upon lowering the magnetic field, such that the system absorbs heat (principle of adiabatic demagnetization). A method for cooling liquid 3He widely used in the 1960s consisted of 3He placing powdered cerium magnesium nitrate (CMN) directly in the sample chamber in order to circumvent problems of heat transfer (CMN is a paramagnetic salt in which electrons order magnetically at a temperature around 1mK). After precooling the CMN-3He mixture in a strong magnetic field to a starting temperature of typically 20 mK, the sample is thermally decoupled from the precooling stage. By adiabatic demagnetization, it is then possible to reach a temperature of about ZmK, just sufflcient to study the properties of superfluid 3He near the transition temperature at higher pressures (Wheatley 1966, 1970, L973, t975). Much lower temperatures may in principle be attained-and have actually

94

4

Basic Experimental Properties

been reached-by adiabatic demagnetization of. a nuclear spin system. Here

a suitable material is copper metal, owing to its good electronic thermal conductivity and the large spin-lattice relaxation. These properties provide the necessary thermal contact of the 3He liquid sample with the copper conduction-electron system in the bulk (usually via the large area of a metallic "sponge" or sinter) and from there to the nuclear spin system. The first successful nuclear demagnetization experiment was performed as early as L956 (Kurti et al. 1956), although the lattice and electron systems did not cool substantially. However, since then adiabatic nuclear demagnetization has been developed into a powerful cooling method, which is now exclusively used to attain temperatures in the millikelvin regime and below (Osgood and Goodkind 1967, Ahonen et al. l974b,c). The ordering temperature of copper nuclei has been determined to be around 60 nK, i.e. 6 x L0-8 K (Lounasmaa 1984), which opens up a whole new temperature regime in the microkelvin range. Since the nuclear moment of Cu is so much smaller than the electronic magnetic moment, the precooling ternperature has to be rather low (80 kG) in order to remove a substantial fraction of the spin entropy. These extreme conditions may be somewhat relaxed if a material with a hyperfine enhanced nuclear magnetic moment, such as PrNi5 or similar rare-earth compounds, is used (Andres and Bucher L968, 1972). 3He nuclear spin systern itself as a Quite a different idea is to use the 3He refrigerant (Pomeranchuk 1950). Unfortunately, the cooling of tiquid by nuclear demagnetization is impractical because the nuclear magnetization of the Fermi liquid 3He is reduced by a factor Tlh from its classical value. 3He attainable in a laboratory is Hence the relative magnetization of liquid onty of the order of 10-3 and is thus much too low to provide any significant cooling poïver. In contrast, solid 3He can be magnetized substantially at a precooling temperature of several millikelvin, at which it still behaves as a classical spin system. On the other hand, to reach thermal equilibration in an insulator like solid 3He takes an increasingly longer time the lower the temperature. Therefore a metal like copper is much preferred as a refrigerant. Nevertheless, 3He may be employed as a refrigerant in yet another, very specific, way. This makes use of the large difference in the entropies of solid and liquid 3He. (In the solid the spins behave classically, the entropy being given by Ssori¿ : R ln 2; in contrast, the entropy of the liquid is Sriq n T lTi owing to the Pauli principle, hence Suo ( Ssori¿ et low enough temperatures.) This anomaly may be exploited for cooling near the melting curve, even without applying an external magnetic field. By simply compressing the liquid adiabatically, solid 3He with high entropy is formed, thus decreasing the temperature of the remaining liquid. This is the so-called "Pomeranchuk effeõt", which may be employed to cool 3He along the melting curve (Anufriyev 1965) until the spins in the solid order (at around 1mK) and the entropy of the solid is reduced to a value below that of the liquid. This

4.2

Thermodynamic Properties

95

makes the method ideally suited for the investigation of the magnetic phase transitions in the solid, as well as of properties of the liquid at melting 3He were pressure. In fact, as already mentioned, the superfluid phases of discovered in a Pomeranchuk experiment designed to investigate the magnetic phase transition in the solid (Osheroff, Richardson and Lee 1972a). The magnetic transition in the solid was only found later (Halperin et al. L974). One of the major experimental problems at these lowest temperatures is to establish an absolute temperature scale. To this day the interpretation of experimental results is hampered by uncertainties in temperature measurement. Various experimental temperature scales have been proposed. The majority are based on the magnetic properties of an electronic or nuclear spin system, for which one postulates the temperature dependence of the ideal model system and a value of ?: at the melting curve. Thus there exists a "La Jolla temperature scale" (Wheatley 1975) and a "Helsinki tempera-

A

second possibility is to derive a thermodynamic temperature scale from measurements on 3He alone. This has been done along the melting curve, yielding the "Cornell temperature scale" (Halperin et al. 1975) and at lower pressures assuming ideal Fermi-liquid behaviour above [ (Greywall 1986). The extremes, given by the La Jolla and Helsinki scales, differ by about IÙVo in the I value and up to 40Vo in the specific-heat results. A critical evaluation of the different temperature scales, including a detailed discussion of the possible origins of these discrepancies has been given by Greywall (1986). In this book we shall use Greywall's temperature scale.

ture scale" (Ahonen et al. L976c).

4.2

THERMODYNAMIG PROPERTIES

4.2.1

Phase diagram

The first indication of the existence of two phase transitions in condensed 3He at temperatures of about 2.6mK and 1.8mK respectively was a barely visible kink and a discontinuity in the pressure versus time curve (see Fig. 4.1) of a Pomeranchuk cell operated at a constant rate of volume change (Osheroff et al. L972a). The two signatures, called A and B, clearly indicated a second-order transition (A) and a first-order transition (B) (Halperin et al. 1973). Using nuclear magnetic resonance (NMR) measurements, these were later unambiguously identified as coming from the liquid rather than the solid (note that liquid and solid 3He coexist in a Pomeranchuk cell). In contrast with the A transition, the B transition did not always occur at the same temperature, but rather showed substantial supercooling. Upon warming and cooling several times, the B transition that occurred during a cooling cycle took place in a wide temperature range, depending on the history, whereas it happened at approximately the same

4

96


Ç 1.8

E (l,

Ë

Oç,

o

2.65 = d 3.0 o

o-

t-o,

ô¡J I

o E (t,

a

lr.0

ãlo ç

fL-

o E X

o L 50

Time Figure

4.1

(m inu

tes

oo-

)

Pressure versus time trace when the volume in a Pomeranchuk cell was changed at a constant rate. The signatures A and B, shown magnifled_in

the inserts, indicate the occurrence of two phase transitions in 'He. (After Osheroff et al. (1972a)-)

temperature during warming. It was also found that an applied magnetic field led to a very small splitting of the A transition into two transitions A1 and 42, whereas the B-transition temperature was strongly reduced (Gully et al. Ig73). In a series of experiments using CMN as a refrigerant, the exploration of the phase diagram was extended to pressures below the *étting pressure, down to somewhat below L0 bar (Greytak et aI. 1973, Paulson el al. L973a,b). Interestingly enough, it was found that the temperature region in which the A phase exists shrinks as the pressure is low,g.#gd, and disappears completely below a so-called "polycritical" point (pCÞ) at approximately 2lbar pressure. These measurements were then extended down to the saturated vapour pressure (which will sometimes be casually referred to as zero pressure) using the nuclear demagnetization technique (Ahonen et al. 1974b). Below L0bar the transition temperature from the usual phase into the superfluid B phase decreases quite rapidly with decreasing pressure to a value of about L mK at s.v.p. A list of values of the transition temperature as a function of pressure in zero magnetic freld as obtained by Greywall (1986) is presented in Table 4.1. Application of a magnetic field has a drastic effect on the phase diagram, in pãrticular below the PCP: (i) a distinctly new phase appears (referred to ur ih" A1 phase); (ii) the ordinary A phase (in this case referred to as the At phase), is stabilized down to zero pressure. Hence the A1 and A2 phases now intervene between the normal Fermi liquid and the B phase (Paulson ef to the above nomenclature, the latter phase, i.e.

4.2

97

Thermodynamic ProPerties

Table 4.1. List of smoothed values of the transition temperatures between the normal liquid and the superfluid state (f) and between the A and B phases (f^r) as functions of pressure in zero magnetic field (Greywall 1986). The "polycritical point" lies at Ppcp:21-22bar and

4(P".") :2.273 mK.

P (bar)

4 (mK)

1

0.929 1.061'

2

1.181

3

1.290

4

1.388

5

t.478

6 7

1.560 L.636

8

r.705

9 10

Ll.

L.769 1.8æ 1.883

t2

r.934

13

1.981

14 15

2.026 2.067

t6 t7

2.to6 2.r43

18

20

2.r77 2.209 2.239

2l

2.267

0

t9

To' (mK)

2.n3

,zll 2.262

24

2.293 2.317 2.339

25

2.3û

26

2.378 2.395

?ß

2.41r

29 30

2.425

21.22 22 23

n 31,

32 33

34

34.338 34.358

2.438 2.451 2.463 2.474 2.486 2.49L

2.242 2.217 2.LgL

2.r& 2.137 2.TLL 2.083 2.056 2.027 1.998 L.969

r.94r L.933 L.932

98

4


the B phase in a magnetic field, is referred to as the B2 phase. In principle, there is also â "Br phase" (namely the so-called planar phase; see the discussion in Chapters 5 and 6), but this phase is unstable with respect to the Ar/42 phases. Therefore in a magnetic field there exist three stable bulk phases: the Ar, A2 and 82 phases.'As found in the earlier experiments along the melting curve, the transition temperature into the A phase hardly changes in a magnetic field, apart from a tiny splitting. Below the PCP it coincides with the transition temperature into the B phase in zero field. On the other hand, the ñrst-order A-B transition temperature is shifted to lower temperatures, and eventually (i.e. in fields of 5-1-0 kG) is even below the lowest temperature attainable so far, depending on pressure. Investigations of the A-B transition in a magnetic field have been reported by Feder et al. (1931) and Hoyt et al. (1981), as well as by Eisenstein et al. (1980b), who investigated the effect of a magnetic field on the flowing superfluid. Most recently, Kyynäräinen et al. (1989) have observed a splitting of the A-B transition at low pressures and magnetic fields of about 2 kG. This has been interpreted as evidence for the appearance of a yet unknown intermediate phase in a very narrow region (width of order l plK at3.4kG and P:2.3 bar) between the Az and 82 phases. The magnetic-field dependence of the At and A2 transitions as a function of pressure has been measured by Israelsson et al. (I98$ and by Sagan et al. (19S4). The complete P-T-H phase diagram as known today is shown in Fig. 4.2.

4.2.2 Specific heat A great deal of information about the nature of the superfluid states may be obtained by thermal measurements. In fact, as discussed above, it was the thermal anomalies associated with the phase transitions that actually led to the initial discovery of these phases. Subsequent specific-heat experiments (using CMN salt as a refrigerant) below the melting curve showed a specific-heat anomaly at the normal-superfluid transition reminiscent of the superconducting transitions in metals (Webb et ø1. L973, Wheatley 1973, L975). Later measurements of the specific heat along the melting curve in a Pomeranchuk cell yielded a similar result (Halperin et al. L976). Precise measurements of the specific heat in the full pressure range have been performed by Alvesalo et aI. (1980,1.98L), Roach et al. (1982) and Greywall (1986). The experimental curve at melting pressure is shown in Fig. 4.3. There exists a sharp discontinuity at T"of relative size L,CIC¡¡ as large as 2. This is larger than the BCS value of L.43, which was expected for the BW state, and much larger than the value 1.19 appropriate for the ABM state. The discontinuity becomes smaller for decreasing pressure, and approaches the weak-coupling value L.43 for pressures below L0 bar. A corresponding discontinuity is observed in the thermal expansion coefficient (Swift and Packard L98L, Roach et al. 1983). The character of the experimental specific-heat curve (Fig. 4.3) does not appear to change drastically at the

4.2

99

Thermodynømic ProPerties

P

(bars)

B

Figare

4.2

phose

nor mo I l.iqu id

T (mK) 3He, showing the normatf:ryi liquid as The P-T-I{ phase diagram of well as the superfluid A, A, and B phases. The superfluid Az and B, phases correspond to the A and B phases in the presence of a magnetic field. The values of the transition temperature shown in this figure are a composite of the results of several groups and should only be taken as semiquantitative (see the discussion in Section 4.1). A complete list of ?i values obtained by Greywall (1986) is shown in Table 4.1.

indicated by an arrow in the figure. In the temperature range shown the specific-heat data approximately follow the law C(T): CQòQ/7b)3. The exponential behaviour C(T)æ.e-Ntc" for the B phase at low temperatures, if it were identified with the BW state (see (3.81)), would only be expected to show up at somewhat lower temperatures. On the other hand, the low-temperature behaviour of the ABM-state specific heat, çABM nT3, given by (3.83), is consistent with experiment, although the data do not extend to very low temperatures. There is a very small jomp in the specific heat at the A-B transition given by (C" -C^)lC^:0.091. Also, the latent heat of the A-B transition is rather small, l¡ø:L5.4erglmol for Ten:2."!.mK at the melting curve. It follows from the Clausius-Clapeyron equation that the associated volume change at the A-B transition is only one part in 10t, the density of the A phase being larger. Substantial supercooling occurs at the A-B transition, whereas only modest superheating effects have been observed (Wheatley L975, Kleinberg et al. 1974). The ptrase boundary between the A and B phases in the coexistence region is well defined, as observed in NMR experiments using a uniform magneticfield gradient (Osheroff et al. L972b).

A-B transition,

4

100

Basic Experímental Propertíes

c o

I

a

o

f, e

E

Y

@"8

E2

o o

g,

d

(l)

.d

al

9 d

o"e"

"8

ctl (u

do

s

oe

.9

ooo

*oo"

'õ (u

o-

tn

ot

04

r.0

0.6

ï Figure

4.3

1.2

lTc

Specific heat in the superfluid phases measured in zero magnetic field at melting pressure. The B transition is indicated by an arrow. (After Halperin et al. (1976).)

As discussed above, the A transition splits into two transitions in an external magnetic field, whereas the B transition is moved to lower temperatures. The splitting is clearly observed in the specific-heat measurements. A discontinuity is found at each of the transitions, with the magnitude of the A1 jomp ¡g(ar)76* :0.'74 at melting pressure being somewhat smaller than the A2 jump. The discontinuities add up, however, to the ju-p observed in zero magnetic field.

4.3 MAGNETIC PROPERTIES The first actual experiments performed on the superfluid phases of 3He 3He nuclear spin system investigated the nuclear magnetic resonance of the in a Pomeranchuk cell (Osheroff et al. L972a,b). By applying a magneticfield gradient to the sample and thereby shifting the local resonance frequency, it was possible to distinguish between contributions to the resonance signal from liquid and solid parts of the sample. Quite unexpectedly, it was found that below the transition into the A phase the resonance signal from the liquid started to shift from the Larmor frequency. The amoung of this shift was found to correspond to a local field of maximally 30

G following the law

o)2: oL+ a'(T) (see also Fig. 4.4). Here

(4.1)

ø¡ is the Larmor frequency and AQ) is a

4.3

Magnetic Properties

¡ o t

0.8 c{

N

I o

o

Þ ?c\¡

06

.Ç 5

o.¿

¡

101

131 kHz 263 kHz 371 kHz 1039 kHz 1510 kHz

a 2760 kHz

ôJ

r

(\¡

I

?

-=

0.2

0ö

0.005 0.010 0.015 0.020

0.025

ÂP (otm) Figure

4.4

Transverse NMR shift as observed in tHe-A in a Pomeranchuk experiment. (After Osheroff et ø1. (I972b).) The pressure difference ÂP may be translated into temperature using Table 4.1.

field-independent but temperature-dependent frequency. At the A-B transition the signal from the liquid appeared to shift back to the original Larmor frequency, but dropped by a factor of approximately two in magnitude. At still lower temperatures, in the B phase, the shift did not reappear, and the signal strength decreased further (but only by a small amount) down to the lowest accessible temperäture. It \ilas later found that the magnetic susceptibility in the A phase is not quite given by the temperatureindependent Pauli susceptibility of the normal state, but is increased by a very small amount (Paulson et al. L974b, Webb et al. I974a, Webb I977a,

; xm

1.0

'=

Þ

o.o

o(u (J

ltl f,

Ø

.:o_

0.2

Ø

0.2

0.4

0.6

0.8 T

Figure

4.5

r.0

1.2

lTc

Magnetic susceptibility of 'He-B as a function of temperature at20bar. (After Ahonen et ø1. (L976b).)

4

r02 Sager et


al. L978a). The nuclear magnetic susceptibility of the B phase

as a

function of temperature is shown in Fig. 4.5 for two different pressures. The data seem to follow approximately a universal law as a function of f lf". As shown by Leggett (1973a,b), an NMR shift of the observed magnitude may be explained by the coherent superposition of the spin-orbit forces caused by the nuclear dipole interactions within each Cooper pair. Given this theoretical explanation, a longitudinøl resonance was predicted to be observable as well (Leggett L973a,b). The expected oscillation of the longitudinal magnetization was first found by Osheroff and Brinkman (1974) and Bozler et al. (1974) in CW resonance experiments; for a review of early experiments see Richardson (1975). It was later discovered that the longitudinal oscillation could be excited by a sudden change in the applied magnetic field, a phenomenon called "ringing of magnetization" (Webb ef al. L974b). The ringing frequencies in the A and B phases are found to scale with the ratio of the magnetic susceptibilities, i.e. as

a'"Q) x"Q) -! d}'o(T) 2 xe. '

(4.2)

except perhaps in the vicinity of the polycritical point (Osheroff et al. t975, Ahonen et aI. I975c, Webb et al. L975a). It also turns out that the longitudinal resonance frequency d2^ in the A phase is equal to the characteristic frequency fJ in the expression (a.1) for the shifted transverse frequency. In Fig. 4.6 the resonance frequency data from longitudinal resonance experiments in the A and B phases and transverse resonance experiments in the A phase are plotted in scaled form, using (4.1) and (4.2). A rich variety of nonlinear behaviour in the spin dynamics of superfluid 3He has since been found. The longitudinal ringing frequency in both the A and B phases shows significant structure as a function of the change in magnetic freld A^FI, which initiates the ringing (V/ebb et al. 1974a, 1975b, 1977, Webb I977b). In the A (B) phase the frequency is substantially smaller than the L,H+O limiting value at one (two) particular value(s) of well above that point. the turni*pff field Â^F/ and increases linearly with large spin-tipping pulse is The nénlinear behaviour after application of a^FI found to be very complex. One observes tipping-angle-dependent frequency shifts, satellite resonance lines and unusual relaxation effects (Osheroff 1974, L975, Osheroff and Corruccini L975a,b, Corruccini and Osheroff L975a,b, L978, Osheroff and Brinkman 1974, Giannetta et al. L98L, Eska et al. 1982). These phenomena will be discussed in more detail in Chapter 8.

4.4

SOUND PROPAGATION

in Chapter 2, a Fermi liquid allows a well-defined density3He zero sound oscillation mode, called zero sound. In normal liquid has been clearly observed for sufficiently high frequencies such that As

discussed

4.4

Sound Propagøtion

103 T/Tc

wvvvvv v

(\

N

o o (\¡

tr

(\¡ NJ

3

(\¡

0.1

I

3

o2

0lã

2.6

Temperoture lmKl

Figure

4.6

Longitudinal NMR frequencies as measured in the A and B phases and transverse NMR frequencies as measured in the A phase plotted in scaled form, using (4.1) and (4.2), as functions of temperature: A, A, transverse shift at melting pressure (Bozler et aI. L974); x, transverse shift at melting pressure (Osheroff et al. L975); E, l, longitudinal shift at melting pressure (Osheroff et al. 1975); O, longitudinal ringing at 33 bar (Webb et aI. L974b); V, Y, transverse shift at 32bar (Ahonen el ø1. 1976b) (open symbols, B phase (scaled); filled symbols, A phase). (After Wölfle (t979).)

the condition for its existence, e)))Ilt, is satisfied (see Fig.2.2). Here r is a quasiparticle collision time. Since zero sound owes its existence to the stiffness of the Landau molecular field, which is the result of interaction processes involving energies on the scale of the Fermi energy, one should expect this mode to persist in the superfluid phases. Indeed, one of the first experiments done on .superfluid 3He in a Pomeranchuk cell showed that ultrasound with a frequency of 10 MHz propagates in the superfluid with essentially the same efficiency as in the normal fluid (Lawson et al. L973, 1974). However, the sound attenuation, though small, develops a sharp peak as a function of temperature closely below [ (see Fig. a.7). Systematic measurements of the attenuation and the velocity of longitudinal sound in superfluid 3He over a wide pressure range extending from about 9 bar up to the melting pressure were made later in an apparatus using CMN as a refrigerant (Paulson et al. L973a,b, L976a,cd). Attenuation peaks were studied near the A-normal and B-normal transitions. The observation of these peaks is indicative of the existence of collective modes in the superfluid. For the three sound frequencies used (5, 15 and 25 MHz) the

4

L04


T (mK) 2:t8

2:t2 2.69

2.75

3.0

2 66

2.8 a a

aa

2ß

aa aa

T

E

2.1-

s¿

o

¿J

2.2

aa

a

a a

2.0

a a

1.8

o oo

a a a

i a a

0.3L2 0.3/.3 0.3u, 0.345 0.346 P - 34 (bqrs) 4.7 Sound attenuation in 3He-B just below ?i as a function of melting pressure in a Pomeranchuk cell. (After Lawson et al. (197a).) The 0.341

Figure

pressure translates into temperature as indicated on the top horizontal scale.

attenuation was found to increase strongly with frequency. Similar experiments conducted in the B phase showed interesting structure in the sound velocity near [ (Roach et al. I975a).In the B phase the sound propagation properties are isotropic. By contrast, in the A phase sound attenuation is found to depend sensitively on the direction of applied external fields, for instance magnetic fields, thermal currents, and also on the shape of the sample cell. This behaviour is interpreted as being due to the orienting effect of the aforementioned external fields on the anisotropic order parameþ¡ in the A phase. A systematic variation of the sound attenuation with the-*ngle between a small magnetic field (approximately 20 G) and the sound propagation direction has been observed by Roach et al. (1975b) and Ketterson et ø1. (L975). These orientation effects are also evident from the fluctuations in the sound attenuation, which reflect fluctuations in the local anisotropy axis (Lawson et al. L975a,b). The latter authors further investigated the sound attenuation in magnetic fields up to 7.5 kG, where the attenuation peaks were observed to split into two in accordance with the splitting of the A transition seen in thermodynamic experiments. The anisotropy of the ultrasound attenuation in the A phase has been used to investigate the dynamics of the local anisotropy axis (Paulson et ø1. 1977, Wheatley L978).In these experiments it was found that the anisotropy axis generally relaxes to the new equilibrium position within a time that is simply related to the spin-relaxation time found in NMR experiments.

4.4 Sound Propagation

105

Later experiments showed the existence of a second small but very sharp attenuation peak at lower temperatures in the B phase (Giannetta et al. L980, Mast et al. L980). The group velocity of a sound pulse in the vicinity of this sharp attenuation feature was found to vary dramatically, the pulse being slowed down to a value as low as one-third of the phase velocity. In subsequent experiments the sharp resonance was shown to split into five peaks in an applied magnetic field of typically 100 G (Avenel et al. 1e80).

Transverse zero sound has also been investigated. Measurements of the acoustic shear impedance show broad features, which may be interpreted as evidence for transverse zero sound (Roach and Ketterson 1976a,b). Hydrodynamic, or low-frequency, sound propagation has been studied in order to obtain information on the viscous properties of the liquid (Eska ef al. 1980). One finds that the attenuation and velocity change of 50-200 kHz sound is dominated by the friction of the liquid with the walls of the resonator cell. The temperature dependence of the viscosity coefficient extracted from these measurements agrees reasonably well with that obtained from shear-flow measurements. It is well known from superfluid 4He that the propagation of heat changes from diffusive to wave-like as one moves from the normal into the

superfluid state. These entropy waves in the superfluid, called second sound, may be expected to be much less well defined in 3FIe, where the entropy at low temperatures is reduced by a factor Tlh from the classical value. This has in fact obstructed the observation of second sound. The flrst successful experimental observation of second sound took place in the A, phase, i.e. the phase that appears between the normal phase and the A2 phase in a magnetic field (Corruccini and Osheroff 1980). There it was found to be strongly coupled to the magnetic properties, measured simultaneously by NMR. Later it also became possible to observe second sound at very low frequencies in zero field (Lu and Kojima 1985). A related phenomenon is the so-called "fountain pressure", i.e. the formation of a pressure head induced by a temperature gradient driving a counterflow of the normal and superfluid components (Shields and Goodkind 1977). A well-defined hydrodynamic sound-like mode is observed in superfluid 3He flowing in a confining geometry, for example inside the pores of a fine powder filling the container (Kojima et al. 1974, Yanof and Reppy 1974). Under these conditions, when the motion of the normal-fluid component is clamped by friction with the walls of the narrow channels, the superfluid component is able to sustain a soundlike excitation mode, socalled fourth sound. The velocity and damping of fourth sound provide information about the superfluid density component and the superfluid bulk viscosity. The existence of low-frequency wave-like motion of the superfluid in a fourth-sound experiment constitutes proof for the superfluid behaviour of the liquid on timescales of the order of the inverse soundfrequency.

4

106

4.5 HYDRODYNAMIC In the early

stages


PROPERTIES AND SUPERFLUIDIW

of experimental research on superfluid 3He investiga-

tions of hydrodynamic flow properties were at the centre of interest, since they were known to elucidate whether or not an anisotropic-superfluid pairing model was qualitatively correct. A number of early experiments concentrated on the measurement of the normal-fluid densitv and viscositv.

4.5.1 Normal-fluid density and viscosity In the first experiments designed to measure the viscosity a vibrating wire immersed in the liquid was used as a probe. The shift in resonant frequency and the damping of the vibrations as a function of temperature may be converted into a viscosity coefficient and the density fraction of the normal-fluid component (Alvesalo et al. L973, 1974, 1975, Guernsey et al. 1976; see also Johnson et al. 1975). Another technique employs a torsional oscillator where either an object of cylindrical shape is suspended in the liquid (Parpia et al. 1976) or else a ring-shaped 3He cell is rotated as a whole (Kiewiet et al. 1975, Main et al. 1976, Archie et al. 1979,1981). The common features of all these experiments are (i) the observation of a T-2 behaviour of the shear-viscosity coefficient in the normal phase (see

Chapter 2), (ii) a rapid drop of the viscosity below I roughly following a Q"- T)rt2 law, and (iii) a smooth decrease of the normal-fluid density to very small values at temperatures below 0.57:. The low-temperature limiting behaviour of the viscosity seems to indicate that the effective viscosity vanishes. Experiments on other transport coefficients will be discussed in Chapter L0. A more precise measurement of the normal-fluid fraction is possible if the motion of the normal fluid is completely clamped by filling the torsional oscillator cell with a tightly packed fine-grained powder. The amount of liquid free to move, i.e. the superfluid component, reduces the moment of inertia of the torsional oscillator and may be measured through a shift of the resonant frequency (Main et al. 1976, Berthold et al. t976). Precise values of the superfluid density fraction obtained in this \üay are shown in Fig. 4.8.

,4.5.2 Persistent currents Perhaps the most remarkable property of the superfluid phases of 3He is, as indicated by their very name, the capability of flowing without friction. This property has been established beyond doubt for superconductors and superfluid aHe, for which persistent (i.e. dissipation-free) currents flowing in a ring geometry have been observed to flow for up to several years. To measure the same property in the case of superfluid 3He has turned out to

4.5 Hydrodynamic

Properties and. Supefruidity

LO7

att'i 1.2

oo

.ttt

ôê oo-

'll . o-tt 'H"

odo

at

08

t aoë

o-

tt

d

eo

oo

oo

-o

ôoo o

ê

Ht

0.t,

oou

0.05

0.r0 1

Figure

4.8

-ïI

0.15

0.20

Tc

Superfluid density fraction p,lp in tHe-A as a function of reduced temperaturc at 27 bar. The measurements were taken at two different orientations of an external magnetic field relative to the flow direction. (After Berthold et al. (L976).\

be very difficult. The basic device used in such an experiment is

a the superflow is

gyroscope, by which the angular momentum associated with measured. In these experiments a torus filled with 3He, packed with powder in order to clamp the motion of the normal fluid, is mounted on top of an axially flexible rod. The cryostat is rotated about the axis of the torus (z axis) at an angular velocity exceeding a certain critical value, such that a supercurrent is set up. After stopping the rotation, the supercurrent continues to flow and thereby gives rise to a macroscopic angular momentum. In principle, the persistent angular momentum is detected by exerting a periodic torque in the direction and observing oscillations of the gyroscope along the y direction, at the eigenfrequency of these oscillations. In this way the existence of persistent currents has been established for the B phase (Gammel et al. 1984, Pekola et al. 1984b, Pekola and Simola 1985). In fact, the persistent current \ilas shown to last for at least 48 h without detectable dissipation. A similar experiment in the A phase faited to detect a persistent current. However, Gammel et ql. (1985) succeeded in enhancing the sensitivity of the measurement by exploiting a subtle effect of superflow on the dissipation of torsional oscillations of the 3He cell found

x

previously by Gay et al. (1983). The Q factor of the torsional oscillations at resonance was indeed found to depend on the preparation of the fluid, i.e. the application of a rotation of the sample prior to the measurement. For angular velocities of rotation exceeding a certain critical value l2-,,,, the minimum of the dissipation is shifted from f):0 to a finite value fJ.. This is interpreted as the rotation speed where the superfluid is at rest with respect to the container wall, i.e. the average superfluid velocity ú,: RfJ,, where R

108

4

Basic Experimental properties

r.0

0.8

aâ {Aa

A A

at,

60

0.6

æ

606o oo q

too.8

E

l¿

>

eo

a B phose,2g bor o B phose, 15 bor a A phose, 2g bqr

0.¿

0.2

'[ rr

Ê

0.2

0.t,

0.6

0.8

r.0

Pt /P

Figure

4'9

Temperature dependence of the critical superfluid velocities in the A and B phases plotted as u! versus the supe{{o ddity i;; ction p"¡p. (F9r tle T dependence o1 ps, see Fig.'4.g.) (eftei 'GaÃ-"r et ar. (1e8s).)

is the radius of the ring channel. l-he behaviour as a function or dzis reversible below a critical value r2". This is interpreted as the onset of processes destroying superfluidity. Hence the so-calleå critical velocity can be deduced as u!: Rf¿.. Similar behaviour was found in the B phase in the earlier experiments. one finds varues for the criticar velocity of ui - 0.2cm/s in the A phase and u!:0.6-0.8 cm/s in the B phase, with a tendency to increase slightly with pressure (see Fig. a.9). The uuí.r", for the e prrase are in agreement with earlier measurements for oscillatory flow through a single lSqT diameter orifice (parpia and Reppy L979). Perhaps the most direct manifestation of ,nu"ror"opic quantum coherence in the superfluid is the Josephson effect, i.e. the tunneliing of condensate particles in a Bose liquid or of cooper pairs through u ,o-"uñ"d .,weak link,, (Josephson L962, Lg6;s).In a supein"iã such a weak link is rearized. by an orifice connecting two reservoirs, whose extension is comparable to the coherence length. The experimental detection of the Josepñson effect has proved to be difficult--æven in superfluid aHe. An unambigûous observation was only reported_almost 25 years after the prediction of the effect (Avenel and Varoquaux 1985, 1986). Subsequently ihese authors were also able to observe a near-ideal Josephson effect in 3He-B at zero pressure (Avenel and Varoquaux L}BT).

4.6

4.6

Rotating Superfluid

3He

109

ROTATING SUPERFLUID 3He

oHe by Andronikashvili beginSince the pioneering experiments on liquid ning in the 1940s (Andronikashvili and Mamaladze L966) it has been known that many fascinating and new features arise in a rotating superfluid. The principal property of the superfluid that comes into play here is that the

superfluid flow fleld is irrotational, i.e. the superfluid velocity ñeld is curl-free (V x u" - 0). Hence a superfluid is not capable of performing solid-body rotation like a normal fluid. Rather, by rotating a superfluid and thus imposing a certain angular-momentum density on it, one forces the liquid into an inhomogeneous state consisting of a regular lattice of parallel vortices, each carrying one quantum of circulation (for an explicit experimental verification, see Williams and Packard 1980, Yarmchuk and Packard L982). The circulation is concentrated in the vortex cores, if the flow of the superfluid is indeed derivable from a potential (u" o V@), as in the case for superfluid 4He. Similar behaviour is observed in a type-Il superconductor, where the role of the mechanical rotation of the container is played by a magnetic field and where the vortices are flux lines. In this case the flux lattice may even be observed under a microscope (see Essmann and Träuble 1967). Experiments on rotating superfluid 3He indicate a rich variety of vortex structures and transitions among them (Hakonen et al. L982a,b, 1983a, Ikkala et al. 1982). The experiments were performed on a rotating cryostat (Hakonen et al. 1983b) using rotation speeds of up to 3 rad/s and magnetic fields of 100-500 G. The experimental 3He cell had the shape of a cylinder of 5 mm diameter and 30 mm length. The principal probe used to detect vortices is the shift in frequency and the shape of the NMR signal and, in particular, the appearance of satellite peaks as a function of rotation speed fJ. In the A phase the sharp NMR line measured in the stationary fluid is broadened upon rotation of the container, as shown in Fig. 4.L0, and a satellite peak is seen to appear at the low-frequency side. In the B phase on the other hand, a large number of satellite peaks are alreâdy present in the NMR signal at d):0 (see Fig. 4.1,1). These peaks aÍe shifted and broadened as a function of Q (see Fig. 8.7). The frequency shifts vary significantly with temperature as shown in Fig. 8.8. At a temperature T:0.6T. (at 29.3 bar pressure) an abrupt change in the frequency shifts takes place-indicative of a phase transition or structural change in the vortex system. Even more surprising, the frequency shifts depend on the relatiue orientation of the magnetic freld (parallel or antiparallel) with respect to the rotation axis. Apparently a magnetic moment is induced by the rotation of the fluid ("gyromagnetism"). The location of the vortex transition as a function of pressure and temperature is shòwn in the phase diagram, Fig. 8.9. In broad terms, the presence of vortices influences the NMR properties

4

110


c

o o-

o tn -o

o

É.

= z

(ul

- rrl¡ I I 2n (k Hz)

Figure4.l0 Effect of rotation on the NMR line in tHe-A: (a) d2:0;

(b)

Q:1.21rad/s. The arrow indicates the appearance of a satellite peak associated with the vortex formation. The absorption scale of the lower curve in (b) is magnified by a factor of 50. (After Hakonen et al. (1e83a).)

by changing the textures in such a way that the torques exerted on the local spins are modified, causing frequency shifts. It may even happen that potential wells are created in which standing spin waves may exist, causing satellite NMR peaks to appear. The various possible vortex configurations and their NMR signatures will be discussed in detail in Sections 7.6 and 8.4 respectively. Other experimental methods used to probe the vortex lattice include ion-mobility experiments (Simola et al. L986, 19S7) and hydrodynamic experiments such as measurements of the critical current (Pekola et al. L984c). More details will be given in Section 7.6. This concludes our discussion of basic experimental observation. Detailed comparison of experimental data and theory is made in the following chapters.

4.6 Rotating

SuPerfluid

3He

Q=0

111

Q = 0.6

rqd/s

(¡)o

c .9 o-

o

(¡)l

UI

.c¡

(¡)

|

Frequency Figure

4.LL

3He-B at 29.3 bar: (a) O Effect of rotation on NMR spectra in A:0.6 rad/s. (After Hakonen et al. (1983a).)

:

0; (b)

FURTHER READING Hall H E and Hook J R 1986 in Progress in Low Temperature Physics, Vol. IX, ed. D F Brewer (North-Holland, Amsterdam), p. L43 3Helium, Part Lee D M and Richardson R C 1978 in The Physics of Liquid and Solid II, ed. K H Bennemann and J B Ketterson (Wiley, New York), p.287 Osheroff D D 1978 J. Physique 39 Colloq. C-6, Vol. III, p. L270 (Proceedings of the L5th International Conference on Low Temperature Physics, LT-L5) rilheatley J C L975 Reu. Mod. Phys. {l 415 Wheatley J C L978 in Progress in Low Temperature Physics, Vol. VIla, ed. D F Brewer (North-Holland, Amsterdam), p.1

Pair Gorrelations beyond Weak Goupling

The experimental observation of two phases, both p-wave pairing states as suggested by the details of the phase diagram, contradicts the result of weak-coupling theory that the BW state is lowest in free energy at all temperatures (see Section 3.4). On the other hand, weak-coupling theory is exact in the limit TlTr-+ 0. Since the various p-wave pairing states are close in free energy, it is not surprising that pressure-dependent corrections to weak coupling, which do exist, may reverse the order of states. At this stage, before indulging in elaborate microscopic theories, it is useful to try to get an overview of the possible states. Since we are dealing with a continuous phase transition, characterized by a continuously vanishing order parameter at the transition point, this can be achieved within the framework of the Ginzburg-Landau expansion of the free energy in powers of the order parameter (Landau and Lifshitz 1959a). In the temperature regime near the transition the expansion may be terminated at fourth-order terms. Quite generally, such an expansion is constructed by identifying those combinations of the order parameter that-for a given power of the order parameter-are invariant under all the symmetries of the system. The number of these invariants will be larger the more complex the orderparameter structure is. The numerical coefficients multiplying these invariants can only be determined from microscopic theory. This will be discussed in Section 5.2.

5.1 GENERAL GINZBURG-LANDAU EXPANSION OF THE FREE ENERGY

As discussed in Chapter 3, one should expect pairing to take place in states of pure relative orbital angular momentum, unless the pair interaction is such that two (or more) interaction parameters are close in value. Systematic discussions of the general Ginzburg-Landau free energy have

5

TL4

Pair Correlations beyond Weak Coupling

been given for p-, d- and f-wave pairing. We restrict attention to p-wave pairing here, following the arguments of Mermin and Stare (1,973) and Brinkman and Anderson (1973). For general / pairing the order parameter is a complex tensor d*¡r...¡, of rank I + l, satisfying certain constraints for I > L connected with the permutation symmetry of the ¡. The frec energy is a real scalar quantity. Gauge invariance requires the expansion to be in powers of dd" (dt is the complex conjugate of d). Rotational invariance in spin space and momentum space is guaranteed by contracting tensor indices in spin space and momentum space pairwise. Thus there is only one second-order invariant term, but there exist several fourth-order invariants; their actual number depends on / (Barton and Moore t974a). The p-wave pairing order parameter is a complex 3 x 3 matrix d*¡ (see (3.64).It is convenient to introduce the normalized order-parameter matrix

Abv

dt"j:31/2a1T¡A*,,

such that

tt

(5.1a)

(AA*):1,

(s.lb)

where Á is the root-mean-square gap (see (3.57)), /.\l/2 (Ao'p&i'ùn'l , AQ): (å) / \u,þ

(s.2)

and tr (. . .) denotes the trace operation on 3 x 3 matrices. There are then five fourth-order invariants, which can be constructed graphically by representing A (A*) as an open (shaded) circle and sums over momentum space (spin-space) indices by a solid (broken) line (see Fig. 5.1). They are

Il ¡t

\J--Y/J) ll ,l ¡l rl

/-\ H/

/f7 //A \¿//

I

\-/

It, Figure

5.1

Graphical representation palnng.

I

¡5

of the fourth-order

invariants

for

p-wave

5.1

General Ginzburg-Landau Expansion of the Free Energy

115

given by

/,

:

ltr (AÆ)12,

Ir:ftr

\:

(AA+)12: I, (s.3)

tr [(AA'XAA')*I,

Ia: tr î(AA*)tl, Is: tr [(AA*XAA*).]. denote the transpose and Hermitian conjugate of A respectively. Note that various alternative conventions are used for labelling the quartic invariants (5.3). The general free-energy functional per volume for p-wave pairing states up to fourth-order terms is thus obtained by adding up the invariants as

Here Ar and

A*

F)

qaz + àao} f =i:-fN rrr + V i:l

'

þ,t,,

(s.4)

where V is the volume. Since in the case of homogeneous bulk phases the quantities F and f are related by a trivial facior, for the rest of this chapter the symbol F will denote the bulk free energy per unit volume. The coefficients a, B¿ have to be calculated from microscopic theory. Note that we have already calculated these coeffrcients in weak-coupling theory (see (3.55) and below). For an investigation of nonunitary p-wave states with equal spin pairing, such as the ABM state in a magnetic field to be discussed in Section 5.4, it is more convenient to express the Ginzburg-Landau free-energy functional in terms of the "spin-matrix gap parameter" Lj (rather than the order parameter A*¡), defined by

L¡

:3r'' a) A*,o*io,

(s.sa)

It

such that (see (3.34a), (3.6a))

A*: 2 l,Ê,

(s.sb)

i

The only second-order invariant is

Io:Ltr,

(Âr.Âr1):3At,

(s.sc)

where summation over repeated indices is implied. Rotational invariance in spin space implies that only products of the type (ÂÂ*) are allowed in any invariant expression. The five fourth-order invariants I¿ are obtained by

5

tL6


taking all possible contractions of momentum-space indices:

ir:

ä2 ft, ({A¿*4.^¡) =9Aa(ZIzjt

1,),

.12 :eaalz, - f.- tr, ({a,t)J Ir: l+4

i.: 12jt [tr, ({-Â,*) tr, (ÂrA,*)] = gA4L, io:

i2jr

(s.6)

[tr, ({.4,*) tr. (4,Â,rl)l-9AoIo,

ir: ä2 tr, ({.A,f^,Ai) it

:9Aa(Iz+ 14- I).

In (5.6) we have also indicated the relation to the invariants d. The free-energy functional thus takes the alternative form F

: rñ + tuls + *2 ß,i,. i:t

(5.7a)

Comparison of (5.7) with (5.4), using the definition of the invariants /¿, yields the following relations among the coefficients þ, and, B,:

ßt: -þ,,, ßr: þt* þr, ßr: þr+2þt) ßo: þo* þr, ßs: -'þs',. j

i, and (5'7b)

In particular, the weak-coupling values of u and B, may be determined by comparing (5.6) with (3.55) derived within the BCS weak-coupling approach; one finds

uQ): -N(OXI - r lr), I þz: þz: þq: -þs: -2þr:Ìþo,J

(5'8)

with B6 defined in (3.56). Here the fourth-order term (tr, (ÂoÁ,f)t)¿

i.

(3.55) is evaluated using

tr, (Â¿Af,)' :

)

ijlm

tro (Â,*Â ,LT L^) Ê,Ê,Ê,k^

(s.ea)

and

*-

,, 1, for the weak-coupling Br values (Barton and Moore l974b,c). The corresponding value of the free energy is given by

F - 4*

: )aAz :

-1artrc-'N(0xkB

n)'h\ å' 7./

I is seen to be determined LC: 12 _1 c* 7ee)" '

The specifrc-heat discontinuity at

(s.r2) by

r as (s.13)

where C¡ is given by (2.L4). Conversely, K may be expressed in terms of L,CICN in (5.10), yielding the useful result

:

^(D "(?X)'''

0,.(t i)'"

(s.14)

There remains the task of finding the minimum of F with respect to variations of the actual (matrix) structure of Ar¡, or, equivalently, of finding the maximum of L/r. The search for stable minima of (5.4), i.e. for possible superfluid phases, can proceed along quite different lines. For example, one may seek the solution of the idealized weak-coupling problem as discussed in Chapter 3 (Balian and Werthamer 1963, Ambegaokar and Mermin 1973); alternatively, one can seek the solution under certain constraints on the order parameter, such as unitarity (Mermin and Stare L973, Varma and V/erthamer 1974), or with the requirement that the resulting phases have certain magnetic properties (Mermin and Stare 1974) known from experiment. A different line of approach is due to Barton and Moorc (L974b, 1975a), who concentrate on those phases whose order parameters do not depend on the (ói¡s¡f"-fheir order parameters do parameters Ér. These phases are called not change their structure upon a change of external parameters, only the overall magnitude of the gap will change. Using methods of differential geometry, which will not be reproduced here, one can in fact derive a general equation for the extrema of the free energy (5.I2) in terms of the matrix A (Jones 1977): þrÍA* tr (AAr)

-

IrAl+ fulAÆA* - LAI + p4lAA+A- I4Al+ psÍA"ÆA- I5Al:Q,

(5.15)

with the constraint tr (AA+) : 1.. In principle, this is a set of 18 real equations for the components A*t. However, only L0 of these parameters

5

118


are actually independent. This is due to the invariance of the free energy (5.4) under separate three-dimensional rotations in spin and orbital space and under an overall phase transformation (see Chapter 6): it implies that seven parameters (three Euler angles in spin space, three Euler angles in orbital space and one phase variable) can be chosen arbitrarily (see Eilenberger and Schopohl 1938). In addition, the constraint tr (AA+) : 1 eliminates one more parameter, such that one is left with only 10 independent parameters. Equation (5.15) is therefore greatly overdetermined. The solution of this nonlinear equation leads to be a surprisingly complicated problem; in fact, for arbitrary Bt the problem has not yet been solved. For unitary p-wave states, however, a full analysis is possible, and will be given below. The spectrum of the mean-field Hamiltonian (3.16) and, in particular, the resulting unitary states have been investigated by Solomon (1981) within a dynamical group model. A classification of f-wave pairing states has been discussed by Barton and Moore (L975a) and Mermin (1,975,1976). Such states have been found to be incompatible with experiment.

5.2

PHENOMENOLOGICAL CLASSIFICATION OF MODEL STATES

5.2.1 Unitary states The signature of a unitary state is the condition

d(k)xd*(ft):g for the order parameter (see (3.37)), implying either (i) a real d(k), or (ii) complex d: c(k)â, where ã is a fr-independent real unit vector. In the case (i) we have A* : Æ, and consequently AA* is a real symmetric matrix, with real eigenvalues )rn, n: t,2,3. The free-energy density may be written in terms of these eigenvalues as

F: F¡r * ua2l t, + +d^lÞ,,(}

^)'

* þ,o,4

(s.16) ^,"),

where we use the abbreviations þn: þr+ þ, etc. The different states are characterized by the number of zero eigenvalues Â,,.

(1) Suppose that only one of the eigenvalues were finite: Lt: )r, hz: hz:O. This is the so-called "polar" state because it is characterízed by a single preferred direction both in spin space, d, and orbital space,

r0:

d(k): doag ' *¡.

(s.17)

Remember that, in the absence of spin-orbit interaction, spin space and orbital space may be rotated with respect to each other without

5.2

119

Phenomenological Classification of Model States

Ás [cds 0¿l

Èrry

Figure

5.2

Schematic drawing of the enefgy gap of the polar state (5.L7). The gap vanishes along the equatorial line. Note that Er and As are not drawn to scale.

changing the energY, so ã and û need not be in any fixed relative confrguration. The polar state has an energy gap lAol: Ás lcos 0¿1, where cos 0ft - îv ' /< (see Fig. 5.2). Its free energy is

g-a

pPolar:R-

21, i

p;,

(s.18)

and K : E, þ¡l þo.

(2) The "planaf'state is realized if two of the eigenvalues are nonzero. It fotlowì from the symmetry of F in )ut and )', that, for example, )q: ]rz= ]r, )rz:0. Minimization of F yields pPlanar:

ú lfr 2þrr*

(s.1e) þror'

and r : (Zþrt+ fu4s)lzBo. The corresponding order parameter has no component in the third dimension, say along I; i.e. it is given by (s.20) d(k) aoR[Ê îtî. Éll.

:

-

Here F is an orthogonal 3 x 3 matrix, rotating spin space relative to orbital space. The energy gap has a torus-like shape, laol: Á6lsin 9¿1, where cos 9e :

l'

k.

(3) Finalty, all three eigenvalues may be nonzero' and by symmetry ht: ]rr: )uz= h. The corresponding State is the BW state, already

5

TZU

in Section 3.3. The free-energy minimum is given by

discussed

FBw: with


r:(3þrr+

ú

4.¡

(s.2r)

- 2(þ'r+ tþto)'

þ:ø)13þo.

These three states exhaust the possibilities for case (i), i.e. for real d(k). By comparing the free-energy expressions (5.18), (5.19) and (5.2L), one sees that ¡Planar hes between Fpo¡" and FBw for any values of the B¿, such that the planar state can never be the equilibrium state. The order parameter for case (ii) takes the form

d(k):

(c . k)ã,

(s.22)

with d a real unit vector and c some vector with complex-valued

com-

ponents. The free-energy expression in terms of c is obtained from (5.4)

F

:

FN

+

\u(c' c*) + r*t[Ér, l" ' "lt *

þror(c ' c*)'1.

as

(s.23)

In order to guarantee stability, we must assume that p2a5> 0. Then, for þrt> 0, the minimum is reached if lc'cl :0, and hence c: Ao(ñ + iñ), where

ABM

ñ:0.

fi

This is the and fi are two real orthogonal unit vectors, ñ . state (or the axial state) discussed in Section 3.5. Its free energy is

IpABM-Ê - rN and

r:

fl2

2þror'

(s.24)

þztslþo.

For B6 < 0, the minimum of (5.23) occurs for real c, which takes us back to the polar state. Note that the polar and the axial states are similar in that the energy gap is axially symmetric (see Figs. 3.4(b) and 5.1). This implies the existence of a single preferred direction in position space. On the other hand, their physical properties are quite different because only in the ABM state do the Cooper pairs have a finite angular-momentum projection. So far we have only discussed unitary states. Flowever, there is no a priori reason why nonunitary states should be unimportant. Such states may possess a spontaneous magnetization when particle-hole asymmetry is taken into account; this has not yet been observed. A detailed discussion of nonunitary states is beyond the scope of this book, but can be found in the literature (Barton and Moore I974b). For p-wave pairing, there appear to be at least five stable states representing (local) minima of the free energy: three of these are unitary (the BW, ABM and polar states) and two are nonunitary (the B and A1 states). Th" order-parameter structures of the latter two are given below. In the B state

dt"j:

An(â* + iêòîj,

(s.2s)

where ãand ê are orthogonal unit vectors in spin space and Á1, as well

as

5.2

Phenomenological Classification of Model States

TzL

An: An and Á¡¡ appearing below, are the matrix elements of the gap matrix Â (see also (6.15), (6.54)). This state may be obtained from the ABM state by interchanging spin and orbital spaces. Only spin-up pairs are formed, and ihe energy gup iJfo"nd to be proportional to lcos 0*l:¡0. Îas in the polar state. In the A1 state

dt"j: t,At¡@* + iêp)(rñi + iñ-)

(s.26)

(see also (6.15), (6.56)). Again only spin-up pairs are formed. The spin and orbital parts of d(k) separate as in the ABM state. In fact, as will be discussed in detail in Section 5.4, this state describes the so-called A1 phase,

which appears between the normal phase and the

A

phase

in a strong

magnetic field. The five abovementioned states have a remarkable property: their structure, as characterized by the order parameter, is independent of the parameters B¡ within their region of stability. States possessing this property have been called "inert" (Barton and Moore L974a,b), in contrast with the "noninert" ones, whqse structure changes continuously as a function of the points of the free energy B¿. Other inert states corresponding to stationary have been found to be saddle points; hence they are unstable (Jones 1977). based on A systematic classification of possible inert and noninert states 3He has been the symmetry properties of the free energy of superfluid derived by Bruder and Vollhardt (1986). It will be discussed in Chapter 6 within the context of broken symmetries. This analysis yields several more states, two of which are the A2 and 82 phases-both realized in superfluid 3He in a magnetic field (see Section 5.4). These states are noninert. Their order parameters are given as follows (see also (6.48), (6.40)). In the A2 state

dt"i: (Aâ* + Bêp + cÎp)Qhj + iñj), where

¿:L(an * Áu),

(s.2t)

6:li(a¡ - au), c:0.

Here Î: d x ê is a third axis in spin space^, perpendicular to both ã and ê, defined in analogy with the unit vector I : ût x ñ in orbital space. Note that i indicateì-the direction of spin polarization S of the Cooper pairs. In equilibrium we expect S and hence / to point along the direction of the magnetic field (cf. the remarks made on the choice of the spin quantization axis in Chapter 2, (2.17)). The sets of unit vectors (d, ê,¡) and (ûr, ñ, f ate orthogonal triads in spin and orbital spaces with arbitrary relative orientation. In Chapter 6 it will be shown that these triads are coup^led by the dipole forces between the 3He atoms such that in equitibrium d and I ate oriented in parallel. In the B2 state (s.28) dt"i: e'o¡A\â*rh, + êr"ñj) - B(ât ñi - ê*ñ¡) + cÎ,îil, where

A: ï(An* Áu), B -\i(Ar - Arr'), C :

A.o.

5

L22

Pair Coruelations beyond Weøk Coupling

5.2.2 Relative stability of model states In order to discuss the relative stability of model states, it is useful to invoke the concept of a parameter space of the B¿. The regions of stability of the model states form subspaces that completely cover this parameter space. In this space the actual physical system is represented by a single point B: (þt, þr, þt, þ0, þt). In the weak-coupling model it has the fixed coordinates B¡¡¡ç:8þo?ä,1,I,1, -1). Corrections to the weak coupling model will lead to a dependence of the position of B on external parameters (e.g. pressure). As these "strong-coupling" corrections can be expected to be small (owing to the smallness of T"ITò, the point B will stay relatively close to Bs¿ç. In Section 5.3 we shall estimate the corrections AB¿ as 0 < lLþ,|þJ< 0.5. We can therefore confine our discussion to model states whose regions of stability are in the vicinity of this point. A detailed investigation of the boundaries of the stability region shows that By6 lies within the region of the BW state but close to that of the ABM state. To reach other stability regions (the closest one being that of the polar state, as indicated by the value of the total fourth-order contribution K; Table 5.1) requires the B¡ to deviate from ÉI" by amounts less than unity. Therefore the BW and ABM states are the most likely candidates for stable states in superfluid 3He. To exemplify such a stability analysis, we now discuss the stability regions of the four unitary states. Among the real states, the stability is governed by the relative size of Br2 and 83a5. If both þrr> 0 and þzus> 0, or else if þt210 and þsos) 3lBt l, the BW state is lowest in free energy. For B3or< 0 but þrz) 0, such that -þrr1þror10, the polar state wins. All other regions are excluded by

stability considerations. There is no regime where the planar state is stable. The ABM state is lower in free energy than the BW state if þrr> 0 and 0 ferromagnetic state (f 0.

(6.1c)

This expresses the fact that there exists long-range order between the spins in the ferromagnetic state. Therefore a correlation function is a measure of the degree of order in the sYstem. A different way of quantifying the degree of order is to look at the

6

Broken Symmetries and Macroscopic Order

139

average spin (S) itself. This approach is closely connected to the investigation of the symmetries inherent in the system. In the paramagnetic phase the ground state of the system is left unchanged if we rotate all spins through the same angle around some rotation axis: it is invariant under spin rotations. In other words, there exists a symmetry under arbitrary spin rotations. Below 4, in the ferromagnetic state, this is no longer true. Here a rotation of all spins leads to a perceptible change in the ground state. So the ground state does not show the same symmetry under spin rotations as the paramagnetic state: the symmetry appears to be "spontaneously broken", i.e. by the system itself, without external influence. On the other hand, the system is in principle not able to pick out any particular preferred direction

for the spin. So, although there exists a preferred direction,

it is still

undetermined. Therefore the average (S), when taken over many ensembles, is still zero as in the paramagnetic state. The ground state of the system is then degenerate. In particular, it costs no energy to turn all the spins into some direction. This allows for a collective excitation of the spins in the ferromagnetic state with energy k@:0 and wave vector 8:0, i.e. a "spin wave", which simply rotates all the spins. Such a mode is usually referred to as a "Goldstone mode" because in any system with a broken gtobal continuous symmetry there exists, in the case of short-range interactions, a gapless spin-zero excitation, a so-called "Goldstone boson". Hence in the present case of a ferromagnet the Goldstone mode is a spin wave connected with the spontaneously broken spin-rotation symmetry. Here it becomes evident that the initial symmetry (i.e. that of the underlying Hamiltonian) has not simply "disappeared". Even though the ground state may have a reduced ("broken") symmetry, the corresponding Goldstone modes supplement it so as to restore the initial symmetry. The concept of broken symmetries and the associated Goldstone particles (or modes) are of great importance in many fields of physics, including condensed-matter physics and high-energy physics (see Section 6.2). Returning to the ferromagnetic state, it is clear that the degeneracy of the ground state with respect to the actual direction of the magnetization will make the system susceptible to the influence of even infinitesimally small effects that lack symmetry under spin rotation. They may be due to interactions so far neglected (because they were unimportant for the actual formation of the ferromagnetic state) or to some external magnetic field. Although these effects can be arbitrarily weak, they are none the less abte to select a particular preferred direction for the spins. In this case the Goldstone mode will no longer be a o:0 mode but will acquire an energy gap proportional to the coupling strength of this interaction. Once a preferred direction has been determined, the average (S) of the spins is no longer zeto. The system then has a finite magnetuation M* (S) and we can conveniently choose M itself as the order parameter of the system (rather than using the correlation function C(r - r')). Above 7" we have lMl:0, while below T",lMl>0 (see Fig.6.1b).

6

140


The above discussion of symmetry breaking in a ferromagnet applies to an isotropic Heisenberg ferromagnet with S: (S,, Sy, S,) as well as to a planar ferromagnet with S: (S,, Sr). The latter system (often referred to as the XY model) is similar in many respects to an isotropic superfluid in that the complex order parameter can be represented by a two-component vector in the complex plane. The connection between a broken symmetry and long-range order has been illustrated in detail for a ferromagnet since this case is particularly simple. Nevertheless, the concept applies equally to a pair-correlated state, i.e. to superconductors and superfluid 3He. In this case the new order is associated with the formation of Cooper pairs below the critical temperature f.. In analogy to the ferromagnet (see (6.1a,c)), this order may be expressed mathematically by introducing a correlation function, which in the present case involves two Cooper pairs (CP) at positions r and r' respectively:

acr'cr(r

- r'): (Vi(") rpiQ)rp{r')ú1(r'))

(6.2)

(for simplicity we restrict our discussion to conventional superconductors with S : l:0 pairing). Here the ñeld operators ,ltit) and rp"(r) respectively create and annihilate an electron with spin ø at position r, and üiQ):E*a[p-i*'r etc., where afo is the creation operator for a particle with momentum kk and spin ø (see (3.LL)). In the superconducting state the correlation function (6.2) approaches a finite value even in the limit of infinite separation of the pairs:

tim acr'cr(r - r')*0.

l¡-r'l+æ

(6.3)

This is in contrast with the normal state, where this expectation value is identically zero. The behaviour expressed in (6.3) is referred to as "off-diagonal long-range otrder" (Yang L962). It expresses the fact that the off-diagonal elements of the two-particle density matrix (6.2) in the position-space representation (i.e. those for r * r') show long-range order. The same concept applies to a.Bose-Einstein condensale', which is the sirnplest example of off-diagonal long-range order (Penrose and Onsager 1956). The particular long-range order implied by (6.3) is a consequence of the fact that in the pair-correlated state the particle-nonconserving operator ø[¡a!¡,¡ has a finite expectation value-this is precisely the pair amplitude F-*n introduced in (3.13b). Hence, rather than using the correlation function (6.2), the degree of order in the pair condensate may equally well be described by the pair amplitude itself, which therefore takes the role of the so-called "order parameter". As discussed in detail in Section 3.2, the existence of such an order parameter implies a spontaneously broken gauge (i.e. U(1)) symmetry.

6.1 Broken Symmetries

6.1

and Group Theory

L41.

BROKEN SYMMETRIES AND GROUP THEORY

To gain a better understanding of the concept of "broken symmetry" and to formalize the connection between order-parameter structures and their symmetry, it is most useful to employ the language of group theory and topology (Michel L980, Mineev 1980). It allows an elegant but straightforward treatment of the problems in question. In particular, we shall see that these methods provide the basis for an understanding of underlying phenomena common to seemingly different areas of physics. From a group-theoretical point of view, "broken symmetry" means that below the transition the lowest state of the system is no longer invariant under the full group of symmetry transformations G, as it was above the transition. Certain symmetries (but, of course, not necessarily all) are "broken", such that this state is now only invariant under a subgroup ,F/ of the full group G of symmetry transformations. While the group É/ describes the remaining symmetries of the order parameter, the coset (or factor) space R defined by (6.4) R: GIH characterizes the broken symmetries, i.e. those transformations that change the ground-state perceptibly. (Although they do change the state, they leave the energy invariant, because the corresponding states are degenerate with respect to the energy.) Note that R will only be a group itself if. H is a normal (or invariant) subgroup of G. (A normal subgroup H of. G is defined by gHg-t:.F/ for all g e G.) None the less, even if this is not the case, R G lH may be called a "factorization" of G with respect to H and one may write G R x É1. So the product R x ^F/ (meant as a product of sets) does not necessarily involve elements of two groups, but, in general, multiplies representatives of the set of transformations contained in R by elements of the group .F1.

:

:

The coset space R is called the "manifold of internal states" or

"degeneracy space". In most systems of condensed-matter physics that exhibit a broken symmetry this symmetry corresponds to rotations about an axis or to threedimensional rotations. Rotations about an axis may be described by the Abelian group U(1), the unítary group of rotations about a single axis. Equivalently, they are represented by the group SO(2), the special orthogonal group of rotations in a plane with determinant *L, having real elements; hence U(1) 4 SO(2). Three-dimensional rotations are described by the non-Abelian group SO(3) (details will be discussed below). Two examples from condensed-matter physics illustrate this point.

Example 1-: BCS superconductor or superfluid aHe. Here the order parameter is given by a single complex parameter p : !)oêtÓ, with amplitude rpo and phase ("gauge") @. Above T" the system is invariant under an arbitrary

6

t42

(ol Figure

6.2

51

Broken Symmetries and Macroscopic Ordcr

(b)

SO¡

Schematic drawing of topological spaces: (a) the representation S' of the group U(1); (b) the representation SOs of the group SO(3), and of the coset sPace S'.

Q-

change of the phase Q' , i.e. under a gauge transformation (see the discussion in Section 3.2.I). This invariance is equivalent to a U(1) symrnetry i.e. G U(1). Below ?i a particular value of r@ is spontaneously preferred; hence the U(1) symmetry is completely broken. The group of transforrnations corresponding to rotations about a single axis, U(1), ìs uniquely characterized by a rotation angle that takes values between 0 and 2n. Therefore this group may be represented by mapping it onto the circumference of a (unit) circle: the topological space 51 (see Fig. 6.2a). Note that we distinguish the abstract group U(1) from its topological representation Sr.

:

Example 2: Ferromagnet. The order parameter is given by the magnetization M, a real three-component vector. In the paramagnetic phas9, i.e. abovè.the transition, the spin system is invariant under arbitrary global rotations in the three-dimensional spin space, i.e. G : SO(3). Below the Curie temperature 7., in the ferromagnetic regime, the spins are aligned along the preferred direction of the magnetization M, i.e. along M : M llMl. Now the lowest state of the system is only invariant under rotations about the single axis û. This remaining symmetry corresponds to the group of transformations U(1), i.e. H:U(1). We see that in this case the symmetry G:SO(3) is not completely broken but is reduced to H:U(l).The broken symmetry is therefore given by R : SO(3)/U(1). Any three-dimensional rotation may be described by a rotation axis É and an angle of rotation 0 about this axis. The group of solid-body rotations SO(3) can therefore be represented as a topological space in the following way: any rotation about an axis ñ by an angle 0 is associated with a point in a solid sphere of radius n; this point is the endpoint of the vector n: Ofi

6.1 Broken Symmetries

and Group Theory

r43

originating at the centre of the sphere, pointing in the direction of fi and of length 0 (see Fig. 6.2b). Diametrically opposite points on the surface of the sphere represent rotations about ¡r and -ir respectively, and are therefore equivalent-hence these points have to be identified. This defines a doubly connected topological space referred to as SO3. In particular, three-dimensional rotations involving a foed rotation angle 0 can be represented by the points on the closed spherical surface of radius 0 within the solid sphere SO3 (see Fig. 6.2b). This surface is itself referred to as 52, the "two-sphere". Consequently, SO3 factorizes as

so(3)/u(1)

: s'.

(6.s)

We observe that the set of transformations contained in 52 is not a group. For example, two successive rotations about the x and y axes by the same angle 0 are in general not equivalent to a single rotation by 0.

The above factorization applies directly to the case of the paralfenomagnet. While /1: U(1) describes the remaining symmetry in

the ferromagnetic state, the broken symmetry is characterized by the degeneracy space R : 52 corresponding to the manifold of orientations of the magnetuation M.

6.1.1 The symmetry group describing superfluid

3He

Owing to the fact that superfluid 3He forms a spin-triplet p-wave condensate, the order parameter Ap¡ has a considerably more complicated structure than conventional order parameters. If we only consider the interactions responsible for the formation of the condensed state, i.e. we neglect small residual interactions like the dipole interaction that are unimportant on this level, then the free energy of the system has to be invariant under separate three-dimensional rotations in spin space, in orbital space and under a gauge transformation (this requirement has in fact been used explicitly to derive the free-energy functional (5.4)). Hence the symmetry group that allows for these symmetries is given by a product of three independent symmetries

G

: So(3), x So(3)" x U(l)r.

(6.6)

Here the indices Z, ,S and Q indicate orbital space, spin space and gauge respectively.

Clearly, this symmetry group is much richer in structure than those considered before. In fact, it incorporates the order-parameter symmetries of liquid crystals, magnets and isotropic superfluids all at the same time. This implies that the brokelr symmetries resulting from (6.6) will also be much more intricate than, say, in the case of the ferromagnet, where only a single degree of freedom, i.e. a single group, is relevant.

6

t44


6.1.2 A two-dimens¡onal model of superfluid

3He

To understand the fundamental difference between the broken symmetries that may occur in the case of a single group like SO(3) and those in (6.6) consisting of several different symmetries, we first consider a somewhat simpler example. Let us have a look at a two-dimensional (d:2) liquid with two different degrees of freedom given by, say, an angular momentum and a spin (Liu L982). Both degrees of freedom are assumed to vary only in the plane; hence they may be described by a U(1) symmetry each. Consequently, if no symmetry were broken then the underlying symmetry would be given by

G:U(1)¿xU(1)s, where

L and,S refer to the angular momentum and spin respectively.

(6-7)

We now think of the particles in the liquid as being characterized by a direction both with respect to their spin (1) and their orbital angular momentum (1). Because of the U(1) symmetries, both directions may point in any direction in the plane. In Fig. 6.3 we see all possibilities of ordering, i.e. symmetry breaking, of these degrees of freedom. Figure 6.3(a) shows the disordered state: the system is isotropic with respect to the orientation of both degrees of freedom. The system stays invariant under separate rotations in spin and orbital spaces; there is no long-range order and hence G :,F1: U(1)¿ x U(l)r, R : L. This case represents an example of an isotropic paramagnetic liquid. On the other hand, in Figs. 6.3(b-e) we see all possible different symmetry breakings, i.e. states with long-range order. In Fig. 6.3(b) only the spin degrees of freedom are ordered ("breaking of rotational symrnetry in spin space"), while the orbital component is still disordered. So, while the U(l)s is broken, the symmetry with respect to orbital rotations is still intact: É1: U(1) r, R: U(1)". If we identify I with the spin of the particle, this system therefore represents an isotropic ferromagnetic liquid. Conversely in Fig. 6.3(c) only the orbital degrees of freedom are ordered while the spin part is disordered ("breaking of rotational symmetry in orbital space"). Here U(1), is broken while U(1)" is preserved: H: U(L)5, R : U(l)r. One might take this as a model for some kind of liquid crystal, where the molecules are oriented along a preferred direction. In Fig. 6.3(d) both degrees of freedom are ordered, i.e. both U(1)¿ and U(1)" are independently broken, so that no symmetry remains: ,FI: 1, R : 3He-4, U(1), x U(l)s- In fact this is a two-dimensional model of superfluid if we associate â with f and i wittr t respectively. (The additional gauge symmetry also broken in the superfluid has not been included here.) So, d and Í are both macroscopically orde-¡'ed ("breaking of rotational symmetry in spin and orbital spaces separately"). One might think that these three states exhaust all possibilities of broken symmetries, but this is not the case. In Fig. 6.3(e) we finally show a

6.1

Broken Symrnetries and Group Theory

1

: Spin degree of f reedom

Ï

: orbatol degree of f reedom

r45

(q)

c/< (b)

*\ ^\\*ç \Þ (c)

f(

(

(

(d

)

"A

(e

)

"B phose "

phose "

ltt Figure

6.3

The possible states of order in a two-dimensional model liquid with two different degrees of freedom. (After Liu (1982).)

t46

6


situation where neither of the two degrees of freedom is orderednf)e, :

At"j.

(6.e)

6

148


Hence the order parameter is seen to be invariant under a simultaneous three-dimensional rotation in spin and orbital spaces. Consequently, the B-phase order parameter still shows an SO(3) invariance.

(ü) 3He-4. The order parameter is given by (3.68), i.e. At"j:2-Lt2â*çh, + iñ),

î

: ñ x ñ.

(6.10)

Note that no explicit phase factor appears: the phase is implicitly contained in the orientation of the triad (ñ, ñ., i¡ atong i. By performing a gauge transformation Ap¡-eiþA*¡, an additional phase factor is introduced. On the other hand, a rotation of the orbital component of the order parameter about î by un angle @' leads to ñ *ifi-+ eia'(ñ +iñ). Hence transformations of the gauge variable and of the orbital part around the z axis with Q' -- -@ just compensate each other. This particular linear combination of transformations thus leaves the order parameter invariant. It corresponds to a remaining U(1) symmetry of the order parameter involving both the gauge and the orbital variable, i.e. a relative symmetry. In addition, the order degrees of parameter remains unchanged under a rotation of the ^spin freedom about the direction of the real unit vector d. This is again equivalent to a U(1) symmetry, but now in spin space. In fact, the order parameter is also invariant under the replacement (d, frr, ñ)- (-d, -ñ, -ñ). This is a discrete symmetry, described by the group Z2 (the group of integers modulo 2, having two elements). Altogether, we find that the A-phase order parameter is still invariant under a U(1) x U(1) x 22 symmetry, where one of the U(1) groups describes a relative symmetry.

(üi) 3He-41. The order parameter is given by (5.59): At

A

j:

+@*

+ iêþ)Qhj + iñj).

(6.11)

Ar,+"taAyi.and a rotation of the orbital component about î:ñxñ to ûr+iñ--eia'(ñ +ifi). Hence, just as in 3He-4, a joint transformation of the gauge and the orbital part (the linear combination with Q' : - Q) leaves the order parameter invariant. This corresponds to a U(1) symmetry involving gauge and orbital degrees of freedom. On the other hand, since the spin part of the order parameter (6.11) is identical in structure with the orbital part, a gauge transformation can equally be compensated by a rotation of the spin component around î:AxêQttiê+eia"(â+i¿)) íf Q":-þ. This is yet another, different, U(1) symmetry, this time involving a linear combination of the gauge and gauge transformation leads to

the spin degree of freedom. (Note that this U(1) symmetry may equally be interpreted as a linear combination involving spin and orbital degrees of freedom, because the two can also compensate each other.) Hence the Ar-phase order parameter is seen to be invariant under two separate relative U(1) symmetries, i.e. under a U(1) x U(1) symmetry.

6.2

6.2

Symmetry and Order-Parameter Structure

L49

SYMMETRY AND ORDER-PARAMETER STRUCTURET

We have now convinced ourselves that the order parameters of the A, A, and B phases still contain a remaining symmetry. Obviously the symmetry group G, (6.6\ relevant to the overall structure of the order parameter Ap¡ of superfluid 3He is large enough to allow phases where the symmetry breaking does not destroy the full symmetry. This we found to be true even in the case of the para/ferromagnet, where the symmetry group G : SO(3) is much smaller but where the ferromagnetic phase nevertheless has a remaining symmetry É1: U(1). On the other hand, the symmetry group relevant to the transition from normal liquid to superfluid oFI", i.e. G : U(1), is too small to allow continuous subgroups, and consequently the symmetry is completely broken below the transition. In the case of superfluid 3He the symmetry group G is exceptionally large, leading to phases with a particularly complex structure. The observation that the A, A1 and B phases still contain a symmetry H c G may lead one to expect that physically relevant phases with a broken symmetry R, whose order parameter has a rich underlying symmetry G, will possess a remaining symmetry (i.e. H +I). Concerning superfluid 3He, this expectation is supported by the following observation: it is easy to see that not only the order parameters of the A, A, and B phases but, in fact, those of. all phases so far identified as stationary points of the Ginzburg-Landau free energy (5.4) (see Chapter 5) possess a remaining symmetry H c. G. Hence in these phases the symmetry G is only partly broken. Consequently, it appears that an'order parameter with a remaining symmetry corresponds to a phase with lower energy than one where all symmetries are broken. On the other hand, the question of which symmetry will actually be broken clearly depends on the details of the corresponding free energy. Therefore one should not expect the above statement to be generally true. Nevertheless, using somewhat simpler order parameters and free-energy expressions than those relevant for superfluid 3He, one may give explicit examples where this is rigorously true (see e.g. Section 6.2.5). Since an exact minimization of the Ginzburg-Landau functional (5.4) for general parameters Ér is not yet possible, it is clear that the symmetry concept of phases is extremely valuable: starting from the underlying symmetry group G, it allows for a systematic_discussion of all phases with broken symmetry whose remaining symmetry H *1is a factor group of G. Once these phases and their corresponding order parameters have been obtained, one may compare their energy to find which is the stable one. Symmetry classifications of this kind have been used before. For

T

This section contains a formal discussion

order-parameter states. 6.3.

It

of the symmetry classification of equilibrium

is not a prerequisite for an understanding of the material in Section

6

r50


example, this approach has been applied to describe a ferroelectric phase transition in perovskites (Jarió L982). Similarly, Kim (1985) illustrated the method by applying it to a Ginzburg-Landau potential describing BaTiO3 in order to explain the three successive phase transitions known to occur in this system. Furthermore, such classifications have been shown to be very useful for studying possible phases of heavy-fermion superconductors (Anderson L985, Volovik and Gorkov L984, L985, Ueda and Rice L985, 3He and aHe (Salomaa and Èlount 1985) and vortex states in superfluid Volovik 1985a,b, 1987). The latter will be discussed in Chapter 7. In the case of superfluid 3He we already know that the energetically stable phases (4, Ar, B) still have remaining symmetries, so we can be sure that they witl appear in a symmetry classification as described above. We now want to present a symmetry classification of the subgroups of the 3He, i.e. of G: SO(3) x SO(3) x symmetry group relevant for superfluid U(1). Such a classification was first considered by Golo and Monastyrsky (1977,1978a-d) and Mineev (1980), yielding several states with a remaining symmetry fI (see also Volovik and Gorkov 1984, 1985). In particular, Mineev (1980) identified all inert states (see Section 5.1) that are obtained by a factorization using continuous groups. This leads to five inert states (note that the "planar" state (5.20) cannot be constructed in this way since it contains a discrete symmetry; see below). A symmetry classification of G, (6.6), including both continuous and discrete subgroups H, may be performed systematically (Bruder and Vollhardt 19S6). It allows for explicit calculation of all bulk superfluid states with a remaining symmetry, i.e. states whose order parameter Ap¡ is still invariant under a subgroup F/ c. G. This classification will be presented next; continuous and discrete subgroups are discussed separately.

6.2.1 Gontinuous symmetries Every group element h e H corresponds to a transformation leaving A= A*¡ invariant (Mineev 1980)

hA:

A.

(6.12)

Introducing T- (T,, Iy,T,), the generators of infinitesimal transformations of the group H, h may be written as

h:exp(-;".1

-r-ïo.r,

(6.13a)

(6.13b) 'i

ii

where ø is an arbitrary vector of infrnitesimal length. Equation (6.12) then

i¿i

;

i

g1

$[ å:

6.2 Symmetry and Order-Parameter

Structure

151

reduces to

(ø or

'T)A:0,

T;A: 0, i:

(6.1aa) (6.14b)

x, Y, z.

The order parameter A*, has the transformation properties of a "bivector": it transforms like a vector with respect to its spin part (fir'st index) and its orbital part (second index). Hence Ar¡ may always be expressed as a linear superposition of tensor products between unit vectors 3(") and ô(t) in spin and orbital spaces:

n:)

hu¡Q)66{;)

(6.1s)

vi

In the present text the sets of unit vectors 3(") and ô(t) are denoted

by and 1î,ñ,ñ'1 respectively. Equation (6.15), combined with the invariance properties of the free energy under the group of transformations (6.6) concerning rotations in spin and orbital spaces and a phase transformation, implies that T can itself be written as a linear combination of these transformations; for example,

(ã,ê, jl

T,: a,V + å,S', Tr: arV + bySY, T

r:

I I

(6.16)

erL + b rS' a tÖ.J

(L', V, L) and S: (S', S', S') are respectively the threecomponent spin and orbital-angular-momentum operators in the threedimensional representation (i.e. 3 x 3 matrices). They are given by

Here

L:

: -ihe¡¡¡,, (S")"^ - -ikerut. (L')¡o

Furthermore, Q :

-ihAlAþ

(6.17a) (6.17b)

is the operator for a gauge transformation, with

þA:

A, þA*: -4.

While S acts only on the first index and Q acts only on the overall phase.

of.

(6.18)

A¡,¡, L acts only on the second one

There are altogether seven generators (L, S,0)-a fact that is reflected in the existence of seven arbitrary real parameters e: (a*, a, e"), b (b,, by, b,) and c. The relations (6.17) define Lie algebras among the L'and SP components: [Lt, L'l

:ih)

[S", S"]

: ih;

e¡¡¡,Lk,

(6.lea)

eu,,rS^

(6.1eb)

k

L

We note that the Lie algebra (6.19) is common to both the rotation group SO(3) and the special unitary group SU(2), although they are different

6

152


groups. The connection between the two groups is given by SO(3)-: SU(2)|Z-,, where 22 describes a reflection, i.e. a discrete symmetry. This brings out the fact that SU(2) is a simply connected group while SO(3) is doubly connected (itt Fig. 6.2(b) opposite points on the sphere are identified). Hence locally the two groups can be identified (and this is precisely the reason why they share the same Lie algebra under infinitesimal transformations). Globøþ, however, the difference is of great importance. Nevertheless, for an investigation of broken continuous symmetries in bulk superfluid states, they can be used alternatively. Equation (6.14b) together with (6.16) and (6.11) yields

a,2

lv

e,¡A¡"t

* bt}

,,*,A",

t iô¿"cA*¡:0 (i:

*, !,

z),

(6.20)

i.e. three sets of nine homogeneous equations for the nine

complex components of the order parametet Ap¡ in terms of the independent parametels a¡r b* and c. Clearly, this system of equations is overcomplete, which in general implies a solution Apj:0. Only by constructing relations, i.e. linear combinations, between the a¿, b, and c can we enforce A*¡*0. This means that we break the underlying symmetry (at least parts of it). By constructing all possible linear combinations of tllre a,, b* and c, we then find all continuous subgroups H*L of G, i.e. we construct all those orderparameter structures that still possess some of the original symmetry (6.6). In other words, in such states the symmetry G is not completely broken. The order parameter A*, in (6.20) is calculated in a Cartesian coordinate system Q.t, j:x,l,z), i.e. the elements of A*, are given bY A,*,4,, etcThis is merely done for reasons of convention. Clearly, any other coordinate system may equivalently be used; in fact, in the present case, a spherical basis is particularly suited. The different representations are connected via unitary transformations. In the case of a vector ec^rt:(e,, ar, ør) this is easily illustrated. Using

U: ,u, 1.

øspher:Uec^n is calculated

(i ï' ];)

(6.21)

as

¿spher:(ä

=ü')

(6.22a)

(6.22b)

6.2

Symmetry ønd Order-Parameter Structure

where the notation in (6.22b) is self-explanatory. order parameter A, the same concept applies, i.e.

153

In the case of a matrix

(6.23) Apr,".: fJAc..,IJ*. Using a notation equivalent to (6.22b), the matrix elements of {o¡.. are given by Á** , A*- etc. We shall now list all possible linear combinations of L', S', Q in the

infinitesimal generators, i.e. all possible continuous subgroups ^F/ of G. Equation (6.20) is then used to calculate the components of A*t. lt should be r¡oted that the order parameters thus obtained are only determined up to a rotation in spin and orbital space and an overall phase factor, because there is no fixed coordinate system (see the discussion below (5.15)). First we start with the case where the linear combination leads to a remaining symmetry under all components of T in (6.16). This corresponds to a remaining SO(3) symmetry. From (6.20), it follows that a solution A*¡ *0 is only allowed if c : 0. In this case the T¿, (6.16), have to transform like the components of an angular momentum themselves, i.e. they have to obey (6.19), which leads to a¿:I,bu:L for all i, p (note that other solutions, a.E. ar: I, ay: a": -I, b*: 1-, may be brought into this form via rotations by ¡ about one of the x, y, z axes, and are thus equivalent). So the first linear combination of the a¡, b* and c is given by the following.

(1)

c

:0,

a¿:

bp: I.

This imPlies

T:L+S.

(6.24)

corresponding to FI: SO(3)r*". Here the order parameter is still invariant under joint three-dimensional rotations of Z and S. In this sense it is the most symmetric of all possible order parameters. Solving (6.20), one finds (see (3.65)) (6.2sa) A*i:3-Lt2õ*¡,

which is a special case of the more general form (see the remarks made above) given in (3.67) (6.2sb) A t"j : 3-Lt2 ei$ R pj(û, e),

where Rr¡ is a rotation matrix describing relative rotations of spin and orbital spaces, represented by a rotation axis ñ and a rotation angle 0. Note that this more general state (6.25b) is invariant under simultaneous rotations described by a matrix O¡¡ in orbit space and (FOß-')u* in spin space. This is, of course, the BW state, describing the B phase. The explicit phase factor eio results from the separately broken gauge invariance. To determine the degeneracy space R: GlH, i.e. the broken symmetries, we factorae G with respect to H: n SO(3)¿ x SO(3)" x U(l)o

tt: :

so(3L*"

SO(3)¿,s

x U(l)o.

(6.26a)

(6.26b)

6

L54

Broken Symmetries and Møcroscopic Order

We observe that the factorization of G with respect to SO(3)¿*s, the group of alt joint rotations of ^L and S, leaves us with SO(3)¿,s, the group of all relatiue rotations of Z and S, as well as U(l)o. Hence in the BW state the relative spin-orbit symmetry (as well as the gauge symmetry) is broken. This is precisely the "spontaneously broken spin-orbit symmetry (SBSOS)" introduced by Leggett (1972) ("f. the detailed discussion in Section 6.L concerning this particular kind of symmetry breaking in a two-dimensional model liquid). We see that the notion of a "broken relative symmetry" (Liu and Cross 1978) occurs quite naturally within a group-theoretical formulation. With the exception of the BW phase with Ë1= SO(3), the subgroups of G are composed of U(1) subgroups only, i.e. there are no other solutions A*¡*0 to (6.20) that involve all three components of T. We may then restrict any further discussion to the invariance properties of Au¡ under T, alone. Equation (6.20) then reduces to

a,(ô¡yA¡"2- ô¡2A¡"r) + b'(ôprAz¡

-

õ¡"2Ay) +

icA*t:0.

(6.27)

In fact, it separates into four independent subsystems belonging to different sets of A*¡ õomponents. For Au¡ to be nonzero, the determinant D of the matrix (6.27) has to vanish: D

:

DtDzD3Da:

D¿ are the determinants components as indicated:

Here the

O.

(6.28a)

of the submatrices belonging to the A*¡

- c'l

(Arr, Arr, Arr., At),

(6.28b)

(Att, Art),

(6.28c)

Dz: a?- c'

(Att, Att),

(6.28d)

D¿: ic

(At').

(6.28e)

Ðt:

- b,)' Dz: b?- ,t l@,

c'll(a" + b")'

This way of writing allows for a simple and systematic investigation of all linear combinations of a,, b, and c in (6.16). Depending on which of the Dt we choose to be zero (nonzero), \rye can immediately determine which of the Ar¡ components are nonzero (zero). From (6.28), we see that only linear combinations of the type a, : *.c, a" - b,: Lc etc. (and not, say, a" : lãc) yield nonvanishing A*¡. This is so becausa a,, b, and c correspond to angles of rotation around the z axis. To accomplish mutual compensation of rotations in the three different spaces (implying a residual symmetry), only particular choices of the a,, b" and c are possible. To construct a linear combination of infinitesimal generators means that we impose conditions on them ("break" the symmetry). As will be shown below, in the case of (6.27) a single condition (s"y, t :0), i.e. the weakest nontrivial restriction on (6./7), implies a'remaining symmetry H: U(1) x U(1), leading to an inert state, discussed in Chapter 5. Their orderparameter matrices only depend on a single overall factor, which is fixed

6.2


15s

by normalization. Hence the order-parameter structure is constant and, in particular, independent of the parameters É,. On the other hand, imposing more than one condition (tuy, c :0 and a,: b,) implies an even more restricted symmetry, H:U(1), leading to noninert states. Their orderparameter matrices depend on more than one parameter, such that even a slight variation of the B, (due, for example, to a pressure change) influences their whole structure. We see that the subgroups with the largest remaining symmetries lead to inert states. In fact, in group-theoretical language they are called "maximal" subgroups .Í1-.*, because there are no other subgroups FIc G containing H^^*. Hence maximal subgroups produce inert states. In the spherical representation of A, (6.23), the invariance property (6.12) determining A*u(p,u:0, *, -), and hence the set of equations (6.21), has a particularly simple structure:

(a,v*b,p*c)A*n:0. (6.2e) This is of course a result of the fact that in this basis the spherical harmonics Yt,^ are eigenstates of L" and S" with eigenvalues mh (see (7.8L)-(7.83)). The different order-parameter structures allowed by symmetry are again obtained by choosing all possible linear combinations of the infinitesimal generatots a,, b, and c (Salomaa and Volovik 1987, Schakel and Bais 1989).

We now complete the list of possible states.

(2) ,

:0,

e", b, arbitrary. This implies T

r:

arL' +

b"S',

and hence 11: U(l)s. x U(L)¿,. Because

(6.30)

of Dt, Dz, Dz#0, Da:O, one

immediately obtains the polar state

A

"i:"''(l ; i)

(6.31)

Its order parameter is still invariant under separate rotations around the preferred directions in spin and orbital spaces, which are both real unit vectors. Accordingly, its broken symmetries are given by

^:ffi'äffxu(l)a, : S?

x S! x U(l)r,

(6.32a)

(6.32b)

i.e. besides the phase invariance, rotational invariance with respect to the directions of the preferred direction in spin and orbital space is also broken.

6

156

Broken Symmetries ønd Macroscopic Order

ftre polar state is unique in that it is the only inert state without a broken relative symmetry.

(3) t

:0, b' :0,

a, arbitrary. This imPlies T

r: arL',

(6.33)

and hence H:U(!)2,. Because Dr,D3+0 and Dz:Dq=O, the order parameter has the structure

Ati:"r"'"(l

3

;r)

(6.34)

A, B and C complex and pt:flAl'+Wl'+lcl'l-t't The condition b,:0 implies that in this noninert state the symmetry with respect to with

rotation in spin space has been completely broken:

o:ffixso(3)5xu(1),

(6.3s)

: S? x So(3)s x U(l)o.

(6.36)

ln this respect it is less symmetric (i.e. more restrictive) than the polar state (6.30), which is seen to be a special case of (6.34), with A : B :0.

(4) t

:

0, Q,

:0,

b, arbitrarY. This imPlies f .: brS',

(6.37)

and hence H: U(l)r,. It corresponds to the previous state with orbital and spin components interchanged, i.e. A

"j:rr"'-(rj

(6.38)

; l)

and R : SO(3)¿ x S! x U(1). Here the rotational symmetry in orbital space is completely broken.

(5) c :0,

e"Tb,:0.

This implies

T,:a,(L'*S'), and hence H:U(l)r"*¿. Because Dt: D+:0 and Dz, Dz+0,

A"j:r*,,(ii where

A, B and C are complex

iî l)

numbers and pt:[2(lAl'

(6.3e)

this leads to (6.40)

+ lBll + lcltl-t''

6.2


r57

This state is a generalued representation of the so-called "Ç phase" (Barton and Moore I974b). However, the name "oblate state" appears to be more useful, because it actually describes a squashed spherical state. In fact, (6.40) represents precisely the order-parameter structure of the B, phase (5.66), i.e. of the B phase in an external magnetic fietd (with A*Øn * Áu), 3 : -|i(Am - Á,,), C: Ár). Furthermore, by flattening the sphere completely, i.e. choosing B : C:0, one obtains the usual planar state. As will be discussed below, B and C are naturally forced to zero by demanding some additional dßcrete symmetry for A*,. Therefore the actual planar state contains both continuous and discrete symmetries. The broken symmetries of (6.40) are found to be

R_

So(3)¿ x SO(3)5

U(1)r,*¿

x

u(l)o

(6.41)

: S? x Sl x U(1) L,,s,x U(l)o,

(6.42)

which includes the broken relative symmetry U(l)r.,", with respect to rotations in spin and orbital spaces.

(6) o, *

c

:0,

b, ørbitrary. This implies

T":a,(L -0) +b,S', and hence

H:U(L)r"-o x U(1)¿.

Because

Dz:0

(6.43) and Dr., Dr,

D¿*0,

one

finds

A":#(s,:

;)

(6.44)

which is of course the ABM state, describing the

A phase. It is still invariant under rotations about one direction (the z direction) in spin space and under a joint transformation of the phase and a rotation around L". The broken symmetry is obtained as SO(3)sx SO(3)¿ x R_

u(1)",

U(l)o

U(L)r,_o

: 53 x SO(3)¿,@.

(6.4s) (6.46)

Here and in the following we interpret SO(3)¿,ø as Sl x U(L).,,r. Hence invariance with respect to rotations of directions in spin and orbital spaces, as well as the relative gauge-orbital symmetry U(l)r",o, is broken (Liu and Cross 1979) (cf. the related discussion in Section 6.1. concerning a two-dimensional model liquid).

6

Broken Symmetries and. Macroscopic Ord.er

We note that in the above classification of continuous symmetries the discrete symmetry of the A phase under the replacement (d, ñ, çd, -ût, -ñ) (see next Section6.2.2). Therefore discussed below (6.10) cannot be obtained the broken symmetries are not yet tully described by (6.a6) (see (6.68b)).

ñ)-

(7) a, * c :0,

b,: 0. This implies T

and hence

":

- 0),

a,(L'

Ë1: U(1) L"-e. Because Dt:

6.47)

Dt:0, Dt, Do# 0, one frnds

A":#(z

rl)

(6.48)

tn this state spin-rotational symmetry is completely broken (á R

:0):

: so(3)" r 99QLl U(lL

(6.4e)

:

(6.s0)

U(1)r,-o

SO(3)s x SO(3)¿,o.

Because of the more restrictive symmetry requirements, its order parameter is less symmetric than either that of the A phase (A B 0) or that of the A1 phase (A iB, C 0). In fact, it is a general representation of the A, Ar), B )i(A¡ phase (5.51), which is obtained by choosing 'q,: T(A¡ Note that the orientation of the spin-coordinate system in Á¡) and C has been chosen differently from that for the the present case (ã.lli,¿ A phase (d ll2), which is, however, of no importance.

:

:

-

:

*

:0.

:

-

lli)

(8) b, *

c:0, a,:0.

This imPlies

T,:b,(S'-0),

(6.s1)

and hence H: U(l)¿-o. It can be obtained from the previous state by interchanging orbital and spin spaces, i.e.

A-#(+

(6.s2)

T f)

Here rotational symmetry in orbital space is completely broken (ø

:0),

i.e.

R:SO(3)rxSO(3)5,r. (9) b" * c :0, a, ørbitrary. This implies

r:

- ,Þ), and hence H : U(l)r" x U(1)¿-4. Because Dr:0 T

e"L' + br(S'

(6.s3)

and Dr., Dr, D¿* 0, one

6.2


159

finds

A':#(l

(6.s4)

S;)

which is called the "B state". Its symmetry properties etc. are obtained by interchanging the orbital and spin parts of the ABM state (6.44) (A-At). Accordingly, in this case a relative gauge-spin symmetry is broken.

(10) c :

a,

å". This implies

r,: and hence 0, so that

a"(L' + 0) + b,(s'

É/: U(1) L,+þx U(l)¿-o.

-

(6.ss)

0),

Here one has Dt:0 and Dz, Dr, D¿*

/t At"j::l _,

\o

i (6.s6)

1

o

;)

which is the representation of the A1 phase. The broken symmetry is given by

tr(L)L,x

U(1), x U(1)* R:S?xSlx" U(l)r"*E x U(1)5,_*

:

S?,

(6.s7)

x S! x U(1) 1.,s,,e.

(6.53)

In this case a U(1) symmetry involving a linear combination of all three symmetries, i.e. a gauge-spin-orbit symmetry, is broken (1,L) a,

i c :0,

b, + c :0. This implies

I,: and hence that

H:

(Liu I979a).

a,(L' + S'

- 0),

(6.59)

U(1)r"*",-4. Here one has Dr, Do#0 and Dz: Ds:0,

A":#$''i

+)

so

(6.60)

where A and B are complex numbers and p3:[lAlt+lB1z7-rrz. This state has been called the "e state" (Barton and Moore t974b); it may be viewed as a linear combination of the A and B states. This noninert state is only invariant under joint U(1) rotations of spin, gauge and orbital variables. In a way it is complementary to the A, phase. Its broken symmetries are R

: S? x S! x U(1)¿" x U(l)s, x U(l)o U(1)r,-r,-o

(6.61)

6

160


: S? x Sl x U(1) L,,ex U(1)"",o,

(6.62)

i.e. a relative spin-gauge and a relative gauge-orbit symmetry are broken.

(I2) a, + 2b, : 0, c + b, - 0. This implies

:

c(ZL' - S' + 0), (6.63) and hence H:U(l)tt"-t"*o. 'We are here dealing with a different linear combination of infinitesimal generators than in case (11) and hence with a different symmetry ^F1. Because D1 : Dz: 0 and Dt, Do# 0, one has T,

(6.64) where A and B are complex numbers and pa:Í2lAlt+lÙl'l-t''. This noninert state is a superposition of the At and B states. As in the case of the e state, it has two separate broken relative U(1) symmetries.

(L3) 2Q, * b,

-

0, c

* a,:0.

This implies

F,:-c(L -zS -0),

(6.6s)

and hence H :U(I)r,-u"-o. It is obtained from (I2) by interchanging the spin and orbital parts of the order parameter (A+ Æ). The last two states are similar to case (11) because the symmetry group.F/ involves particular linear combinations of all three symmetries, i.e. particular joint rotations. Consequently, their broken relative symmetries will all be very similar. We have thus found L3 order-parameter structures that still contain the symmetry of a continuous subgroup H of. G: one with a remaining So(3) symmetry, four with a U(1) x U(1) symmetry (they all lead to inert states) and eight with a remaining U(1) symmetry, yielding noninert states. In Fig. 6.4 the consecutive factortzation of the symmetry group G with respect to coritinuous subgroups is shown schematically. An arrow running diagonally always indicates that the factorization involves more than one group, such that the resulting state has a broken relative symmetry. The figure also allows for a straightforward identification of the maximal subgroups, i.e. those subgroups of G that are n-ot contained in any other subgroups themselves; they lead to inert states. There is one So(3) subgroup, generating the B phase, and four U(1) x U(1) subgroups corresponding to the polar, A, P and A1 phases. All other subgroups derive from these five maximal subgroups and lead to noninert states. Note that the classification yields all phases known to exist experimentally.

6.2

r6t


so(3)saSO(3)¿AU(1), \/

so(3)¿*s

B; (1)

I

U(1)r,*", U(1).'.

A

Br,

U(1),-,

\

J

U(l).',

oblate,; (5) polar; (2)

\á-

U(l),-,-t

A

E,

A; (6)

F; (e)

A,; (10) U(1)r,-o

Ar; (7)

I U(1),-,

(3)

I

u( 1).r. - o

(8)

I

u(

(4)

U(1)r,*r,-o

Figure

6.4

e; (11)

U(1).r.-rr-,-o

(12)

U(l)rr,-,-,*ç

(13)

Consecutive factorization of the symmetry group G: SO(3)¿ x SO(3)s x U(l)r with respect to continuous subgroups. Broken relative symmetries arise in factorizations involving more than one group, indicated by diagonal arrows. The numberJon the right refer io the states as they appear in the symmetry classification; conventional names of states are also included (Bruder and Vollhardt 1986).

6

162


6.2.2 Discrete symmetries There is no a priori reason why superfluid phases in a spin-triplet p-wave state should only display broken continuous symmetries-they may also have broken discrete symmetries. As discussed above, the continuous subgroups of G may be specified by the infinitesimal generators of G: each element is written as an exponential of a linear combination of generators. There is an analogue in the theory of discrete groups: a discrete group is generated by a subset of elements, i.e. each element can be written as a product of powers of generators. The groups C" for example, i.e. the rotations through Znkln,k:L,...,Í!-I, are generated by one element only. However, because of the large number of different discrete subgroups of G, these generators turn out not to be so useful, i.e. it is more convenient to resort to a different approach. Let Ho be a discrete subgroup of G, (6.6). Each element of G can be describing a rotation r" in orbital space, a written as a triple (I", ,", "to) rotation f in spin space and a gauge transformation etÓ. Clearly, r" and r" will have to be elements of discrete subgroups of SO(3), and eio an element of a discrete subgroup of U(1). The discrete subgroups of SO(3) and U(1) are well known (see e.g. Landau and Lifshitz 1958). For SO(3), they are given by

k:

(i) c"

the rotations about the z axis through Znkln, with '1.,...,n- L; the elements of C" are denotedby CÍ;

(ii) D"

generated by C, and an additional rotation through ¡r about an orthogonal axis;

(iiÐ T, O, Y the point groups of the tetrahedron, the cube and the icosahedron respectively.

For each discrete subgroup H¿ of G, there exists a minimal direct-product subgroup Hox H"x Hr of G that contains .F1¿. "Minimal" means that there is no smaller direct-product subgroup of. Ho x ¡/" x H* that contains //¿. It is easy to construct Ho, H" and Hç by projection of. H6 onto the factors SO(3)¿, SO(3)s and U(1) of G. This construction may also be applied to continuous subgroups. If, for example, we take the symmetry group of the B phase, SO(3)s*¿, the trivial result is that SO(3)¿ x SO(3)s is the minimal direct product containing so(3)s+¿. It turns out that F/a, the discrete group of gauge transformations, is essentially determined once ^É1" and H" are given. It is therefore sufficient to elucidate the structure of the orbital and spin parts of FI¿ with respect to Ho

x H".

The discrete subgroups H¿ of Hox,FI" can be thought of as follows: there exist normal subgroups HL of Ho and H'" of ,F1" such that the factor groups H"IHL and H"lHl are isomorphic, i.e. in one-to-one correspondence. (Note

6.2

Symmetry and Ord.er-Parameter Structure

163

that in the case of normal subgroups the coset space R: G lH is itself a group.) The group Hd is then given by the set of all pairs (å", /r,) with h, e H,l Hi and h, e H"l Hi, where ho and h" are isomorphic elements. Since Ho and Ë1, were chosen such that Hox ^F/" was the minimal direct product containing H¿, we do not consider direct-product subgroups of .F1" x Ã1, (for details, see Bruder and Vollhardt 1986). Following this recipe, we can write down all discrete subgroups of SO(3) x SO(3): the invariant subgroups of C,,, D,, T, O and Y are known, and it is easy to calculate H"IHL and H"lHl. Once all discrete subgroups of SO(3) x SO(3) have been found, we can proceed to determine the order-parameter states invariant under their action, together with the appropriate gauge transformations. The states are obtained by solving systems of linear equations as in the case of continuous subgroups. If A*¡ is to be invariant under the transformations of the discrete group

H¿: {(r?, f., "tÓ*)},

where

/
-þt+ lÉtl, the order

parameter is given by

B*":;(;

ì l)

(6.7e)

which has the form of the A1 phase, (6.56), and whose energy is F - -Êlzþ,;

(ii) for -6þr< þt< 0, the matrix has the form 0

B*u:

#(i

e-'2rril3

0e

i,",),

(6.80)

6

172

which is the

s


phase; its free energy is given by

F:-szlÍZ(þ"+

iþ,)h (iii) for þr lV : f¡,. For this, we may use the results of Section 3.5, where the expectation value of two-particle quantities has been calculated (see (3.Ll9a,b)). Inserting (6.86) into (3.119a) or using the Fourier transform of (6.86),

e'o''Q*u(r) : -!n(ôr"- 3Êrfr,),

Q*"(*):

I

o',

(

6.87a)

(6.87b)

together with (3.119b), we obtain (Leggett I973a,b,I974a, 1975a)

fo:

-L(yoY

- -*(yh)

I o'r IlF(")l' - i lî. r(")l'l

(6.88) (6.8e)

¡tv kk'

We note that 8*"(k), (6.87a,b), only depends on the direction of k. Therefore it might appear as if the sums over the magnitudes of /c and k' in (6.89) only involved the pair amplitudes and not Qr"itselfl However, this is not quite true, since the argument of Quu in (6.89) contains the difference between the two momenta, i.e. it involves (k - k')llk - k' l, over which one has to sum. Nevertheless, to lowest order in the small quantity T"lTr, the dependence on the magnitudes of /c and takes the form

r':T

eh)'àI

# Ï*

k'

may be neglected, so that (6.89)

8*"(k- o')(ã o;")(à

o") (6 e0)

Inserting (3.43b) for F¡np into (3.118b) to obtain Fo*, and making use of Q3aa) to replace Ao*uby the vector d(k), the lÉl integral over F¡,u in (6.90)

6

176


in the absence of a magnetic field is written

A,(r)=

as

-Ztkt Fo*

:4

ourUt

)

tå(oe

o*)**l[",0,

1."

(6e1)

"Wl

The last factor in (6.91), i.e. the Ç integral, has already appeared in the gap equation (3.46b). For unitary states this term is independent of lt, and in fact, even for nonunitary states, the ¡r dependence of .E¿" is expected to be negligible (Leggett I975a).In this case (6.91) reduces to

A*(r)

:

¿ *€e)t

(l Ak(T)|, T),

(6.e2)

where

1: N(o)

I"

ot

tanh (E¡,lzkBT)

(6.e3)

Ek

:

ÍE' + l ak(T)l2ltt2. As mentioned below (3.46b), for e. >> l¿o(f)l the 6 integral is dominated by the E-L dependence of the integrand at large E. This leads to a logarithmic dependence on e", making the other energy scales, which are of order 7i, unimportant. Indeed, (6.93) hardly depends on Ap at all, as can be verified by calculating the limiting values for T+ 7. (see the evaluation below Q.a7c)) and T-+0 respectively: and E ¡

I(0, T"): N(0) rn (cr¿f|), 1(lÁ*(0)1, 0)

: N(0) tr

(6.eaa)

[,]h]

(6.e4b)

T+

7", (6.93) does not depend on the gap at al[, it acquires only a weak logarithmic dependence for T+0. To a good approximation, we may therefore neglect its fr dependence altogether and use its value at T" for all T. In this case the only /c dependence of (6.92) comes from d(r),leaving us with Hence, while in the limit

A*(ò:

d*(frY(o, ?:).

The dipole energy density (6.90) is then finally obtained

fo:4n(yh)zl2(0, r.)

x {a.(t) - d(r')

-

(6.es) as

I# I*

- 3l1. . d.(Ð[q . d.€Ðl],

(6.e6)

where Q: (k k')llk - k'1. In the case that the order parameter is in a definite / state, such that d(k) is a superposition of spherical harmonics of order / (see the discussion below

6.i

177

Orientation of the Order Pørameter

(3.46b)), one of the angular averages in (6.96) may be performed explicitly. Furthermore, using the TL relation (3.48) to eliminate /(0, 7:), (6.96) reduces to (Takagi 1974, Leggett I975a) .fo

: ÂoN.(3 lie . dÊ)l' - l¿@)l) a.

rc.e7)

Here we have introduced a dimensionless dipole coupling parameter

3tr 1 y2h2la3 (6.e8) T(/ + Ð E, (N(o) lv,l)'' where V¡ is the pair-interaction constant for given / and ¿ is the average ^ hD:

atomic spacing, o : 73n2)t'tkrt , such that y'h'lat is the dipole energy for two atoms at distance a. In particular, for p-wave pairing (/: 1) the dipole coupling parameter has the approximately pressure-independent value

)ro-5 x L0-7

if values of N(0) l[l-0.2

(6.ee)

and 0.3 are assumed at low pressure and at

melting pressure respectively. As discussed by Leggett (L974a,I975a) and Takagí (1973), there may be 3He atoms renormalizations of (6.97) due to the quasiparticle nature of the involved in the dipole interaction. This will lead to a prefactor, which is expected to be close (or even identical) to L. Equation (6.97) for the dipole energy density of a p-wave condensate can be expressed in terms of the order-parameter matrix A*¡, (5.L), as .fo

: !so(r) > lAiA¡¡ + AlAit - ?,qie,il.

(6.100)

,l

The temperature-dependent dipole coupling constant go follows from the above as

so(r)

-

LoNeat(T),

(6.101)

where AQ) is the average energy gap, (3.57a). It corresponds to the average dipole-dipole interaction energy of the two 3He quasiparticles in the Cooper pairs at relative distance a (the average interparticle spacing), per unit volume. Indeed, using the definition of Âp in (6.98), (6.101) may be rewritten as

80(z)

-#l#l'",

(6.t02)

where the first factor is the average dipole-coupling energy of two quasiparticles at distance a, (AlEp)' measures the probability for these two quasiparticles to form a Cooper pair, and n is the overall particle density. In the normal state the dipole energy density "fo is zero since positive and negative contributions cancel on the average. As one enters the superfluid phase, S"(T) increases linearly with T"- T, and at low temperatures approaches a temperature-independent value of approximately hDN'.(kBf)z.

6

178

It is a remarkable

Broken Symmetries and Macroscopíc Order

of

property

anisotropic pair-correlated states that the dipole interaction energy of atoms correlated into Cooper pairs adds up. This effect is due to the coherent orientation of the preferred axes in orbital and spin spaces of all the pairs in the superfluid state, i.e. it is a direct consequence of the spontaneously broken symmetries. Indeed, the broken symmetries set the stage for the amplification of the dipole interaction to a macroscopic size. All that the dipole interaction has to do is to select the energetically most favourable relative orientation of the already long-range-ordered preferred directions. Let us now evaluate expression (6.100) for various model states. Note that in both (6.97) and (6.100) the last term in the angular average yields an orientation-independent contribution, àince (l¿G)f) a: A'(T), tr (,44+) - 1. Hence

/p

:

3ÂnNr

$( ' d€c)l') n+ const

(6.103a)

: !gD(r) 2 Ítit,, + AfrA¡)* const.

(6.103b)

: !ÂpNr 2 0. The coefficients a¿ and b¿ may be determined in analogy with the calculation of the change of the order parameter in a magnetic field (see the last paragraphs of Section 5.4). For small values of the field .8, the orientational energy density is found to be (6.r23) LfE: lbBspf(tr R)(ERE) + ÊRzEl.

An analogous effect occurs in the presence of superflow. There the gap component in the direction of % is suppressed. From the energy contribu-

6

184


tion lp"ul, one obtains an order-parameter distortion õA p¡ :

) [o,ô,, t

buû

",iî, ",)A ¡"i,

(6.t24)

which gives rise to an orientational energy density of the form (6.123), with bB and.E replaced by -å" and u. respectively. In the presence of a magnetic field, cross-terms will appear connecting u, and H, as well as Ê and H, i.e. orbital vectors and spin vectors (Fetter 1975a, L976b, Dahl 1977). The corresponding orientational energy is obtained by substituting the gap distortion from superflow, (6.124), into the magnetic energy (6.116a), or vice versa.

these orientational effects in the B phase may conveniently be collected into a single expression, provided that the fields are not too large, by scaling the fields H, E and u. appropriately (Brinkman and Cross 1978):

All

Lruo,"

: -,

"l(T)' .?(ffi )' . (l)'

-?(ffi)'-(T)',1

(6.rzs)

Here g'z" At- Al -'"-5 -2 Lxæ ar

Lx H7:?so, P"u1:8o, (vh)' 4u2Ûf;: Lx: ,lN - xs. R, ,

(6.126)

At intermediate pressures and near [, when e" tends to a finite value, the critical fields (see (6.99), (7.27), (7.28), (7.23Ib) are H.:2[]u"(l+ Fi)l''tío:!4G, u.: u.,p(m"l6m)12:"1'.6mm/s, E.:15 kv/cm. For an experimental determination of es, it is useful to relate e" to the transverse NMR shift Aø, discussed in (8.38), which yields

et:

L,oty --, 4 ,, Xsflé, 5

(6.127)

ør

is the Larmor frequency. Let us now discuss the orientational energy density (6.L25), but without the electric-field terms, whose effect has not yet clearly been observed. The remaining terms in (6.125) tend to orient rû: (i) along H for H>>Ë1" and u"11u.; (ii) along u" for u.))u" and H1H"; and (iii) for u"))u., H))H.,

where

along a direction characterized by the property that a rotation about fi through the Leggett angle carries u, to A,ãL In the case that HlH"))u,fu", but arbitrary otherwise, only the terms involving H are important in (6.I25). For parallel magnetic field and flow, the vector ñ is oriented parallel to both. For perpendicular configuration of If and u", sây II along â and u. along f, the orientation depends on the ratio u"lu", which determines whether the first or the second term in (6.125) dominates. Thus in the

6.i

Orientation of the Order Parameter

185

that u"1u", ñ is oriented along fI, but for u"]u", ñ moves out of the common plane of H and i into the direction fr: (n*, -(Z)'''n,n", ft,), where n, : f,l(L - n)lQ + Zn?)lt'' and n,: l'l'1 + t(u"lu,)'f'''. At intermediate values of u" the component of ñ perpendicular to H is given by n2r:3(1 - uTlu?). The four orienting forces acting on the order parameter in (6.125) generally compete against each other, i.e. each of them tries to force the vector Ê locally into a given direction. However, rapid spatial variations of the order parameter are energetically unfavourable. So the actual energy cost is described by the "bending energy" of the order parameter, which will be discussed in the next chapter. It turns out that the bending energies involved in spatial variations of the order parameter on the scale of a few millimetres are of similar magnitude to the orientational energies in (6.L25), i.e. of order 10-1s Klks per atom, corresponding to 10-e erg/cm3. Therefore the order-parameter configuration will be smooth and long-range case

rather than rapidly varying.

6.3.6 Surface energies and boundary conditions Finally, we want to discuss the influence of a wall on the order-parameter structure and on the macroscopically defrned preferred directions. The interaction of the atoms in the liquid with a surface (for example with a solid wall, a free surface or an interface between different phases) is a complex subject, whose details are not yet well understood. Nevertheless, the overall orientational effects on the order parameter in the bulk due to surfaces are well established. At a surface the many-body correlations in the system are perturbed on a truly microscopic scale. After all, the surface potential and the binding energy per particle in the liquid are usually of the same order; furthermore, the surface potential changes over distances of the order of a few interparticle distances. As a consequence, any theoretical description of the influence of a surface necessarily involves the full range of energy and momentum transfers at the surface, making such a theory very complicated. This gives rise to considerable changes, for example in the gap equation at and near the surface. One may attempt to calculate these changes within a weak-coupling model calculation (Ambegaokar et al. 1974, Buchholtz and Rainer L979, Buchholtz and Zwicknagl 1981, Buchholtz 1986). The important result of these calculations is that the longitudinal component of the order parameter di:E¡î¡dr¡ (where.î is the surface normal) falls off more rapidly than the transverse components as the surface is approached. This has been shown to occur for nonmagnetic surfaces in the case of specular scattering as well as for completely diffuse scattering. In fact, it may be expected to hold quite generally. The suppression of the order parameter only takes place within a layer of thickness f(f) adjacent to the surface, where f is the temperature-dependent coherence length (7.18c).

6

186


However, even outside this layer, where the magnitude of the order parameter has recovered to its bulk value, its orientatio,r¿ is still affected by the surface. This is so because the orientation for z > 6 will minimize the longitudinal component of the order parameter to give a smooth match of the respective structures at z 11 f and z )) Ç, where z is the distance from the wall. The A phase

The fact that the order parameter is anisotropic in this case makes it easy for it to avoid having a longitudinal component at the surface: it may simply orient itself in such a way that. the vector I is normal to the surface (Ambegaokar et al. L974). The additional suppression of the order parameter due to surface roughness does not seem to change this qualitative effect much. The energy per surface area gained by orienting the order parameter in such a way that the longitudinal component vanishes (rather than assuming an arbitrary orientation where the auerage order parameter is substantiatly suppressed), is of order f,f, where f":f -"f" it the condensation energy den^sity. On the other hand, the loss in bulk energy due to bending of the Î net¿ is of order f"(EolD'f, where L is the distance over which the bending occurs (size of the sample) and Á(6ol L)2 is the typical order of the bending energy density. Since E'"lEL I

[ô,¡

-

$.î¡l(z)]R p¡(z),

(6.128)

where z is the distance along the surface nonnal .î (assuming a locally plane surface), R*¡(z) is a position-dependent rotation matrix and /(z) is a test function varying along the surface normal from unity at the surface to zero in the bulk. The variation of Rr¡(z) is governed by the change in the dipole energy (note that in the absence of spin-orbit coupling the spin part of the

Further Reading

L87

order parameter would be unaffected by a nonmagnetic surface). One may show that the rotation angle 0 between spin and orbital spaces will stay at the Leggett angle 0¡ (Brinkman et al. I974b). Indeed, if 0 + 0'- then the maximum possible gain in dipole energy in the surface layer would b" go5, while on the other hand the energy cost, i.e. the energy needed for 0 to recover to the bulk rotation angle 9¡,- would be go$oi here åp: (NF^'z(T) l6g'.)'''Eo is the dipole heating length (see (7.27)). Since Eo>> Ë, 0 will stay at its bulk value. The rotation axis ñ, on the other hand, ¿s affected by the surface. In order to calculate the orienting effect of the surface on Ê, one has to substitute the model expression (6.128) into the general form (6.103) for the dipole energy density. This gives

a/o,,

: -Isofff + 4)[( s

.

û),

- Th(s

.É).],

6.Lzs)

f (z) is still a position-dependent quantity. However, since the coefficient of the last term varies between 0 and -T, the minimum of A,fp,s is attained by parallel orientation of ñ and.î, irrespective of the precise value of f (z). Hence the surface dipole energy tends to align ñ with the surface where

normal ^î. The surface energy per unit area derives from (6.L29) by integrating along the surface normal. Assuming the simpl e form f (z) : e-''E , the integral may be performed to yield the total dipole surface-energy contribution

^f Fo,s: -äs"E I d2r[(3 - ñ)' - *(.î . ñ)']. Jsurface

(6.130)

The distortion of the order parameter near the surface also affects the orientational energy in a magnetic field. By substituting dr¡(z), (6.L28), into the energy contribution due to the magnetic field, (6.113), one frnds the following result, which depends on the orientation of the order parameter with respect to the surface normal .î and the magnetic field ff (Brinkman et al. 1974b):

F¡¡,s: -€a,s

I

a'r6mt¡',

(6.131)

where €H.s: E LXH' and L)(: XN- Xs is the spin-susceptibility anisotropy. While the form of the surface-energy expression (6.t29) and (6.131) follows directly from the symmetry of the phases (Fomin and Vuorio 1975), the exact values of the prefactors will depend on the details of the surface structure. In the case of nonplanar surfaces there are additional energy contributions depending on the spatial variation of ñ. The orienta-

tional energies (6.129) and (6.131) define a characteristic field Flra G"ln7¡'/2-25G; see (7.L75). For H11H", the surface effect (6.129) dominates and û is oriented perpendicular to the surface (,ñ ll .î). In the opposite case, H)>H", û will be oriented in such a way to make (6,131)

minimal. An explicit expression for the total magnetic field surface energy in terms of the vector ñ is obtained by substituting the form of the rotation matrix ß

6

188

given by (6.106) with

Fa,s: -(å)'.r,,

0:

Broken Symmetrics and Macroscopic Order

0r into (6.131), yielding

I a'r¡1s . ñ)(H.ñ) + (?)'''ñ. (3 x n¡ -å(s .ã)l'.

(6.t32)

For later applications, we determine the minimum of (6.132) for general

field orientations, perpendicular and parallel to the surface. For II along the axis of the surface normal, the dependence of Fr¡,s on ñ is the same as that of the surface dipole energy (6.130), i.e. the surface tends to orient ñ perpendicular. On the other hand, for H in the plane of the surface, û is oriented such that a spin-orbit rotation by the Leggett angle 0¡ about ñ takes.î into É. fnis is achieved when ¿

: a(å)t''7L, 3'/', r.L¡,

(6.133)

where we have chosen S ll 2, and l* ll.û. For II making an arbitrary angle p with the surface normal, minima of (6.132) are attained for the following directions of ñ (Ahonen et al. I976c, Lin-Liu and Maki 1978):

ût: Ií-rrz73trz latätþ, (1 + 4 cos ç)1/2 tanlq, (1 + 4 cos rp)1/2), (6.134a) ( lS-'''r-3t/z cotLrl,, -(t - 4 cos ,lù''' cotlp, (L - 4cos rp)1/2) tor cos-1 (å) < !, [tñj(vñ) - ñj(vrh)l j

(7.e) l

Hence, under a Galilean transformation, when the phase @ transforms according to (7.4), the quantity on the right-hand side of (7.9) transforms like -(Zmlh)u, i.e. like a velocity. This allows for the definition of a superfluid velocity Us,i:

h

(7.10)

Z**'Y¡fl.

It

is sometimes convenient to express the orientation of the orbital triad (ñ, û, i) of the A-phase order parameter relative to some fixed coordinate system (î,t,2) in terms of Euler angles e, þ, T:

¿

:

(cos øcos B cos y

- sin a sin y, sin

¿

:

(-cos øcos

B sin y

úy

cos B cos y

* cos ø sin y, -sin B cos y), (7.lla)

- sin u cos y, -sin ø cos B sin y *

i

cos ø cos y, sin B sin

(7.11b)

(7.IIc)

- (cos ø sin B, sin ø sin B, cos B).

While

y),

u

and þ are the usual azimuthal and polar angles respectively of spherical coordinates, the angle y describes rotations øbout Î (see Fig. 7.1). Hence the orbital part of the A-phase order parameter may be written as

f,-2-rrzQît+iñ)

:

e-ivÂly:o.

(7.IZa) (7.12b)

In using the Euler angles, one has to beware of unphysical singularities in ø when sinB:0 (Mermin 1978a). The superfluid velocity (7.10) then takes the form rl.

- - ftfrr*

cos þvu).

(7.t3)

This shows that in the A-phase u, is not simply given by the gradient of the overall phase of the order parameter as in superfluid oHe or 'He-B, but is more complicated owing to the orientation of the order parameter. Clearly, the rotation angle y plays the role of a phase of the order

7.1

r93

Superfluidity

Figure

7.L

Euler angles u, þ, T characterizing the orbital triad

(ñ, ñ, î¡ in 'He-A.

parameter. Hence it may be identified as y - -ó, so that, lor þ : LîÍ, (7.I3) leads to the usual definition (7.5) of the superfluid velocity for isotropic superfluids. The fact that a quantity % may be constructed from the order parameter at all is a strong indication, but no proof, of the existence of dissipationless flow of the condensate, i.e. superflow. The relevant quantity is in fact the current density associated with u,. It may be determined as follows. We perform a Galilean transformation of the condensate free-energy density f by an infinitesimal velocity z, which yields

f-f

* 9". u.

Q.\aa)

The coefficient of ø in (7.l,aa) is the superfluid current density 9.. The effect of the infinitesimal Galilean transformation is to multiply the order parameter by L -i(Zmlh)u'r.This amounts to the replacement of.Yd*¡by lY-i(2mlh)uld*t. Hence 8. may be obtained from an expansion of the free-energy density in terms of gradients of the order parameter, the so-called "gradient free-energy density" fs,â, discussed in Section 7.2. 'to this end, we assume a fictitious gauge field u with charge m on the He atom. To obtain a gauge-invariant free energy, the gradient operator has to be replaced by the gauge-invariant operator Y -i(Zmlh)u. Note that for a system of charged fermions the role of the velocity field z is played by the vector potential A (Ambegaokar et aI. L974). The mass supercurrent is then given by the term in the gradient free-energy density fe,^¿ that is linear in u: o:: ös

ö.f*,.¿

ôz

(7.14b)

7

194

Supefiow and Textures

In order for Galilean invariance to be guaranteed, the total mass current g, which is given by

I :8, * 8r,,

(7.15a)

with 9',: pnu' obtained in (3.86), should change by the amount pz under transformation to a frame of reference moving with velocity ¿. This implies that g, is associated with a fraction of the total density p, called the "superfluid" density p., such that the normal-fluid density p" (given by (3.87b), (3.91)) and p" add up to p:

Pr: P1 -

(7.1sb)

P,,'

In complete analogy to the above, one may introduce a spin superfluid velocity usp as the gradient of the relative phase of spin-up and spin-down Cooper pairs, multiplied by hl2m. By introducing a gauge field for each spin component, one may show that the spin supercurrent is given by

jì1,:_þ, ô(vro;)' where

7.2

ô9| is a rotation angle in spin space about the axis

(7.16) ¡r.

GRADIENT FREE ENERGY

The free energy of a weakly nonuniform state is well represented by the lowest-order term in an expansion of the free energy in terms of the spatial gradients of the order parameter. This is referred to as the "gradient freeenergy density" fe,^¿. Since odd powers of the gradients are not compatible with the rotational and gauge invariance of the free energy, the expansion starts at second order. In the Ginzburg-Landau regime, close to the transition temperature, this expansion is particularly simple: only terms of second order in the order parameter need be kept. The general form of the gradient free-energy density is then obtained by combining the vector indices of the gradient operator V¡ in all possible ways with the orbital tensor indices of the order parameter d*¡: re,^¿:

å) [K'(v¡dpù(vidiù + Kz(vidpù(vik) + K3(vrdpi)(v¿iù\. jltt

(7.L7)

The simplifying feature of the Ginzburg-Landau (GL) expansion is that the coefficients are normal-stafe quantities, i.e. they do not depend on the order parameter, and hence are isotropic. In the general case, i.e. outside the GL regime, these coefficients will be anisotropic in orbital and spin spaces. The coefficients K¿ have been calculated in weak-coupling theory, with the result (Wölfle 1974)

Kt: Kz: Kz: å¡f.53,

(7.18a)

7.2

Gradient Free Energy

where

195

ç- 17 EQ)f'" so: I agnz J k"T (:120^4, at P*.rt) hu,

(7.18b)

is the zero-temperature coherence length characterizing the spatial extension of a Cooper pair. (Hence in superfluid'H" 5o is seen to be significantly shorter than in conventional clean superconductors, where 6o-103-100Ä.) For later use we also introduce the temperature-dependent coherence length (7.18c) EQ):60(1 _TlT¡-rrz Actually, K2 and K3 are not exactly equal but differ by terms of order Q.ITÐ' (Blount et al. 1974). The coefficients Kr need not be calculated directly. In fact, they follow from the expression for the superfluid current

to be discussed below and the result for the normal-fluid density obtained in Section 3.4, by requiring that the sum of the superfluid and normal current contributions must obey Galilean invariance. Note that the difference of the second and third terms in (7.17) combines to a pure divergence and thus integrates to a surface term in the total free energy. None the less it may not be discarded when the supercurrent (a local quantity) is calculated from (7.I7).In the weak-coupling regime the Ginzburg-Landau free energy may be derived to all orders in the gradients of. d*¡ using the kinetic equation for quasiparticles presented in Chapter L0 (Combescot and Saslow 1977).

7.2.1 The A phase In the A phase the spin and orbital parts of the order parameter factorize, i.e. dpj:d*A¡, with A:AoA given by (7.L2a); note that this is a particularly simple form of (6.15). The gradient free-energy density in (7.L7) may easily be expressed in terms of Á and d. In weak coupling (Kr: Kz: K3) one obtains (Maki and Kumar I977a,b)

fî,ô: lrcr$ V . al'+ lv x al' + 2lØ. Vdl' + lill' [v x ãl'+ F . ù'l +zV[(A. V)^* -

^(V.^*)]],

(7.re)

where the last term is a pure divergence. Let us now concentrate on the so-called London limit, i.e. the case where the order parameter attains its equilibrium structure everywhere in the bulk and where weak perturbations will not change its structure but will merely influence the preferred directions in orbital and spin spaces as well as the phase variable. In this case the general gradient free-energy density (7.L7) may be rewritten in terms of gradients of the symmetry variables only, and is then referred to as the "bending energy" density^ ft"na.The corresponding variables are (i) the gradients of the orbital axis t, (ii) the gradients of the angle describing rotations about i, which may conveniently be replaced by

7

196

Superflow and Textures

the superfluid velocity u" as defined þV (7.10), and (iii) the gradients of the preferred direction in spin space d. The bending free-energy density is quite generally^given !y ttt" sum of terms containing all possible combinationsõf ,,, V.Î, VxiandYdrintheform(Blount etal.t974, Cross L975)

f*'¿:

äp"r(î xu")2 +äpo,(î'u,)'

* ![c,-%. (v x D ¿m

G,-

c,,xr" .

î¡î'g

x î¡]

.:(*)'{",,o -D'+ K,Íî .(v x i)1'z + ¡(bli x 1v x i)1'? (7-20) + p"p,- ) tt xvâ)z* p"o,, )ltt 'vâ,f\. Here a pure Oin"rg"i"e term Vt(i. Ðt - Î(V . i)l has been omitted. The subscripts s, t and b on the coefficients K", K, and K6 in (7.20) indicate that the respective energy contributions are due to a splay-, fwist- or åend-like spatial change of. I, as illustrated in Fig. 7.2. This terminology is well established in liquid-crystal physics (de Gennes 1974). It should be noted that the apqearance of I in the last two terms of (7.20), describing the bending of Q, does not impty any physical coupling (e.g. dipole coupling). The vector I is only used as a reference direction to which the notation p,or, p"p¡¡ refers. In thé special case of dipole^locking @ll[, Q.20) becomes much simptel since the terms involving d may then be incorporated into those due to I by the replacement

Kr+ K. *

Prpr

, Kr+

Kt

* P"pr, Ku-

Ku

* Prptt'

Q 'ZIa)

Hence the bending energy of Î is seen to increase. This reflects an increased stiffness of the Í texture due to the spin part of the order parameter! It should be noted that the first two terms in (7.20) may also be written as

Lp"r(î x

a")2

+ äpr,(î 'u")' : tp,'u?

- ä(p"'- P",,)(Î' %)'.

(7.zIb)

There also exists a slightly different notation in the literature (e.g. Mermin and Ho L976, Bhattacharrya et al. t977, Hall and Hook L986), which is related to that used above by

P"= P"t, Po-

Pt:_

- P"rr'l

C=Ct, Co-CL-Cu.

(7.2Lc)

)

In the Ginzburg-Landau (GL) regime the coeffrcients in (7.20) may

be

obtained by comparing with (7.L7).In the weak-coupling limit one finds

p"t:Zp"r: r(T)'

*,az:?(T)'*^,tt,

Ct: -C¡:äprr, K": K, : äKt: ).P"s,

(7.22a)

(7.22b) (7.22c)

7.2

r97

Gradient Free EnergY

'-- I

\l?

(a) Splay,

(V'iF

\ \ ¿ (b) Bend, (i x (v xÐ)'

(c) Twist, (Î.(v Figure

xò)'

7.2 Three different kinds of deformation of a vector freld i and their corresponding contributions to the free energy (see (7.20))-

where Ás is the maximum value of the gap. In this limit /6.,,¿ takes the form

(^ ^ : r.,,{r3 - +(î . u,)' + .fu",,¿

.ie)'[(o

k

,

zm[Lr,-(v

4.

x i)

- (u" - Dî -(v x i¡]

. D'+ti. (v x i)]'+ 3t(i'v)il'

* o4[Q x\â,l',+2]

rti

.vâ-rl].

(7.22d)

Outside the GL regime the coefficients in (7.20) have been calculated by Cross (1975), Fetter (1979) and Fetter et al. (19S3) in weak-coupling theory, including Fermi-liquid corrections. The coeffrcients P"r,¡¡ may in fact be

7

198


deduced from the result for the normal-fluid mass tensor in Section 3.4, as explained below. The remaining coefficients may be calculated along similar lines by considering the superfluid mass current and spin current as well as the orbital torque ô.fo."o/ôi discussed in Section 9.3. Keeping Landau parameters with / < 1. only, the result of this calculation is given by

c

:

p": (4)-tp3, Lp?u(D.)-'(l

P"p:

-

Zî

(7.23a)

ø î¡,

It 1+åFî 2^-r;ir.:r(D')-tp3, Lm

*":ifi

o?',

3P3,, \

'1..

m K,: 12m*

(n3, + 2n3,,

t + t¡z p?,,t p )'

Kt:I#(rog,, + Epy-

*fffi¡,

(7.23b) (7.23c)

(7.23d) (7.23e)

(7.23f)

where

D',u:1

+ l4"roú ,

^/Yo(Ê; D@. v: r\--TVlr

(7.23s)

i)o\

p3: p1 -

p3.

I

^,

(7.23h) (7.231)

(Note that (7.23c) is only valid in the case that th9 spil polanzation-is perpendicular to ã; lor the general case see (L0.60).) Here I Ø I is the tensor product, pfl is the normal-fluid mass density tensor in the absence of the Feimi-liquid interaction given by (3.87b), and Yo(fr; T) has been defined in (3.88). Strong-coupling corrections to the above weak-coupling results have been calculated by Serene and Rainer (1977,1983), Fetter (1979) and Fetter et al. (1983), and the effect of a magnetic ñeld has been considered by Brand and Dörfle (1981). The coefficient y in (7.23h) is found to diverge as ln 7 in the limit T+ 0. This signals the breakdown of the gradient expansion at low temperatures. The physical reason for this divergence may be traced back to the existence of nodes in the energy gap of the 3He-A order parameter. (Indeed the denominator in the angular brackets in (7.23h) has precisely the form of the square of the A-phase gap parameter and is thus the origin for the abovementioned ln T divergence.) As a consequence of the nodes, pair breaking, i.e. excitation of quasiparticles, is possible at arbitrarily low tenperatures and, quite generally, forbids a well-defined analytic lowtemperature expansion of any quantity in terms of. T 17.. We shall encounter this problem again in the discussion of the nonlinear contributions to the superfluid current.

7.2

Gradient Free Energy

199

7.2.2 The B phase In spite of the apparent simplicity of the order parameter of 3He-8, the corresponding bending free-energy density is considerably more complicated than in the A phase. This is of course due to the fact that in this case At i:3-Lt2eiþRu.(fr, e) is proportional to a general rotation matrix charac-

terued by an axis of rotation ñ and a rotation angle 0 (see (6.106)), where now both ñ and 0 vary with position. The full expression for Aut has to be inserted into (7.L7) for the bending energy density. Using the relations (7.23j) fi.Y¡ñ,:0, (7.23k) ñ.(ñ.v)fi:0,

Y¡R¡,t: (Vrg)[sin 0 (ñ*ñ¡

-

ôrr)

- cos 0)

earñ*l

+ (1 - cos 0)[(V¡ñ)ñ, + ñp(V iñr)]

- sin u 4

r*o(Y ¡ño), Q.231)

and expressing LK,A' in the weak-coupling Ginzburg-Landau regime in terms of the isotropic superfluid density without Fermi-liquid corrections, p!, (t.zst),

(Ll'og. LK,a':! t,st - to\2m) where (see (3.92) with Fì:0 and (3.89)) ^0 P":r-Yo

(7.23m)

(7.23n)

p

7eQ) / \2 :#kr/ ^

(r4+--+..^\.--=..*__-

\ l'

\-

-\\ù ---.t

-'+

Figure7.7 Annihilation of a defect at P, with N:1and one at P, with N:-1 (vortex-antivortex pair). The resulting texture may be embedded into uniform background.

a

(d:3, ft:2, i.e. D : I) allows line defects, which are the well-known vortex lines. In the case of Ising spins (n : I) defects in d -- L,2,3 are oIJe

pointlike, line-shaped and surface-like respectively (Fig. 7.8). In contrast, Heisenberg spins in a plane (d:2, n:3, i.e. D> 6o )) r, we already know that there are only two topological classes N:0, L. This implies that the vortex core structure wilt be continuous or discontinuous depending on whether Nr * Nz is even or odd. In the case where Ni + N, is an even number the vortex thus has a "soft core" of extension $r. Examples of possible vortex structures

--î,

-

Figure

--_,)¡.

ã

*î

--

7.17 Vortex in a weak magnetic field.[l with d ll I L H and winding

number

Nt:2 in the ã, î n ta. The change in orientation of the orbital triad, which determines the winding number N, of the phase, is not shown.

7.5

Linear Defects

233

AT texture and the MH texture. In

general, the vorticity is distributed over the cross-sections of radius 6r. In the opposite case, where N, + N, is an odd number, the outer and inner vortex core regions 6o< r<En and Ç 111Çp belong to the class N:1, and there is a "hard core", i.e. a normal-fluid region for r30 G.In the outermost shell of the vortex core, lor Ç¡¡ ( r ( fo, the dipole orientation energy is unimportant, but á is confrned to lie in the plane perpendicular to the magnetic field. Therefore the degeneracy space is

Rl:

(S' x SOr\lZr.

(7.75c)

According to (7.57b) and (7.59b), the first homotopy group of this space consists of the groups Z and Za, i.e. øt(Rl) :ZxZo,^describing the defect classes of disclinations of winding number^N i" the d field and the orbital triâd. In the inner core region, r 1Çr, the d field in a disclination may avoid the singularity by escaping into the third dimension. The loss in magnetic orientation energy is then overcompensated by the gain in bending energy. It may be shown that^the energetically most favourable core structure in the class of nonsingular / textures is given by the AT texture, rather than the MH texture (Volovik and Hakonen L981, Zotos and Maki 1985, Fetter 1985a).

The vortex containíng one quantum of vorticity again has three core regions. In the outermost region, r ) Eo, Î and ã are dipole-locked and confined to the plane perpendicular to the magnetic field. Inside the outer core, i.e. for Eul-,-*rî" * b, lci*c:,,, -*l' + bz(c*^cl^)z +

(b 3c p^cl*c > ", v,n

_

^cI,

_,

+ b 4Cl^C + b sC p^C - *,Cî*C:,,)], "*ClnC p, is,â:

n- p-m

.Z_rl@tlcu^-m

' . Here

(due

(7.es)

(ttt

*rrrr-

kn

'.^)

- u, .;-)]

( à llue,r.-l'.

(7.e6)

is the integral over the relevant orientational energy densities dipole interaction, magnetic field etc.) in the (E, r) plane.

Øonent

to

Furthermore,

Ur:-:**, where þ, are the B parameters of the free-energy density (5.4), and

(7.e7)

r

and Bs

in (5.11) and (3.56) respectively. The order-parameter components C**in (7.95) and, (7.96), defined in (7.86), are dimensionless. The prefactor p, in (7.94) is a superfluid density in the Ginzburg-Landau are defined

regime:

7EQ)m lao\' P': 4¿ *. \k"r") P'

(7.e8)

The minimization ol Ø with respect to the components Cu^ generally leads to a system of nine coupled Euler-Lagrange equations, each a

7

240

Supeflow and Textures

nonlinear second-order ordinary differential equation. These equations may only be solved numericalty. An alternative approach, requiring less computational effort, is to employ a variational method. For this, one chooses a

set of test functions for the C*^(í), which depend on a number of variational parameters Â¡, substitutes these into the functional and calculates the minimum of the resulting free erlerg] g"*({}'}) with respect to variations of Â,. Both methods have been employed to determine the Cu* for those vortex structures considered in the symmetry classification scheme. In the case of nonaxisymmetric vortices the energy is calculated from the general expressions for the free-energy densities in the A and B phases, i.e. the bulk free-energy density (5.4) and the gradient free-energy density (7.17).

7.5.8 Vortices in the A phase Axisymmetric vortices According to the above classification of axisymmetric vortices, there exist four different types, which may be distinguished by the remaining discrete symmetries.

This is the most symmetric vortex, since it pos^sesses all the discrete symmetries P1, P2 and P3. The symmetry P1 requires I and d to be uniform, oriented along the vortex axis far from the core region. In this case one has a pure phase vortex. If we consider a vortex whose orbital angular momentum in the asymptotic region (where only Co* and Co- are nonvanishing) is characteruedby the quantum numbeÍ n, (7.83a), (7.84a), then a parity operation implies (7.ee) PCo* : einnCo+.

(i) o-uorfet.

Therefore we shall require Cu* to be invariant under the operation P1 : P(U*)". This would leave us with four real components C¡,^ for this type of vortex. However, in the region r 1Eo, where the dipole coupling may be neglected, the symmetries in spin and orbital spaces in (7.8L) are also uncoupled, so that the most symmetric vortex has two separate continuous symmetries with the generators: Lã- + L' - nþ and S'. The latter implies ã ll t, which leaves us with only two components: Cea and C6-. The general expression for the order parameter of this vortex structure is given by (Salomaa and Volovik 1985b)

Api:2-rt22tlco*Q)(i¡

*

ii)ei("-r)e + cs-(r)(i¡

-

ifr)ei{"+tre1. (7.100)

The two possible boundary conditions at infinity are

(i) Co*(-):l, C6-(co)-g for the vortex wilh n':n-L (ii)

quanta of

circulation and with asymptotic orientation i1*¡ : i; and the vortex with n' = n * f quanta and i(-¡ - -2 and where Cs., and Cs- are interchanged.

7.5

Linear Defecæ

241,

:

n:

These vortices have a normal core except for the cases n' -2, -L and n' :2, n l. Vortices of the above type belong to the classes with topological quantum 1. They are not allowed within the class of numbers I vortices, because these vortices necessarily involve a nonuniform â field, which is incompatible with the P1 symmetry. However, even within the 1 given (7.100) by are energetically unstable. classes, pure phase vortices as is since in this case a continuous deformation In the case N:0 this evident, lowers structure the vortex energy considerably. nonsingular into a

:

N:0,

N:

N:0,

P1 symmetry is broken. A classification accordP2) symmetry has been performed by Salomaa and

(ü) u-uortex. Here the

ing to the residual (i.e. Volovik (1983b) and Seppälá et al. (1984). An example of a nonsingular v-vortex of the AT type (see (7.71)) is given by

i

- sin q(r) 2 t

u,- -

q(r) î,

(7.101a)

,^rl, *sin q(r)lq,

(7.101b)

cos

h

where nQ): -|n and ry(*) : Ln. The vortex carries two quanta of circulation. The fact that in the abo'¿e state the Pt symmetry is spontaneously broken leads to the existence of two degenerate states (indicated by the A signs), which transform into each other by a P1 operation.

A

nonsingular vortex of this type, where P3 remaining symmetry, is given by the twisted Í texture

(äi) w-uortex.

Î = sin q(r) 2 * cos q(r)

us-

is

the

Q,

(7.I02a)

L - ,*rfl*sin q(r)le,

(7.102b)

with ry(0) : -Ln, ry(*) : Ln.

(iu) uuw-uortex. An example of a nonsingular texture of this type where all discrete symmetries are broken is

i us

-

sin q(r) 2

i

- - fiV*

cos 4(r)[cos

sin

ne)le,

u(r) ç

t

sin

u(r)

î1,

(7

.L03a)

(7.103b)

with the same boundary conditions on 4(r) as in cases (ii) and (iii). The vortex state is fourfold-degenerate; the states transform into each other under the action of P1, P2 and Pr, which form the group Zrx Zr. Non axisym m etri c vo rti ces

In spite of their high symmetry, the axisymmetric vortices in 3He-A do not have the lowest energy and therefore are not stable. Hence the symmetry

7

242


classification approach is not very helpful here. To calculate the energy of vortices in the A phase in the London limit (see above (7.19)), one has to consider the free-energy density composed of the bending free-energy density (7.20), and the energy densities (6.104) and (6.115), due to the dipole interaction and an external magnetic field respectively. As first noticed by Volovik and Mineev (L977a), the pure phase vortex with â : î : ?, for example, is unstable against a nonaxisymmetric deformation into a pure phase vortex with I : d:f owing to the anisotropy of the superfluid density tensor. In fact, the energies per unit length, Øou(n, d, l), of the pure phase uortices (sufñx "pu") with n quanta of circulation (see e.g. (7.65) with y : -nE) in the dipole-locked region Ëo 11 1R (here R is the distance between vortices or the size of the container-whichever is smaller) are found from (7.20) as

ã: î : 2): fl'p",tko, Q.I0aa) (7.104b) Øou(n, ã:î:.û): Ln'(p",r* p,,')ko, where ko:l(hl2m)z\n(R/5o). Since the components of the superfluid Ør,(n,

density tensor parallel and perpendicular to I are related by p,,rr ( p,,r the energy of the nonaxisymmetric vortex is seen to be lower than the energy of the o-vortex (Volovik and Hakonen 1981). The reason is that superflow tends to align the vector Î, and hence an Î configuration perpendicular to the axis (in the plane of the flow) is preferable. In the soft-core region riSo-ttre ã-vector field is decoupled from the Î texture and may assume a uniform configuration even for highly nonuniform i textures. On the other hand, the gradient energy of the orbital part of the order parameter may be lowered by allowing the vorticity to be carried over from the phase field E to the l'texture in the form of a disgyration (see e.g. (7.73) with y - -nq:^0). The energies per unit length, Ø¿¡(n:0, d,l), of the disgyrations with l: Ê and I: Q, i.e. for singly quantized vortices, are given by

ã: const, î : ù: K"kz, Øoo(n :0, ã: const, î : Q): Ktkz, Øoo(n:0,

(7.10sa)

(7.10sb)

where K, and K6 are the stiffness parameters in the expression for the bending free energy (7.20) and k2-- )(hl2m)ttn(8"16). I. the Ginzburg-Landau regime K. and Kb are given by (7.22c); since K": åK¡, the radial disgyration costs only one-third of the energy of the nonaxisymmetric phase vortex. The broken axisymmetry of the singly quantized vortex is hence restored inside the soft core. As shown by Muzikar (1978,1979), the radial disgyration changes continuously into a polar state in the core regime r - E, before the liquid goes normal inside the hard core. The (singular) vortices discussed above belong to the topological class with quantum number N: L. Let us now turn to the class with N:0, containing vortices without a normal core. At first one might think that

7.5

243

Linear Defecß

in energy, since they do not have a normal core. However, the texture required to avoid the normal core is rather complicated and thus implies an increase in bending energy. Even in the most favourable case (the nonaxisymmetric AT texture; l: î) this leads to an energy per unit circulation quantum that is twice as high as the bending energy of the nonaxisymmetric phase vortex outside the soft core. Were it not for the energy of the soft-core region, the singular vortex would always be lower in energy. The energies per unit of circulation outside the soft-ðore region (r ) max (6o, 6r)) of (i) the MH texture with Î: ¿ (ii) the MH texture with î: Q, and (iii) the nonaxisymmetric doubly quantized AT texture, which are all nonsingular textures, have a ratio of 4.5:3.5:3. (Salomaa and Volovik 1987). Approximate numerical calculations show that the singular nonaxisymmetric vortex has a slightly lower energy than the AT texture for vortex separations /g2 ) tsz., i.e. for low rotation speeds f2 < O. (see Section 7.6 on the rotating system) (Seppälä and Volovik L983, Vulovic et al. 1984). The simplest representation of doubly quantized nonsingular vortices in large magnetic fields (5r < fo), thought to be stable for f2 ) Q", is the these vortex structures should be lower

w-vortex characterized by

i:sin q(r)î tcos 4(r) [sin Ei - cos gâ], ñ + iñ: s-ir[sin 4(r) (-sin q i + cos q 2) t cos q(r) I + i(sin q 2 + cos gi)1,

ã:i,

,,:**r[L with 4(råo):|r and Î1*¡:i. A v-vortex vortices give

structure is also possible. A calculation of the energy of these preference to the w-vortex (7.106) (Seppälä et al. L984, Zotos and Maki 1984, Maki andT-otos 1985a).

Consequences of the broken discrete symmetries The spontaneous breakdown of discrete symmetries in the soft core of the 3He-A vortices is associated with the appearance of (i) a spontaneous electric polaruation D. along the vortex axis (in the case of broken P1 and P3 symmetries), (ii) an axial supercurrent (broken P1 and P2 symmetries) and (iii) an axial spin supercurrent (broken P1 and P, symmetries) (Volovik L9S4a). Thus an axial supercurrent is flowing in the w-vortex (7.106). On the other hand, a v-vortex may carry electric polarization via a polaruation of the 3He atomic shelts. In particular, at the boundary where one of the two doubly degenerate solutions (see (7.101)) having opposite orientations of the electric dipole moment switches over into the other, there resides a charge e*, which has been estimated as e* -L}-se. The electric polarization as well as the axial spin current may lead to observable effects if

7

244


the interaction between the vortices leads to an ordering of the usually randomly distributed orientations. The energy of an array of w-vortices (7.106) may be lowered by forming a helical arrangement of the preferred direction in the (x, y) plane along the axis of the vortex (Salomaa and Volovik 1987), i.e. the I texture outside the vortex core is described bv

:

(7.107) (qz)Í + cos (qz) 9. Again the interaction between the vortices forms a "ferromagnetic" arrangement, i.e. the vortices are all of the same type (+ or - sign in (7.106)). The existence of this structure would also be reflected in the dispersion relation for "Kelvin waves", i.e. the wave-like excitation of a io,,,n

sin

vortex along its axis (Sonin and Fomin 1985).

Half-quantum vortices As discussed in Section 7.5.3, vortices in the topological class N: å carry half a quantum of circulation. Their existence is made possible because the ã-vector field compensates for the missing half quantum. A different, but somewhat simpler, interpretation of the half-quantum vortex can be given in the weak-coupling limit, where the subsystems of up and down spins are uncoupled: it corresponds to a vortex with a single unit of vorticity for one of the spin components and zero vorticity fo^r the other component (Leggett 1989). The corresponding textures require d and I to be nonparallel over a region of macroscopic size; as a consequence of the dipole interaction, this region is confined to a domain wall bounded by the half-quantum vortex. The half-quantum vortices are very costly in dipole energy. The domain wall may be avoided altogether in a restricted geometrY, for example a narrow stab with spacing d < Eo between the parallel plates (Salomaa and Volovik 1985c, Volovik and Salomaa 1985b; see also Maki 1986b). The interaction with the walls orients i parallel to the surface normal, whereas application of a magnetic field of strength H8 - 30 G parallel to the surface normal forces the d texture into the (*, y) plane. In this way, the dipole energy is fixed at its maximum value. With these boundary conditions, a half-quantum vortex (7.74) is possible. Estimates of the energies show that the half-quantum vortex can compete with the singlequantum phase vortex, but the energies are very close. Another confrguration in which half-quantum vortices may be stabilized is that of pairs of vortices with opposite values of the period of the d field (i.e. with &' :0 and k' : L in (7.7\). Since these are connected by a domain wall, their energy is proportional to their linear separation R, just as in high-energy physics, where quarks are bound by gluons. On the other hand, the bending energy prevents the vortices from coming too close together; therefore there exists an equilibrium distance R6, which has been calculated by Salomaa and Volovik (1985c).

7.5

Linear Deþcß

245

Yet another consequence of the existence of half-quantum vortices is an increased attentuation of certain order-parameter collective modes, to be discussed in detail in Chapter 11. One of these modes, the so-called "clapping" mode, represents an oscillation of the ñ and ñ vectors. In the presence of a vortex of circulation n' the amplitude of the clapping mode obeys the equation (Khazan L985)

# :,,(io - "Í r - @3+ u),t,,

(7.108)

where øs and c are the energy gap and the velocity of this mode and U and A are external potentials generated by the vortex. The "vector potential" A is given by

n:Lr q.

(7.10e)

Hence the motion of the collective boson is similar to that of a charged particle in a magnetic field H :Y x A:2nn'2 ð(t)(r), confined to the vortex core where 6{zl(r) is the two-dimensional delta function. The familiar Aharonov-Bohm interference effects (Aharonov and Bohm 1959) lead to an additional scattering of the collective bosons in the case of half-integer quantum number n', which should be observable in the width of the corresponding collective mode peak of the ultrasound absorption (see Section tL.4).

7.5.9 Vortices in the B phase Axisymmetric vortices

The most symmetric vortex consistent with the asymptotic boundary condition ø*-(*) : øoo(-) : a-*(*) - A(T)ei"e necessary to generate the bulk B phase far away from the core is the o-vortex. It is characterized by five real components C*- , Coo, C-*, C** and C--.

The vortex with n: I circulation quantum has a normal core because, as discussed below (7.86), the phase factors multiplying all the five components Cu^ have singularities at the vortex centre. The radial dependence of this vortex has been calculated in full generality by Ohmi et al. (1983) and by Salomaa and Volovik (1983b) and for the simpler version with C++: C---0 by Passvogel et al. (1983). It is found that C*-, C6s and C-* decrease monotonically as one moves towards the centre. Until r>2.58 they stay at their bulk value (r-+æ), while for r L) discussed above. However, in the present context the symmetry breaking relates to a single vortex, in contrast with the former situation, where it was associated with the collective behaviour of the vortex array. The transition from the uniform ñ texture to a nonuniform one at H: H" may be detected through an effective weakening of the orientational force

7

256


constant Â. At H >> H" the area around the vortex cores where Â(r) is nonzero is only a small fraction of the total area. This implies that the vortex cores are essentially decoupled from the bulk of the sample as far as the orientation of fi is concerned. N on

axi sy m m etri c vo rtices

The breaking of axisymmetry in the core of the v-vortex discussed in the previous subsection introduces additional orientational effects owing to the èxistence of a further preferred direction 6 in the plane orthogonal to the vortex axis. The magnetic-anisotropy energy of a v-vortex (axis along 2) with completely broken axisymmetry (i.e. with a vector symmetry along Í, say) is determined by the spin-susceptibility tensor

ri...) )(t

": (R-tyB)n"- (r::.":';

(7.133)

For a v-vortex with C- symmetry (see Section 6.2), i.e. a "director symmetry", one has 7,,:0, since in this case a contribution involving both preferred directions, 2 andå would be odd under the parity transformation. The magnetic-anisotropy energy for the vector case is given by (Theodorakis and Fetter 1.983, Salomaa and Volovik 1.983a)

filn"*:?Lfr,,, {hll(ililA¡z + t rr(Ênã>20 bar) and the double-core vortex discussed above (7.112) at low pressures. These results apply to the

Ginzburg-Landau regime, i.e. not too far from the phase boundary, and to the low-density vortex system. The vortex lattice is presumably of hexagonal symmetry in both cases. It is conceivable that the phase boundary between the above two vortex states bends upward and meets the melting curve, but this has not been calculated yet. In this case, the two vortex states might be identified with the two regions observed in NMR experiments (see Fig. 8.9). A comparison of these data with theory is made in Section 8.5. Given the large number of possible vortex types, there may exist further states in

7.6

Rotating Superfluid.

3He

261

regions of the phase diagram that have not yet been experimentally investigated. In particular, it is likely that a different vortex state is stabilized at high vortex density, or equivalently high rotation speed. The A phase

Here the texture depends sensitively on the strength of an applied magnetic field. For low magnetic fields, continuous nonsingular vortex textures appear to be more favourable in energy than singular vortices. The boundary condilion imposed by the periodicity of the Î RetO in the rotating frame requires I to be along the axis of rotation for axisymmetric vortices in a lattice with one vortex per elementary cell. One possible texture of this kind is a triangular lattice of doubly quantized AT vortices (Volovik and Kopnin 1977). Another possibility is to combine an MH and an MT texture into a pair of interpenetrating square lattices with four vortices per unit cell, carrying four quanta of circulation (Tsuneto et al. t978, Fujita et al. I978a, Ho 1978a,c, Nakahara et al. 1979). In this wây, the MH and MT textures, which do not satisfy the boundary condition of i beittg paratlel to the axis as required flr a Bravais lattice of MH or MT textures, may form a continuous periodic i texture where, at the cell boundary, i is normal to the boundary and in the plane (see Fig. 7.20). Fujita et al. (1978a) have numerically evaluated the dipole-locked free energy of such a vortex lattice by explicitly

/l\ìl¿ \1/ _/1\ -_l__'

__

rlz' -+o rlz -/1\i -/1\i +.--o.-+ .e+- :.(F-c-+

/1\

Ð(_,![j\1/

¡l¿lJ\ /l\ \ll

Figure 7.20

/l\

---+e+, +-c +

Lattice of continuous nonsingular vortices in 3He-A made up of alternating MH and MT textures. The unit cell, indicated with dashedlines, conlains four vortices and carries four quanta of circulation (O, Î up; @, t down).

262

7

Superflow and. Textures

incorporating the fourfold lattice symmetry. A simpler, approximate, evaluation, in which the MH and MT textures are treated separately, each confined to circular cells of area *A""¡, has been carried out by Fetter et al(1983), and yields similar results. For currently available rotation speeds f2 ( fo- For , < 6o, hoióuer, it is energetically advantageous for i to deviate from the

uniform â orientation and to titt towards the centre into a radial disgyration pattern. As mentioned above, the singularity in the i net¿ is avoided by the changing its structure from the axial into the polar state for òrder parameter -Th" uttgnlar velocity above which the z-vortex lattice should be stable r < Ç. is ternperaturé-dependent and has been estimated as 30rad/s at T:O'99\, with increasing uál,t"r at lower temperatures. The theory of vortices in the neighbourhooá of the upper critical angular velocity has been considered by Schopohl (1980). Thè situation is entirely different in a strong magnetic fleld H, parallel to the rotation axis, such that the vector d is forced into the plane perpendicl¡lar to H. T'he bending energy of the â n"tO acts in such a way as io t""p â uniform (along i, tui) "*""pt for q small region of radius Çfi, (l.Zga,b).Inside a soft core of radius Ço the lfiet¿ may deviate from the uniform orientation along ã et), in order to transfer some of the kinetic energy of floW stored in tñe phaie variable field into the Î fietd. Vortex structures of this type have been discussed already in Section 7.5. There is a singly quantizèd singular nonaxisymmetric vortex and a doubly quantized noñsingular vortex (given by (7.106)). Detaited numerical calculations show the singular vortex to have slightly lower energy than the nonsingular one for angular velocities f) less than about 3 rad/s (Seppälä and Volovik L983, Vulovic et al. t984; see also Passvogel et al. I9SZ). As will be discussed in the next chapter, NMR experiments indicate the presence of nonsingular vortices. More work is needed to clarify this question' It is worth mentioning that a doubly quantized nonaxisymmetric vortex may be viewed as a bound pair of an MH and an MT texture (Zotos and tvtat 6p). Recalling that the vector Î is in the (x, y) plane at the Wigner-Seitz zone boundary, it is seen that the magnetic field tends to flatten the texture. At some point, when the magnetic field is of order H* :30 G (see (7.28c)), the MH-MT structure might undergo a first-order transition into the high-field phase. However, other types of transition involving a distortion of the lattice are also possible. Ion-mobility experiments show the existence of a low-field regime at H> 6D), the degeneracy space is R SO¡. Since ø2(SO3):0, point singularities are unstable in the dipole-dominated A phase. In the dipole-free situation, as is valid for surfaces surrounding a point

defect

at

distances r11Çp,

(see (6.68b)). The mapping Brinkman 1977)

the degeneracy space is R:(S'xSOr)lZ2

nr(R) is (Volovik and Mineev L977a, Cross and

nrÍ(s' x So)lzr,l

- nr(S'x Sog) =

nz(Sz)

:2, since ø2(SO3):0. Hence

it is only nontrivial

+ ør(So3) (7.147) because of the topology

of,

â,

i.e. the spin part of the order parameter. So we find that a stable point defect may only exist in the d- but not the l-vector field, and only at lengthscales smaller than 5o. At larger lengthscales it becomes unstable. These findings do not contradict each other. Similarly to the situation in the B phase, where line defects in the ñ field may establish themselves as edges of planar defects, the point singularity in the d freld in the present case must beìhought of as the termination point of a nonsingular line defect in the ã field as shown in Fig. 7.21. A sphere enclosing P at a distance less than fp will only see the singularity at P, while a large sphere, besides cutting the

7

266

Figure

7.21 Point


singularity in the á-vector field of-3He-A. point of ã nonsingular line defect in the á n"t¿.

It is the termination

line defect somewhere above P, essentially passes through a uniform á texture. As discussed above, the orbital part of the 3He-A order parameter does

not allow stable poinJ singularities. Hence any (hypothetical) pointlike defect in the i nel¿ ("Í monopole") is topologicaity .rnitubt". In-fact, any î

monopole would always be connected with a line singularity carrying two quanta of circulation (Volovik and Mineev I976a); see Section 7.9.s. (Equally well, one may say that a singular vortex line with two quanta of circulation (i.e. a 4n vortex) may terminate at a singular point to escape into the third dimension.) Hence an l.monopole does not appear in a topological classification of point defects since it cannot actually be surrounded by a closed contour without cutting through the normal-fluid vortex core, where the order parameter is zero (Brinkman and Cross 1978). Such a (hypothetical) object is unstable since it transforms into a nonsingular 4¡r vortex, which may end in a nonsingular point defect and which can be removed by continuous transformations (Anderson and Toulouse L977). Finally, the 4¡u vortex connected with any / monopole is energetically unfavourable. It is expected that the system gets rid of it by contracting the vortex line to zero, for example by pulling the i monopole to the wall of the container, where it becomes a surface defect now generally referred to as a "boojum" (Mermin I977b,1978a,b). This will be discussed further in Sections 7.9.5 and7.I2.3. As discussed by Volovik and Mineev (L977b), the order parameter of the A phase also allows stable non-singular point-like defects, which may be embedded in a uniþrm inel¿ ("particle-tike solitons"; see the discussion in Section 7.8.1). They are of ñnite size and thereby carry finite momentum, energy and angular momentum. This isfn analogy to (singular) vortex rings in superfluid aHe. Hence there is a Mag\s force acting on the soliton. The same is true for a solid body of size R >50 immersed in 3He-A (see also Section I0.4.7).

F$ 1

7.8

7.8

Planar Deþcts

267

PLANAR DEFECTS

7.8.1 Topological classification Although planar defects have not been explicitly addressed so far, they have already made their appearance in Section 7.5. There we saw that certain line defects may be viewed as the edges, i.e. termination lines, of domain walls- In this situation the domain wall is characterized by the same topological invariant as the line singularity (votovik unä Mineev L977a, Mineev and Volovik 1978) Without having to go into any detailed topological study of planar defects in superfluid 3He, *" ure able to conclude thai in the absence of external fields and dipole coupling, planar defects may not exist. To have stable defects of dimension D : 2 (see Section 7.4) in a three-dimensional space, order parameters with only a single degree of freedom, i.e. with z : 1 as for an Ising spin, are necessary. Since in superfluid 3He the order-parameter space is at least two-dimensional, singular surfaces may not be foimed. on the other hand, external fields (e.g. a magnetic field) and the dipole interaction will introduce further symmetry bréakings thát may, in fact, make order-parameter degrees of freedom Ising_like. The A phase

For example, in the A phase the dipole energy aligns ã and, i uut does not distinguish between íilli and ¿u-î.This ã¿screte symmetry breaks the invariance with respect to arbitrary relativ e ã, î rotations but still allows a discrete degeneracy of the ground states. So we see that tt á-u""tor field may act as a backbone on which the i-vector field becomes"Ising-like, and vice versa. In this situation stable planar defects may exist by viiue of the finite dipole coupling. Indeed, it is now possible foi the ,yrí". to assume different ground states at different positio;s in real space. ín this case there will be a boundary region where the asymptotic configurations meet (see Fig. 7 -22), in analogy to a domain wall in u f"rro-ugnet. These defects are nonsingular and hul: asymptotically uniform texture. Hence they are _an localized objects, which may be added together without creating a substantial interaction between them. Domain walls in superfluid 3He" have been introduced and extensively discussed by Maki und .o*orkers (see Maki !?7\b, 1986a), who also coined the term "(planar) soliton" to deìcribe this kjn{ of nonsingular planar defect. The termüotogy originates from the close similarities in the properties of such a domain *ãit unã those of solitons, as known from the theory of nonlinear dynamics (e.g. the sine-Gordon equation; see below and Section 8.4.4). In both cases the corresponding differential equations have localized iolutions, which resemble classical particles and whose stability is guaranteed by the conservation of a topological charge. such solitons may be formed in both the A and the B

7

268


of superfluid 3He. They will be discussed in detail in the end of this iection. In fact, topologically stable defects are sometimes grouped into solitons and singularities, irrespective of their dimensionality, in order to differentiate between nonsingular asymptotically uniform textures and defects causing a long-range distortion of the respective vector freld. Therefore the line defect in Fig. 7.2I is also referred to as a "linear soliton" 3He-A discussed at the end of Section 7.7.2 as a and the point defect in "pointlike" or "particle-like soliton" (Votovik and Mineev L977b, Mineev and Volovik 1978). As in the case of line and point defects, planar defects may also be classified in terms of their topological properties (Mineev and Volovik 1978). To this end, we consider two points P1 and P2 on either side of the planar defect and map the line between"Pr and P2 into the corresponding order-parameter space (see Fig. 7.22a). This leads to the definition of a topological chargó N that meãsures the reorientation betw""n â and i on the line Pr-P". Assuming that a magnetic field H along 2 confines A to the (*, y) plane, this topological charge (ot "winding number") is deñned by phases

fy':

:

i+1"

dE,

H. [Î r (v,Î) - ã x(v,â)I,

(7.148)

x, ! and where the integral along any path joining P1 and Pr. îs Owing to the existence of both an I- and a d-vector field, several kinds of domain walls appear to be possible, depending on which vector field changes its orientation across the wall (see Figs. 7 -22b-g)Fifures 7.22(b,c) correspond to i solitons ,1.e. î does the winding while â stays (more or less) constant. According to (7.L48), the two domain walls topótogical charge. Note, however, that by turning both i "uíry^àpposite and ã by n at every point in the entire sample, such that their relative angle is conserved, one is led from Fig. 7.22(b) to Fig. 7.22(c). Such a transformation does not cost energy as long as there are no walls to fix the the direction of i somewhere in the sampte. This implies that a planar defect with N:1is topologically equivalent to one with N_-1. (This can also been seen by simply turning Fig. 7.22(b) upside down, thus changing N: 1 into N - -1.) Mathematically speaking, the mapping of contours P1 and P, through a planar defect in 3He-A into the appropriate degeneracy space corresponds to Z2: there are only two classes of domain walls in the A phase, corresponding to N eZ2, i.e. N:0, 1 (Mineev and Volovik 1.978). In þarticular, this implies that in an infinite system â and i solitons are equiualent, as can be verified for exampl^" by observing that the topologically-Fig. 7.22(c) may be transformed into the ã soliton in Fig. 7.22(e) isontJn in by aligning i but keeping tire angle between ã and i nxe¿. Furthermore, the structures in Figs 7.22(f ,g) do not carry a topological change because they can be continuously transformed into the homogeneous texture.

with

6,

7.8

269

Planør Deþcæ

1' It^

td

(ol

*-J *--/*--._. _- *\ (bl N=I

--+

--+

-\l/ t--

---+

--+

t

---+

(c)

\ll

.-i

N =-l

. | *t {d) N =-l

>lz.l, the dipole-free magnetic-field-free case is realized, so that

Rs:

51

x SOr.

g (ol

Figure

(+ + o)

(b)

(- - o)

o

o

(cl (+ - 0)

(dl (- + o)

(e) (-+ l)

(f) (+ - ll

(gl

(hl

(+ + l)

(- - t)

mappings of the path PrP, in Fig. 7.22(a) into the degeneracy 3He-B, space of corresponding to different domain walls.

7.25 Different

7

272


The topologically stable classes of planar defects in the B phase are then obtained by starting at one of the points n: tH0¡ and also ending at one of them. The possible paths are shown in Fig. 7.25. There are altogether eight different classes of paths, i.e. eight different domain walls. The classification is represented by a triplet, (+ -0) etc., where the first two indices indicate the direction of û relative to H at the starting point and end point respectively, and the third index, N: 1 or N:0, shows whether or not the path went through 0 : n. The paths in Figs . 7.25(a, b) may be contracted to a point and thus do not form a stable domain wall. Figures 7.25(c, d) correspond to a transition from a configuration with ñ ll ^üf to one with -ñ ll H ("ñ soliton"), i.e. a domain wall due to a reorientation of fi (this structure is topologically equivalent to one where I varies between 0¡ and 2n - 0L by going through 0 :0; however, they differ in energy and extension). The contours in Figs. 7.25(e, f) imply a planar defect in which the angle g goes from 0y on one side of the wall to 2n - 0¡ on the other side by passing through 0 : n in the middle of the wall. The structures corres_pond3He-B ing to Figs. 7.25(c-f) reflect the degeneracy of the dipole potential in (see Fig. 6.6). Finally, in the domain walls corresponding to Figs. 7.25(g,h) g also goes through n but only passes through one of the points 0¡- or 2n - 0¡. They may be constructed from the domain walls in Figs. 7.25(f, d) and 7.25(e, c) respectively. The index N:0, L characteruing the domain wall is the same as the index classifying the line defects in 3He-8, where the homotopy group ø1(SO3) :Zz is relevant; hence N eZ2 Therefore two domain walls with N: L can annihilate each other by virtue of N1 + N2 :2- 0 (mod 2). The

A, phase

The order parameter of 3He-Ar, (5.5a), does not allow the kind of solitons discussed above since the dipole energy (6.105) is minimized whenevet Í ües in the plane defrned by the complex spin vector â + iê. This implies a continuous degeneracy of the ground state rather than a discrete degeneracy as encountered in the A phase. Consequently, domain walls in the A1 phase lack topological stability.

7.8.2 Planar solitons in the A phase The term "soliton" was first introduced in the present context by Maki and 3He-A and 3He-B coworkers in their investigation of magnetic exóitæions in (Maki and Tsuneto L975, Maki and Ebisawa L975, Maki and Kumar L976a). Indeed, the spin dynamics in the A phase (i.e. the dynamics of the vector d) allows localized magnetic excitations, i.e. sine-Gordon so^litons (see Section 8.4.4). These 3re propagating^ domain yalls in the d freld where the orientation of. d changes from d ll I to -d ll f as shown in Fig. 7.2L. They

7.8

273

Planar Defects

have the following properties: (i) they are localized objects; (ii) for small perturbations, they behave as classical particles; (iii) they carry a topological ðhutg" whose conservation is guaranteed by topological conservation. The soliton solution for the â domain wall only holds if the underlying Î texture is uniþrm^throughout the system. Clearly, ^the dipole coupling of the â soüton to Î will distort any given uniform I texture-, and thereby eliminates the necessary condition for the existence of the á sohton itself. Hence tt e â sohton is unstable. As discussed in Section 7.8.1, a domain wall in ttte i freld is topologically equivalent to a â. domain wall. Which one of the domain walls will actually be formed then depends on the details of the energies involved (this is so at least in an open system, where boundary conditions may be àeglected). In turns out that in an unbounded system a domain wall in the i fieid wittr a slight adjustment of á ("composite soliton") has the lowest energy (Maki and Kumar I977a,b, 1978). Although such a domain wall is not a soliton solution in the strict mathematical sense, it is nevertheless referred to as a "soliton" because it is also localized and topologically stable.

The composite soliton is thought to be the stable end product of a decaying â soliton (Maki 1977b, 1986, Kumar 1977), where the latter may be created by pulsed NMR (see Section 8.4.5). For such a decay to occur, the ã sohton has to transfer its winding number to the I texture. Details of the creation of composite solitons and of the scenario of the transfer of the winding from â'to î are, however, not yet known. In fact, â sohtons themselves have so far not been observed at all. A more detailed study also has to take into account the effect of boundaries. Since Î is n*ed on the surface of a container, the creation of a domain wall in Î with given winding number, starting from a more or less uniform i texture with zero winding, can only take place if the texture carrying the opposife winding is somehow spread over the rest of the samPle.

Energetics of solitons

To determine the spatial structure of a soliton, we follow Maki and Kumar (L977a,b) and start from the bending free energy ft",,¿: I d3rfv.,,¿ io weak òoupling, (7.Ig), amended by the dipole interaction energy ÄFo: I d3r Lf',, (6.104). The more general case with Fb.nd, Q.20), valid at all temperatures and including strong-coupling and Fermi-liquid corrections,^ has been discussed Uy Vtati (fqgOa). The orbital triad (frr, fr, I and ã are parametrizedby Euler angles, with râ, fi and Îgiven by (7.1L) and with y : -0 (see the discussion below (7.I3)); the vector d is written as

/:

(sin g cos r/, sin

I sin r/, cos 0).

We now assume a strong static magnetic field

.GI ll

(7.14e)

â to confine á and (by

7

274


dipolar coupling) i to the (*, y) plane, i.e.

i:

(cos ø, sin

a,0),

À:2-rrze-iv(-i

/:

(7.150a)

sin ø, i cos ø, -1,),

(7.1s0b)

(7.ls1)

(cos r/, sin qt,0).

The total free energy is then given by

F:

äAl .J

a'r IlY al' + 2( .Y n)'

+

4lYyl2

-2(î -Yy)'+

4lvVlz

-2(î.vrþ)'

+ 2a,(î. vy)

- 6Ì"(î

-Y

-z\"t

cosz

(a

-

o) (7.rsz)

4ù1,

a":ôslðz

etc. and A:+KÁ?o and where the pure divergence in (7.t9) has been neglected. We are now looking for planar solutions of (7.I52). To tlris end, we note that (, y and rþ arc functions of only a single variable s : É . ¿ with Ê: (Ê*, Êr, Ê).Then (7.I52) reduces to where

p

r : )Ao(ÉU

-

4k"au"y"

[(1 +

2o')ú + 2(2 - a')(y? + V?)

- 2E;' cosz (u -

,þ)1,

(7.1s3)

o:

î . î(: É, cos u + Ê., sin ø, and o(Îr) is the surface area of the domain wall, with normal vector /c. The free-energy functional (7.L53) has to be minimized with respect to the functions ø(s), y(s) and ú(s). This may be done by solving the Euler-Lagrange equations

with

ô a/(s) _ ô/(s) :u' _.,

ôta(dy,/d") W

(7.rs4)

where F : I ds/(s) and ¡ : u, T, r/. Clearly, the overall phase y is a cyclic variable (since / only depends on %) and may be eliminated, yielding

#:+ol"

[(t

+za2

-#)a!

+2(2-

o'),t? -zE;'cosz (u

-,þ)). (7.lss)

If the Î texture is uniform, with Î ¡¡* (owing to Í being clamped in a narrow

slab, or imposed as an initial condition in a time-dependent situation), rp is determined from (7.155) through

d'rl, d"t

- ,*sin}r/ : 01

(7 -156)

This is a time-independent sine-Gordon equation (see (8.139)) with the

7.8

275

Planar Defects

._Yt*

(b) Figure

5D

7.26 (a) The angular variable r¿ describing the orientation of â in a ã soliton. (b) The change in ,lt, i... ,1t", across the soliton clearly shows the localized nature of the soliton.

solution

tt:2tan-1 ["*o (å)]

(7.rs7)

sin r/:sech(s/6ír) and EL:Í2(1,-+k?)ft''F¡.. The solution (7.157) describes a â soliton at rest, i.e. a kink in the d field, where the angle r/ varies from 0 at s: -@ to n at s: *æ. The change in the orientation,

i.e.

q,": Jse"h $, 5p 5p

(7.1s8)

occurs in a narrow localized region of width 6i, (see Fig. 7.26). The energy of the á sohton comes out as

#:

2s/2

A(I -

+k1)''' E;'

(7.Lse)

.

Clearly, tne ãsohton assumes the lowest energy for îe ll i i.e. Ê,:'l'. The á soliton carries a spin current with polarization along 2 (Maki L977b), i.e. a magnetization pulse.

Composite solitons We shalt now consider planar textures where both

i

and

ã cooperate in the

reorientation, which have been named "composite" solitons (Maki and Kumar L977a,b). For a planar texture with its surface normal parallel to H

7

276

("twist" structure),i.e. îe: (0,0, L) and hence


o:î. î(:0,

(7.L55) reduces

to

Ff +A

"(,r):

J

dz ÍtuT + tu? - 2E;' cos2 ul'

(7.160)

where we have introduced new variables

u:

a + 4rlrr|

(7.161)

u:g-1þ.) The Euler-Lagrange equations for u and u,

?r,, - àE;'

sin 2u

: o,

ur,:0,

Ì

(7.162a)

)

are seen to decouple. They have the solution

u:2tan-1'l t"*o

l- /s\rtz z-ì)

L*6/ 6"1j, I

u: constant.

(7.162b\

)

For the general solution, valid also beyond the Ginzburg-Landau regime, see (8.146a,b). Hence the relatiue coordinate u changes from 0 to n, while u, the centre-of-mass coordinate, is constant. This means that i only has to do f of the reorientation, with á doing the remaining ! as one goes through the domain wall. This reorientation is shown in Fig. 7.27. The correspondz

,/

ã-"

,/

+--

,/

'-5

,/

?' ,/ I

,/.1 +---i

Figve 7.27 Twist composite soliton in 3He-A; the change in orientation occurs along 2, with I doing Ë of the reorientation.

7.8

Planar Defects

277

ing free energy is found to be

Fg$ -

5-ttzPrutist,

(7.163)

and is hence considerably lower than that of the pure d-twist soliton given by (7.159) with Æ, : g. This is due to the fact that rhe width of rhe composite domain wall, (?)'''Eo, is itself a factor 5-1/2 smaller than that of the pure d domain wall, which is also reflected in the energy, given by 4AÇo2 times the domain-wall width. Twisting the ãtexture is obviouily more costly in energy than twisting i. This is clear from the free-"."tgy expression (7.20), where, according to (7.2L), the coefficients of the splay and twist term of the d texture are about a factor of p"o, lK",t- 4 larger than the corresponding coefficients for the I terms. The equilibrium configuration of a twist composite soliton in the immediate vicinity of the transition into the A, phase has been investigated by Gongadze et al. (1978b). In general the vector lr, i.e. the orientation of the domain wall, is expected to be determined by the shape of the container in conjunction with the orientation of the magnetic field (Maki and Kumar L97S). For example, in a cylinder the domain-wall surface is smallest for & along the cylinder axis. If the magnetic ñeld is also along the axis (É ll ^[I) then the twist composite soliton will be favoured, while for H perpendicular to the axis a composite soliton with É llÍ l.ãI and a "splay" õonhg,rrution in i (see Fig. 7.36a) has a lower energy. Since the free energy (7.155) can no longer be minimized analytically in this case, these configurations have been investigated variationally. Using the parametrization (7.L61), the splay composite soliton is well represented by

sinu:sechr¡2,

s:

tt) + |n(L

-

t),

(7.t64)

of all domain walls, the twist composite soliton has the lowest energy. The effect of boundaries on domain walls in a restricted geometry of width of order fo has been investigated by Lin-Liu and Maki (1980b, 1981) and Maki and Bruinsma (1980) for parallel-plate and circular-cylinder geometries respectively. A soliton in 3He-A represents a localized reorientation of the preferred directions of the order parameter and thus carries a local deficit of nuclear dipole energy. Consequently, such a structure acts as a potential well for with r:0.681. and 4 - 0.706o.

spin oscillation. This leads to well-defined satellite peaks spectrum, which will be discussed in Section 8.4.5.

in the NMR

Soliton lattice We have so far considered domain walls as single entities in an otherwise homogeneous texture. Clearly, there may be more than one domain wall in

7


a sample. In particular, if composite solitons do indeed originate from d solitons then many such solitons will be created after a magnetic pulse. Since solitons experience a strong mutual repulsion, they are expected to form an array of equally spaced domain walls (Bruinsma and Maki I979c). Indeed, the Euler-Lagrange equations (7.154) for the free energy (7.160) have a stable periodic solution

:." f / 5\r/2 z L\t/ k E" "1.

cos u

-],Ì (7.16s)

ø: constant, where sn is a Jacobian elliptic function (Abramowitz and Stegun 1965) of parameter m: k2. This solution describes an evenly spaced soliton lattice with lattice constant

Lo:

($)t'2g,',krc1k'¡,

where K(k) is a complete elliptic integral of the first kind. In the it reduces to

cosu:tanh[(;)'"

*],

(7.166)

limit k-> | (7.167)

i.e. the equation for a single soliton, which is identical with (7.162b). For

k+0,

a harmonic solution cos u

: sin

l()'" ;;l

(7.168)

I and d rotate oppositely at constant rate. For other orientations of the magnetic field relative to the domain wall, soliton lattices with different î, d configurations may be determined is obtained, where

variationally as in the case of a single soliton.

7.8.3 Planar solitons in the B phase As discussed earlier in this section, domain walls in bulk 3He-B are only stable if the vector ñ is asymptotically aligned, say by an external magnetic field. The symmetry of 3He-B then allows qualitatively different domain walls, as shown in Fig. 7.25 (Mineev and Volovik 1978); they are nonsingular planar defects ("planar solitons"). As in the case of 3He-4, planar solitons in the B phase were introduced and extensively discussed by Maki and coworkers (Maki and Tsuneto L975, Maki and Ebisauta 19'75, Maki and Kumar l9'76a,b, L977c, Maki and Lin-Liu 1978, Maki L977a,b,c, 1986a).

Of all orientational energies in 3He-8, the dipole energy is by far the

7.8

Planar Defects

largest. Therefore it is energetically favourable to keep the rotation angle 0 at its equilibrium value 0 : 0v The remaining degree of freedom is then the rotation axis ñ itself, which may change its orientation between the asymptotic configurations ñ ll ff and -ñ ll H. This leads to a "planar ñ soliton", whose structure is determined as follows (Maki and Kumar 1976a, L977c). We parameftize ñbY

¡:

q sin 4, sin g sin 4, cos 4) (7.16e) and assume that E and 4 depend only on a single variable s : É'r, with Ê:(Ê*, Êr, Ê) the unit vector perpendicular to the domain wall. Then the energy Fa: Î dtr(.fo",,o + Lfù due to bending, (7.24b), and magnetic orientation with H ll2, (6.116b), takes the form (cos

4: !' ld"{;{ o(k)- I\mzôg Y"

tt? +

J

q?sinz 4

- å[(3t'to + 5r/2b)r¡, + (5r/2a - iE-; cos2 4),

3t''b)8, sin 4]2] Q.I70)

where

c

:

cos q (Ê,cos

g + Æ, sin q)

- É, sin 4,1

b:Ê,sinE -ÊrcosE,

J

(7.r7r)

o(iÛ) is the surface area of the domain wall, and $¡¡ is the magnetic healing length for 3He-8, as defined in (7.29). In the case of k ll2, implying a splay-like structure of the domain wall, i.e. for É parallel to H, (7.170) reduces to

k2 Frt" Lo(t

): *æ

,.,

r

o?J az {Ztq?

+

E? sin2 4

- ft sinz q (3'oq" * 5t''q" sin 4)21

- IE;' cos2 4).

(7.t72)

The Euler-Lagrange equations (7.L54) then yield

?,

: r(å) "gr'll,- *s,'t ql''' sm 4' L l-r,siF l (P":

I5r/2

sin 4

16-5sin24 4r'

(7.t73a) (7.t73b)

which give the explicit profile of the fi soliton: fi changes from antiparallel to H at z: -æ to parallel at z: *æ and is perpendicular to H at z:0; this splay-tike behaviour, described by T, is quite similar to the reorientation shown in Fig. 7.2(b) and occurs in a confined region of width fo. However, superimposed on this profile is a reorientation of the xy component of fi,

7

280

Superflow ønd Textures

described by q, which changes from q--0.33 at z ---æ to q:0.33 at z : *æ, with g :0 at z :0. This gives an additional wave-like component 3He-4, to the reorientation of fi. Hence we see that, unlike thç solitons in the û soliton has an internal three-dimensional structure. We also note that the width of the ñ soliton is determined by i¡¿, which is much larger tHe-4. Hence the Ê soliton is a very than the corresponding lengths in extended object. Other configurations of Ê solitons, as well as the influence of surfaces on fi solitons in a parallel-plate geometry, when the surface orientation energies (6.130) and (6.131) dominate, have been discussed by Maki and Kumar (I977c). The contribution of solitons in the B phase to the thermodynamic properties has been considered by Kumar (1978).

7.9

SURFACE-INDUCED TEXTURES

We now want to examine the role of surfaces and boundaries in determining the textures in a container of given shape. After all, any experimental sample has a finite size, so that surfaces will always have some effect. In particular, in the absence of external fields, boundaries of a sample will be essentially responsible for the equilibrium texture. As in the bulk situation, textures at a surface may contain defects. We shall therefore first consider the types of defects and singularities related to the existence of a boundary. This involves three different aspects (Anderson and Palmer 1977): first, one has to classify the topologically stable defects that are permitted by the order-parameter structure; secondly, while in the infinitely extended bulk defects always correspond to excited states of the system, the presence of a surface may force the existence of a defect even in the ground state. One therefore has to ask whether in a given container a defect-free texture is possible at all. Thirdly, one has to know how a surface defect connects with the interior of the sample. The first aspect may be answered by using topological methods (i.e. homotopy theory) similar to those employed in the bulk situation (Volovik and Mineev I976b, Volovik 1979, Bailin and Love l978a,c,e). This requires a mapping of a contour on the surface, surrounding a defect, into the degeneracy space ^ñ of the surface. In general, the latter is a subspace of the degeneracy space R of the bulk, depending on the restrictions on the order parameter imposed by the surface. For the second point, the topological properties of the container itself have to be investigated. For this, it is necessary to know the "connectedness" ¿ of the container, i.e. it is simply connected (n:0, e.8. a sphere), doubly connected (n--L, e.g. a torus), triply connected (n--2, e.g. a pretzel) etc. More precisely, one needs to know the so-called "Euler characteristic" of the surface, given by

E:2fi,-n).

0.r74)

7.9

28r

Surface-Induced Textures

can be proved that the sum of the topological quantum numbers of all point singularities of a vector field on the surface (for example the Î field in the A phase) is equal to E (Mermin L977b). For example, for a sphere (E:2), the total topological_quantum number is 2. Hence in a sphere

It

itte equitibrium texture of 3He-A must have two quanta of

surface hand, textures On (see the other in a torus below). may circulation a is the only container with this property! In fact, torus be defect-free. Concerning the third aspect, a surface point (line) defect ffiây, but need

(E:0)

not, be the termination point (line) of a vortex line (domain walt) in the bulk. This in fact depends on the details of the energetics involved; although a bulk defect will cost additional energy, it may help to reduce the bending energy in the neighbourhood of the surface defect, as is the case in tiny spheres (see below).

Even though most experimental cells have a simply connected geometry and hence necessarily contain surface defects, the importance of these defects for the overall configuration of the bulk texture depends in an essential way on the shape of the cell. For example, in a flat disc-shaped cell the volume occupied by surface defects may essentially be neglected, in contrast with the case of a sphere. This situation will be discussed in Section 7.9.5. From the discussion of the various healing lengths in superfluid 3He (see Section 7.2.3), we know that in the B phase the lengths are relatively large, i.e. often comparable to the usual container sizes. (We note that the much shorter dipolar tength 5o is almost always unimportant in 3He-B because 0t)). In addition, the relevant one is in a dipole-locked situation rather small, so that boundary conditions on ñ are not surface energies are strict. Consequently, an fi texture is usually smoothly varying over the entire sample. Textures in the A phase are generally more intricate than in the B phale because the order parameter is characterued by two vector fields I and d. Furthermore, the characteristic lengths involved, i.e. the dipolar length Çp and the magnetic healing length ffi (with Efr> 6o) may be described as follows: in the bulk of the sample th; ñtexture is uniform, and is oriented parallel to the axis of the freld; this uniform region is connected to surface layers of width Ç¡¡, where ñ changes over into the surface-dominated orientation' The behaviour at low magnetic fields H > E¡,), â and î are dipole-locked (B -- 0), i.e.

p

: ]Ao(Ð[ a, @? + ?E-,]cos'B).

(7.r84)

We note that while the orientation energy is of order (þlEò' for small B and thus goes to zero proportionally to H2 as Ë1+0, the bending energy is at least of order (Þlf¡'. Consequently, for small .F/ such that Er) L, bending is too costly and the texture remains uniform. The Euler-Lagrange equation for B then yields B,,

+ sin 2þ

:0,

(7.18s)

where t: zl5rt2En Considering f as a dimensionless time, (7.185) describes a mathematical pendulum subject to the boundary condition þ:0 at

7.9


t*: t*Llí''tEr.Clearly, a solution þ+0 is only possible for "oscillation periods" T : 2nl2r'' u 2(r* - t-) : 2Ll5rt2fr. This means that for magnetic fields H

1Hr,

where

ç

b

(Z)",nT H/', Hp: r\¿/L

(7.186)

with I/o defined in (7.28b), the Îtexture is uniformly oriented perpendicular to the surface, while for H ) He bending sets in continuously (Fetter 1976b). Note that the critical field decreases with increasing slab width as 11, n LlL. The implicit solution for B as a function of z is given.by an elliptic integral

dþ'

r"' [sin'zB(o) -

sin2

/z * : t\Z p'l'o

1\ H

il u.'

(7

'rsz)

just abova Hp, one finds B(0):2(HIHF-t)''', i.e. þ has infinite slope and, for H->oo, tends to ln. The transition has been observed by Hook et al. (1986, 1987) in two different experiments: by use of a torsional For

^FI

oscillator and by zero-sound propagation. In general, and in particular for slab widths L € 5o, â and i will not be dipole-locked. In this case both â and Î huue to be treated as separare variables. The Euler-Lagrange equations for Î(B¡ anO ¿@) follow from (7.183) as (7.188a) - þ) : 0, 0""(3-2cos 2P) +20"þ.sin2|+ Eo'sin2(B- 0) + f;2sin 20 :0, (7.188b) with the boundary conditions þ(X.äL) :0. In contrast with the dipolelocked case, a general analytic solution of these coupled nonlinear

þ",(2 + cos 2P) + @7 + |f;) sinzþ

-

6õ2 sin 2@

differential equations is not available. On the other hand, certain limìting cases may easily be discussed. In the limit of plate separation L1fp, land d must be uniform with f perpendicular to the plates (B : 0). In this case the free energy in (7.183) reduces to

, * I d, (Eo' - E;ðsin2 o.

(7.18e)

For small magnetic fields, such that Er>Ço, the minimum is given by 0^:0, i.e. d ll l, while for fields H> H* (i.e. Er< Ço) the configuration d L I is favoured. The corresponding critical freld is of course given by H*:28G, (7.28c). For general magnetic fields and plate separations, the phase boundaries of the two configurations ãUî and ã t î may be determined from a linear stability analysis of (7.188) (Ambegaokar and Rainer 1974, Fetter 1276^b), as shown in Fig. 7.30. Variational calculations of the nonuniform d, i configuration in this phase diagram have been performed by Fetter (1977) (see also Bruinsma and Maki 1979).It is found that for strong magnetic fields H >>.F/* and L 1 L* , where L* :3'''nEr,

7

288


H/H*

ñl l- slab

2.0

¿

tî

1.0

dll t

0 Figure

7.30

1

2

3

4LlL*

5

ã, î confr,gurations of 3He-A in a slab of width L (in units of L*:3rEnEo) with the magnetic field Il Phase boundaries between different

oriented perpendicular to the slab. (After Fetter (1976b).)

a uniform configuration with ã t î ll ^H is stable (see Fig. 7 .30). For somewhat larger plate separations, a configuration with ¿ ll î I H (except for a narrow surface layer) is formed. On the other hand, for small magnetic fields H > 6(f) the boundary conditions force I to point radially outwards at the wall, i.e. along a curued surface.

7.9


289

The wall configuration being fixed, the question is now how Î is oriented in the interior of the cylinder and how the vector d behaves in this situation. For cylinder radii EQ) H*, a discrete transition to a planar structure is expected, i.e. to an i disgyration with ãilî, to minimue the dipole Flowever, "n"tgy. these two structures are topologically distinct (cf. the difference in surface circulation) and cannot be continuously deformed into one another. As discussed by Nakahara and Ohmi (1979), there is a barrier between the two

structures, making

the (deformed) MH structure metastable even for

magnetic fields where the energy of the disgyration is smaller. The response

292

7


of the MH texture to a time-dependent rotational perturbation has been studied by Ohmi et al. (1981). As usual, the nonuniformity of the i and â configurations induced by the cylinder will show up in an NMR experiment via shifts in the signals. This will be calculated in Section 8.4.3.

7.9.5 3He-A in a sphere In a spherical container the surface (with normal S) has an even stronger effect on the orientation of the f-vector field than in the geometries

considered so far (plates and cylinders), since the boundary condition i ll í must now be satisfied simultaneously in euery direction. As discussed at the beginning of this section, topology alone tells us that in a sphere (Euler characteristic E:2) the equilibrium texture necessarily has to contain surface singularities with a total of two quanta (i.". 4n) of surface circulation (Anderson and Palmer 1977, Mermin t977a,b, Volovik 1978b). This is easily understood from the surface geometry of a sphere. Consider a three-dimensional sphere with the orbital triad (ñr, ñ, i) positioned at the north pole and pointing upwards. We now move the triad to the south pole on lines of constant longitude (i.e. azimuthal angle). When we arrive at the south pole, i points downwards of course, but at the same time ñ and ñ have an orientation that depends on which path on the sphere rvas chosen, i.e. on the azimuthal angle. The total difference in orientation of the triad is equivalent to a rotation of the triad about î by 4n, which is precisely the surface circulation quot'ed above. Since a rotation about I corresponds to a phase change, any I texture in a sphere (or in any closed simply connected container, for that matter) has to have a singularity somewhere. If this singularity \ilere located in the bulk liquid then there would have to be one or two vortex lines with a total of. 4n circulation connecting the singularity with the wall of the container, as shown in Figs. 7.32(a, b) (Volovik and Mineev I976a, 1977a,b, Blaha L976, Anderson and Palmer \977, Mermin L977a,b,Hu et al. 1977, Maki L977a). What then does the Í texture in a spherical container look like? Intuitively, one might expect that<wing to the spherical symmetry of the problem-the equilibrium configuration would have to be the one where f points radially inwards (or outwards), forming a "monopole" at the centre of the sphere (Fig. 7.32a). However, this is not correct-in fact, even the assumption of spherical symmetry is incorrect, because the order parameter in orbital space is not just given by Í but by a triad, which is the reason for the 4æ circulation connected with the I texture. The necessary phase vortex line(s) discussed above in fact break the spherical symmetry. Furthermore, these vortex line(s) have a normal core and are thus associated with an energy that increases as Rln(zRlÐ, where R is the radius of the container. So they are energetically disadvantageous. To reduce their length, the monopole has to move to the wall (Fig.7.32b), thus forming a surface

7.9


293

-\.li

\ *\ '{ \-{

> 6o) and no external magnetic field, the degeneracy space is R : SO¡ and hence the homotopy group is ø1(SO r) : Zr, (7 .57c), as discussed space cotrerpóttding

in Section 7.5.4. The topological charges ate N2:0, 1, implying that a stable As state of superflo\il may at most carry a single quantum of circuløtionl 3He-A is already expressed by tÍre Mermin-Ho relation (7.53), superflow in not quantized.It may therefore be continuously decreased by an appropriate spatial variation of the three-dimensional orbital triad, until it is reduced to zero, or at most one quantum of circulation, depending on whether the originat state carried an even or an odd number of circulation quanta. Thir t"tolt is not changed much if there is no dipole locking, as for example in a very "thin" torus with channel thickness r < Eo- The degeneracy space is then a little largel,leading to the mapping (7.59a) with four topotogical charges: N: 0, +.t, L. This allows two more currentcarryinf states: both with hatf a quantum of circulation. One iherefore finds that, in the absence of an external magnetic field and if surface effects are neglected, superflow in 3He-A is topologically, i.e.

7.10 Stability of Superflow and Related Textural Transitions

297

globally, unstable-it dissipates. Possible dissipation mechanisms will be discussed in Section 7.L2. The situation is changed if an external magnetic field ^tl is applied, which confines âto the plane perpendicular to it. Because of the dipole coupling, Î will also lie in this plane. Now the orbital triad is no longer able to perform three-dimensional motion, which implies V x u. - 0, i.e. the current is quantized. Indeed, the degeneracy space is reduced to (7.75a), i.e. two circles. The topological charges given by ø1(S1), (7.57b), are characterized by Z as in superfluid oHe or'He-B. So a magnetic field stabilizes superflow in 3He-A via the dipole-dipole interaction of the 3He-nuclei! The surface and its orientational effects ott i have been neglected so far. Owing to the boundary condition that Î be normal to a surface, the orbital triad cannot move freely at a surface and hence a continuous decay of surface superflow cannot take place. Indeed, it is easily shown that right at the surface the circulation is quantized. To this end, we consider the inner or outer circumference of the torus in Fig. 7.33. It lies in the plane perpendicular to 2, and the vector i will always point radially outward. The orbital triad may then be parametrized in cylindrical coordinates as

râ:cos qE-sinø2,1

ñ:sinqQ+cosq2, I î:î.

(7.202)

)

The superfluid velocity u,, (7.10), along the circumference is given by

u":

ha 2m

r'

(7.203)

just as in superfluid 4He, and consequently the circulation r, (7.55),

is

quantized:

*:N

h 2m

(7.204)

This result holds for any contour along the surface. So, while topology guarantees the stability of superflow at the surface, it fails to do so in the bulk. This suggests that the current will decrease in magnitude as one goes from the surface into the interior of the liquid. How this is achieved is a very delicate matter and is still not fully understood (see Section 7.12). While it is clearly advantageous for the system to reduce the current to lower the flow energy, the contortions through which the orbital triad will have to go in order to achieve this will add to the gradient energy. Besides that, the topological arguments cannot tell whether or not the orbital triad can freely move to reduce the current. For example, the anisotropy of the superfluid density prefers i pointing along us, as expressed by (6.L20). In this case it is not clear whether the system first has to climb a

7

298


small barrier in energy space, i.e. whether it needs an activation energy, before the decay can eventually start. If so, this would imply "local" stability (in energy space) of superflow. Such behaviour is indeed found for certain parameter values of the prefactors in the gradient energy and will be discussed next. Local stabitity of the uniform î terture Superflow in a superfluid with anisotropic superfluid density introduces an n"* preferred direction to the system. It will thereby influence the orientation of Cooper pairs (Glassgold and Sessler 1961). In particular, in the A phase a superfluid velocity field us may be expected to orient the vector iparatlel to itself (see (6.120), (7.zlb)), since the component of the superfluiã density tensor along î, pr,, is smallest (de Gennes and Rainer Lm4, Leggett L975a). Furthermore, this orientation may be taken to be uniform because then there are no contributions to the bending free energy. Atthough these arguments, taken by themselves, are all correct, the 3He-4, (7 .29), is clearly complicated enough to defy bending fiee energy of un unJyris of ttrJitaUitity of the i ll u" configuration that simply discusses individual terms. The detailed investigation of this problem was initiated by Hook and Hall (1977, unpublished), who tested this hydrodynamic stability of the uniform Î texture, finding an instability of the I l¡ u" texture in the Ginzburg-Landau regime. Even though their analysis was based on incorreci equations, it was of considerable significance since it provoked further studies of the problem, which finalty led to its clarification (Bhattacharyya et al. 1977a, Cross and Liu L978). Following Bhattacharyya et ø1. (1977a), we study the stability of an equilibrium configuration in the presence of (arbitrarily small) uniform flow alãng 2. Periodicity along 2-'as realized in a torus-is assumed. (In superfluid aHe this doubly connected geometry leads to the topological stáUitity of superflow as discussed above.) The flow is assumed to take place between plane-parallel walls of separation d whose surface normal is along i. At the walls î ll i: Far away from the surfaces \ile assume that the flow has aligned the vector I:

îo:2, us:

h

ûsz.

(7.20s)

1:10 U,m being of We note that ú. defines a characteristic length, with [r, the order of the dipole coherence length Çp for u"': 1 mm/1. For u"t > d, the surfaces will dominate the direction of I, such that I ll i. Only for ú,-t(( d, í.e. u">>(h|2ry)d-', which is assumed here, will alignment of I into the flow direction (Î ll 2) be possible. Although this condition limits the superfluid velocity from below, us can be made arbitrarily small by making d large enough. On the other hand, in the case that u, becomes so small that 1

7.10 Stability of Superflow and Related T¿xtural Transitions

l,;'>R,

where R is the diameter of the torus (see Fig. 7.33), the effects of curvature must be included (Bailin and Love 1978b, Harris 1980). The stability test should only consider deviations from (7.205) that vanish at the surface. This is achieved by limiting the range of transverse wave vectors from below, which leads to a second-order stabilitv test. Introducing small fluctuations ø, u and w,

with w - -T(ut

ôi= î _ îo: (u, u, w),

(7.206)

* rt), (7.206) may be writren as ôt: /r - Tt1t,

(7.207a)

where

It:

(u, u, 0).

(7.207b)

The deviation ôi leads to a contribution ôu, to the superfluid velocity via (7.53): V x ôu"

using the identity Yu

xVu:vx

-+(Vø ¿m

x Vu).

(7.208)

(uYu), ôu, is found to have the general

form

ò%=

ffVr+l(uYu-uYu)f,

(7.20e)

where y is a rotation angle about Í. Equations (7.207) and (7.209) are now substituted into the bending free energy f1",,¿: I d3r.fu.,,a, given by (7.20), which is then expanded in terms of the fluctuations. Assuming that fluctuations are characterized by a wave vector k: (k*, k, k,), we express 11 and y by their Fourier transforms, and thus-after minimization with respect to y(/cþ-obtain the following quadratic contribution to f'6.,,¿:

(L\') :!t\zil

.

+ ln"lq ' t'l' + K'18 x t'l'

ôr1.'¿ =

f

+ (Kbk|+ pot?)t?+ (zco* p,,,)tr"k.Im (tf x tr)

# r

c\k?tqx t,t'

êÍp\t?tq-t,t'+

p¡Csu"q'k.Im (tf ,, tr)lÌ.

(7.2r0)

Here q = (k,, kr,0) is the transverse part of fr with respect to u" ll â and we have used the notation of (7.21c). It should be noted that (7.210) is correct in both the dipole-free and the rigidly dipole-locked situations. (In the latter case the coefficient-s K", K, and K6 have to be interpreted as indicated in (7.2La).) If the â, îtexture is dipole-locked at u,:0 then this remains true

7

300


even for (infinitely) small flow velocities us as are considered in the present

investigation. For higher flow velocities, however, this is no longer true since then â and i will acquire different directions. This case will be treated in the next subsection. Equation (7.210) now has to be minimized with respect to 11, i.e. the fluctuations ø(Ic) and u(Ic). This is considerably simplified by taking into account that l4l :d-', such that we may neglect q relative to k'. The minimization of (7.210) then only involves the third and fourth terrns in (7.2L0), and ôF*,,¿ is found to have a minimum at

,o-Ct-Cl*LP"l "z Kuú.

(7.zrt)

For ôF6.,,¿ to be positive at k:k9â, this leads to the stabitity criterion for the i f u" texture (Bhattacharyya et al. I977a, Cross and Liu 1978) f

(7.212)

By contrast, if s ) 1, one has ôF1",,¿ < 0, indicating an instability of the Î ll u, texture at lÉl : k, characterized by a circularly polarued vector 11. The denominator of the stability parameter s in (7.212) is determined by the prefactor of the two contributions to the bending free energy li.ZO¡, (1.2rc) that favour the i ll ". configuration. Indeed, for strong anisotropy of the superfluid density p. and large stiffness against bendilg. of I, these two parameiers make s small and thereby stabilize the uniform , ll u. texture. On the other hand, the terms äpr,(î ' u")' and -(C' - C,,) (i. uJti. (V x Í)] are found to disfavour this configuration, their prefactors appearing in the numerator of (7.2I2). The cause of this behaviour will be discussed in the next subsection. For transverse dimensions of the system d>> Eo and for u">> (hlLm)-r (e.g. for u, >> 1.0-2 mm/s at d:1 mm) the vector Î is dipole-locked to â (see the discussion below (7.20I)). Using the weak-coupling results for the parameters appearing ín (7.2L2) in the Ginzburg-Landau regime as given by (7.22) with (7.21.a), one finds

'l*""*coupting,-å L) a ¡rrent-carrying state is totally unstable, such that a catastrophic decay of rperflow sets in, or whether there is a "displacive" transition to a new, onuniform-but still stable-state of superflow. In the dipole-free case ,hattacharyya et al. (I977a) explicitly constructed a continuous family of onsingular configurations that lead from the original state of flow to one 'ith no flow, with the free energy monotonically decreasing. On the ther hand, since i and â are dipole-locked, at least for small superfluid elocities, the additional stiffness of the Î ll ," configuration due to the râ :xture obstructs such a monotonic decay (a similar argument applies in the ase of a finite magnetic freld, which fixes the vector d; see Liu 1979b). rdeed, Fetter (1978a,b, 1979) and Kleinert et al. (1978, L979) showed that, eyond the threshold given by (7.2L2), the texture undergoes a transition to static helical configuration with I ll d winding afound the direction of uperflow. This implies that superfluid currents may flow even beyond the

tabilityregions(L. The ultimate reason for the formation of such a helical texture lies in the resence of the term -(C ,- C,,Xr, . î>tî^. (V x Î¡¡ in the bending free-energy ensity (7.20). Since the factor , ' (V x I) describes a torque on I (which is rhy the contribution to /6s¡¿ corìtaining this term is sometimes called the torque term"), one has competition betwegn a configuration with % ll i onstant in space, making lr,^' Il maximal but / ' V x l:0, and one where u, nd i are not aligned, with i varying in space. In the latter case l4'il < 1 ut now li. (V , i)l > 0, so that the torque term leads to a finite negative ontribution that decreases the overall free energy. A spatial variation of I dll therefore lead to a (perpendicular) torque on ¿ causing a helical onformation. We note that the torque term is the only term with a linear oupling between Î and u., which may therefore distinguish^between i ¡¡ u, n¿i ll -%. While in the uniform state this is irrelevant (V x i: 0), a spatial ariation of i as in the helical texture will lead to a breakdown of chirat ymmetry. Following Fetter (I978a,L979), we initially assumeâ uniform I texture nd superfluid velocity u",s as given by (7.205); I and d are supposed to be tipole-locked. Using the parametrization of I in (7.tL), the Euler angles ø nd y are found to increase linearly with z,

g:-knz, T:-krz,

(7.218)

o that (see (7.13)) Us,o:

h

Úr,o: kn + k".

(7.21e)

^0",o2, For superflow in a torus, the periodic boundary conditions imply opological constraints on the order parameter to guarantee its single-

7.10 støbility of superflow and Reløted Textural rransitions

303

In particular, the "winding numbers', kn and k, are in general quantized (Fetter 1979, Kleinert et al. Lg7g, Kleinert 1979a,b, Fetter and Williams 1979,1981). Transitions between states of superflow in a toroidal valuedness.

geometry with different winding numbers are thus prohibited for topological reasons. while for þ *0, n the values of fr. and k, are separately quantized, for B :0, n only the combination k* xk" is quantized. In añ ..open,, geometry, with a relative superflow generated for example by applying a thermal gradient, the situation is quite different: in this case theie are no topological requirements on the order parameter and it is the current g, itself that is a fixed. quantity. This difference has been discussed in detail bv Fetter and Williams (1979, 1981). Below the instability temperature f, discussed above a helical texture develops. With B as the opening angle of the helix at a given temperarure T < T, the superfluid velocity (7.13) id given by

Us:

Au,:

Us,o

- ÂU'

*o-rt -

(7.220) cos þ)2,

i.e. it is found to be reduced by Âq; u. is still along â but is no longer

quantized. On the other hand, the superfluid current (-7.39) takes the form g":9",,2 *9,_rlr, where l, : (sin B cos ø sin B sin u,0) and

8",,: ftW","{r,rr * Bsr

:

po sin2 B)

ftVu".opocos

-

B + k,(C L

cokncos B sin2

-

Cosin2

Bl,

(7 .221a)

(7.221b)

B)],

with Po:P"t-P.lr and Co:Ct-Cr. The current is

seen

to

have

a

solenoidal direction. The opening angle B is now determined by minimization of the bending 'with free-energy density (7.20) fixed ú" and ko. Just below 7,, i.e. for temperatures T with 4 - f 0 for all k, the uniform texture with î U ¿ ll u" is tocally stable.

(2) When gt(k)>9t(k^):0 and the equality is satisfied by a single k^, and furthermore 1f 92(k-) against the appearence

) 0, the uniform texture becomes unstable of the helical texture with wavelength

Â- : 2nk^r. Therefore this condition gives between the uniform and helical textures.

the phase

boundary

(3) For gr(k^)O, the helical texture with wave vector

k*

and u?o:

lgrll% is locally

(4) For &r(k^) er k of a and rp, (7.235b), must itself be assumed to be a variable, with 'espect to which the energy should be minimized. (This is in contrast with he toroidal geometry discussed below (7.219), where k is quantized and ;annot simply adjust.) The possible helical solutions with constant þ and 0, rnd the optimal k are then obtained from

ffi:o'

Úur:o'

fo:o'

(7.244)

fhe stability criteria for such a texture are given in terms of the eigenvalues !r of the eigenequations

Mf

: )uf,

(7.24s)

(xr, x2, x3) and M,,: (ð291ðx, ôxi)o, with x, : ô8, xz: õ0 and fu: õk, and the subscript zero implies that the value is taken at the rptimal values of P, 0 and k. The texture is stable if all eigenvalues are ron-negative. In this way, one obtains the stability region II of the helical :exture shown in Fig. 7.35. The outer phase boundaries at which the úability regime terminates have a roughly triangular shape with a sharp tip ú p-L.16, h:0.59. For h> l),,) this reduces to (Greaves 1976; see

't?:{ît.2&

(8.37)

[rì

(8.38)

From the general result (S.17) on the spectral weight of these resonances,

8

348

Spin Dynamics

we conclude that the resonance at o¡? - 0 carries no weight. The resonant shift Aar, : (Ðt ttty of the Larmor frequency in (8.37) has been observed experimentatly (Osheroff and Brinkman 1977). It may in principle be used to determine yet another combination of free-energy parameters, B3a5.

8.3

NONLINEAR NMR PHENOMENA IN UNIFORM TEXTURES

Even in the simple case of spatially. uniform spin motion, the nonlinear Leggett equations (8.8a,b) are complex enough to defy general solution. In the following we shall therefore address the A phase and the B phase separately, and we shall discuss a few special cases that allow an analytical solution while still revealing the underlying physics.

8.3.1 The Leggett equations for the A and B phases Using (8.9b) and (3.68b) to determine Rp, the Leggett equations in the A phase take the form

i:ys xH*s""#lgxî¡g-î¡,

(8.3ea)

fi: yãx("- ?),

(8.3eb)

the preferred directions in spin and orbital spaces and f)o ^t" is the characteristic frequency of the A phase, defined in (8.20). It follows

where

ã

and

î

from (8.39) that S . d is a constant of the motion. The nonlinear spin dynamics of the B phase is more complicated than that of the A phase owing to the spin-orbit correlation of the order parameter d*(k): L,Rr¡ k¡. For most purposes, it is convenient to parametrne the rotation matrix by the unit vector ñ specifying the rotation axis and the rotation angle 0. Alternatively, one may employ a representation in terms of Euler angles. The rotation matrix in the axis/angle variables is given by (6.106). It is straightforward to write the Leggett equations (8.8a,b) in terms of ú and 0, using the general expression for the dipole torque valid for unitary states,

(Ro)*

> €¡"ut(AurAþu+ A"uAþt+ A"BA\B), - - ggo(r) ß],v

(8.40)

(6.8) and the relations below (6.106). The result is

i : yS x H * + A'"4sin 0(1, + 4 cos 0)fi,

(8.a1a)

8.3 Nonlinear NMR Phenomena in Uniþrm

Textures

349

.v2 e-'i.,s" x

.

1^/

ñ- - ;:tñ ¿)(-

(8.41b)

x S' + cotl? ñ x (ñx S,)1,

(8.a1c)

where .S' :,S - y-')(H is the deviation of S from instantaneous equilibrium. It should be noted that the dipole torque is along ñ, (B.4ta), and that the spin-orbit rotation angle is driven by the projection of ,s' on ñ, (8.41b).

8.3.2 Spin dynamics in zero magnetic field In the absence of a magnetic field there are no external forces acting on the spin system. If we neglect dissipation for the moment, this impties that the total energy is conserved, i.e. is a constant of motion. The spin motion is then in many \ryays similar to that of a spinning top in classical mechanics. We shall see that the present problem presents a nice example of the usefulness of differential geometry. It provides a simple mathematical framework within which it is easy to show that the spin dynamics in zero field of both phases can be solved exactly (Vollhardt 1984b). We shall focus attention on the motion of S, i.e. its space curye, when some parameter f is changed. Here t can be, for example, the time. There is, however, another parameter-the so-called "arclength" of a segment of the space curve-which is even more suitable for such a discussion and which leads to the so-called "natural parametrization". The arclength s(r) of a segment is given by (see e.g. Struik 1961)

s(r):

I,or'l#1,

(8.42)

-

and hence s(r) lSl. We can then take S to be a function of s(r), rather than of t itself, because the connection between s and r is given uniquety by (8.42). In this wây, the characterization of the space curve is considerably simplified. For example, the derivative of S with respect to s, denoted by ,S' dS/ds, i.e. the tangent vector to the space curve, is given by

:

,S'

:

s : s

s :isiso ,S' =

of

i

(8.a3a)

t'

(8.43b)

is automatically a unit vector. Furthermore, the second derivative

,S,

defines the "normal"

S":î'=no ns to the curve. It is directed

(8.44)

perpendicular to the

8 curve, i.e. to i (this is a consequence called the "curvature" of the curve:

ns:

of

lî12

-

Spin Dynamics

1). Its magnitude Insl is

Kû¡,

(8.4s)

r-r,

the radius of curvature, is the radius of the circle fitting the curve at that point. The plane passing through the tangent Î and the normal zs is called the osculating plane, because it fits perfectly to the curve at every point. A third vector can then be defined by where

6:îxño,

(8.46)

which is perpendicular to this plane and is called the "binormal". The three unit vectors form an orthonormal triad that is a new svstem of reference moving along the curve. Any change of 6 during this motion (which must be parallel to ño) describes a tilting of the osculating plane and defines the "torsion" z of the curve.

6'

:

(8.47)

-rño.

It is a measure of how strongly the curve winds out of the osculating plane. Using (8.43)-(8.47), one easily finds that r is given by T:

S' . (S" x,S'')

(8.48)

K,

The A phase

In the

case

of the A phase one obtains from (8.39a)

tño:

sin

dxl sin

, [i#

where 0 is the angle between

(8.4e)

g'

î x Qx s) + ¿],

(8.s0)

ã and i. fne binormal 6 is then found to be

6:îxfis:i,

(8.s1)

i.e. it is given by the preferred direction î. Let us

assume i to be time-independent, i.e. to be a constant vector in space. Then the torsion (8.48) of the curve described by S, being the measure of the change of 6, is identially zero.In other words, ,S moves in such yuy that the projection of Son 6:î is a constant of motion (because d(S." Ðtdt -S. 6':0),

S'Î- Cn: const. This is also evident from (8.39a), where S . l:0

(8.s2)

for H

:

0. Thus there exists

a constant vector

Ca: (S.i)i

(8.s3)


Textures

351

of magnitude Ca and direction i, which enables us to find an exact solution of the spin dynamics of the A phase for H: 0. For this, we make use of the vector identity

to

exDress S in cos one d which yields .

l:

I,

Sr: (S .î), + (S x î), (8.s4) terms of the angle 0 only. Taking the derivative of finds e: -ytx-t(S. î); furthennore, Sxî:Coâlsin0,

s,: l4)'r, \y" /

*l=q)' \sin 9/

(8.ss)

The total energy, as determined by (8.1) with H:0, is a constant of motion and may then also be written in terms of a single variable only: E e.

where e e:

:

#

(* . #*

d)'orin, o),

(8.s6)

Q'lx)cn. we are then lead to the equation of motion for g: ë

- e',-# +

äd:z',*sin

29

:0.

(8.s7)

The problem of solving the five coupled differential equations in (8.39) has thus been reduced to the solution of a single equation. Once thîs equation has been solved, every information about the motion of S and d can be obtained by elementary techniques. In qarticular,^S(r) is specified bV (i) its magnirude (8.55), (iÐ its projection on ô : î, . 6 : Ca, ^S and (iii) its projection into the plane perpendicular to 6, where it precesses with the angular velocity

.

9s: ld ldr ls

,Sxó

-

"a

dt2e.4 c,o

v-

(8.s8a)

cos' e

s-e"

(8.s8b)

The corresponding precession frequency of d around i is given by Ea: e olsinz 0. It is interesting to note that (8.56) and (8.57) are very similar to the corresponding expressions for a spinning top, where C^ represents the vertical component of the angular momentum, which is a constant of motion. Equation (S.57) describes a periodic motion of â (and therefore S) and can easily be solved analytically by an elliptic integral. The solution depends on the magnitude of Ca. In a turn-off experiment, when an external field Íl is switched off, Ca can be determined from (8.55), putting 0 :0. For this, the initial conflguration of ã and S relative to Î and thè external field ^tI (see Fig. 8.1) has to be determined first. (In fields H < d|oly the spin S is not aligned

8

352

Figure

8.1

Spin Dynamics

Configuration of ã and S relative to Î and a fixed external magnetic field

H.

along .ãr because the dipole coupling favours S r Í such that á is not perpendicular to.ãL) Note that once the system has reached equilibrium in the presence of the magnetic field, â and,s are stationary, lying in the plane of I and .tf. This initial condition results solely from the exclusion from (8.27) of relaxation effects that damp the motion of ã and,S, which were originally precessing under the influence of ÊL Minimizing the energy with respect to lSl and @0, the angle between S and i yields

:LIl

S

cos

where 7 is the angle between .õl and

tanZQo: When the field

ÊI

(þo-

(8.se)

X),

î, and Øo (:

Ln

-

0ù is determined by

sinZy cos2y

(8.60)

- ({2olyH)''

is suddenly switched off, ó¡, is given by e

where

e:

d2oz cos þo,

(8.61)

,:*cos(@s-,) g¿nand

(8.62)

o4: yH.

The resulting periodic motion of ã and s is determined by (g.57) and depends on whether z S 1. Starting from gs, 0 initially increãsès in both cases. However, for z L),

(8.63b)

8.3 Nonlinear NMR Phenomena in Uniþrm Textures

Figure

8'2 Precession of S about i

3s3

and simultaneous nutation, after turn-off of a

magnetic field FI >> eo/y.

where K(x) is the complete eiliptic integral. At z : the energy due to field ] turn-off is just sufficient to push â perpendicular to L It takes infinite time to attain this (metastable) position, i.e. ihe period is infinite.One expects that fot z - | relaxation effects become uery i-portant because of the long timescales involved. The resulting motion of the spin vector for yH >> fJA is shown schematically in Fig. 8.2: ,S precesses about i with an instantaneous angular velocity øs, (8.58b), while nutating at the same time with twice the frefiu"n"y of the 0 motion. This nutation is due to the finite spin-orbit coupting, which wants to have and. i perpendicular: the spin .s i, ,,o longer ^s a conserved quantity' As the projection of S on i and the total energy ãre constants of motion, the component of perpendicular to i will thðiefore change. Its ^S oscillation is a consequence of the periodic tunnelling of Cooper pairs between the two subsystems with spin projection ü : *-t (for a discussion see Section 8.3-3). Note that the precession frequency of S about t tr-Ñ; different from that of the 0 motión. It is mainty oetá*ined by the size of yHld¿" and the initial direction of H relative to î. rne two frequencies are generally unrelated. The B phase

In spite of the somewhat more complicated nature of the spin dynamics in phase, (8.41), the field-free case can also be solved^"*u"íly, B lhe as was first shown by Maki and Ebis awa (r976a) using an Euler-angle approach (see also Golo 1981b). within the present framãwork, we again study the

8

Spin Dynamics

space curve of S and determine the moving triad (î, ño, ¿î) UV taking the derivative with respect to the arclength s(r). V/e obtain from (8.a1a)

î-S':fi,

(8.6aa)

: ñg: $" -1. , .t

(8.64b)

or,-sñ-'i'ñ

(8.6ac)

¡Jô,

s-

where s : * ç*"Xy-' sin g (1 + 4 cos g). Interestingly, the (unit) tangent vector to S is given by the direction ñ. The binormal ó is then given by

b- ñxñ tñt

(8.6s)

Calculating the torsion z of the curve via (8.47) or (8.48), one finds that z is identically zero at all times. This implies that 6 :0, i.e. the binormal (8.65) is a constant vector in space, and hence the osculating plane is constant too! As mentioned before, zero torsion of S means that S rotates about 6 in such a manner that its projection on 6 is constant:

s.6:liÊxsl

sin]9

: Cs: const'

(8.66a)

(8.66b)

where we have used (8.41b) to obtain (8.66a). At the same time, û also rotates about 6 but with zero projection on n @ - 6 :0): its motion is planar, taking place in the plane perpendicular to 6. We have thus identified three constants of motion. They can be combined into a constant vector

Cs: (s.¡i)¿i. (8.67) This is essentially the constant vector ,4 (: -(y'lÐCs) introduced by Brinkman and Cross (1978), whose square Az is identical with the corresponding constant of motion found by Maki and Ebisawa (1976a). Making use of the vector identity (8.54) and of (8.66a), we can again express 52 in terms of one variable only-the rotation angle g:

*:(#)'(,'*(#)'

(8.68)

We note that the expressions .(8.55) and (8.68) for S are almost identical! The total energy may now be expressed in terms of 0 and 0 only, as E where e

s:

:

+lu' * ffi+

Q'l ÐC".

E aLG* cos Ð']'

(8.6e)

8.3 Nonlinear

NMR Phenomena in Uniþrm Textures

355

The equation of motion for 0 follows immediately as 0

aisin g(1 * 4cos e) :

- +ei#$ - +

o.

(8.70)

Again, the problem of having to solve a set of (here, six) differential equations has been reduced to the solution of a single one. Once this has

been done, .S can easily be determined as described below (8.57) in the case of the A phase. Here the frequency of precession of S about 6 is given by

es:tt*rTrr#, Y-uvB

(8.71a)

and that of É by

E^:!+. t0' 2sinz

(8.71b)

8.3.3 Nonlinear spin dynamics in the A phase In the presence of an external magnetic field the equations describing the spin dynamics of superfluid 'Ffe are considerably more complicated than those for the field-free case. So far, only certain limiting situations have been solved analytically (see Maki I975c). one such case is the purely longitudinal spin motion, s: (0, 0, s"). The vector á is then confined to lie in the (*,y) plane, /:(cosr/, sin qt,O).Taking i along the x axis, the minimum of the dipole energy is located at qr:0. Eliminating ,S,, one finds for rþ the equation of motion of a driven mathematical pendulum:

,i+*At, sin2p:-TÈ..

(8.72)

As suggested by Leggett (r975a), (8.72) may be considered as a Josephson oscillator equation. For this, we imagine the A phase to consist of two subsystems, the .spin-up and spin-down particles, described by order parameters .Ár * ei? and Á, o e-ie. The dipole interaction introduces a weak coupling between the subsystems, allowing pairs to tunnel from one to the other. The associated change in magnetic energy provides a restoring force for the pair current, giving rise to oscillations in the relative particle number tt.¡ - ny and hence the spin S, (see Maki and Tsuneto 1975). The "pendulum" may be excited by means of an impulse-a sudden change of magnitude of the magnetic field by LH. The subsequent "ringing" of magnetization is a special case (þo:ln) of the spin dynàmics in the field-free situation. The oscillations of ,S and, at half the frequencies, of. â, occurs at a frequency a4:ZnlT. where f, is given by (8.63á,b) with z: o4f d2^, a)L: y LH (Maki and Tsuneto L974b):

,:

[

4AÃL

K(arl

dZ

) (o\ < d2o),

l+o¡txloolo) (en< ør).

(8.73)

8

3s6

Spin Dynamics

3 LJr

QA

rectongulor covíty

l-TlTc=0.010

o

porollel plotes I

-TlTc=0.013

o

2

t:r a

t'tc . l

o I

012

¡ AH /Q4 Figure 8.3 Nonlinear parallel ringing in 3He-A at P:22.0bar in two different experimental geometries. (After Webb et al. (1975b).) Solid curve: theoretical results by Maki and Tsuneto (1974b).

In the limit ø--+0 the ringing frequency ø, tends to d2o, the longitudinal resonant frequency discussed in Section 8.2. For increasing or, i.e. larger amplitude of the pendulum oscillations, @, decreases until it approaches zero at o)L: Oa. This is the point where the impulse is just right to push the pendulum into its (unstable) maximum position. Beyond this point ú), increases with art, finally following the law o),--2e)yror ør>>l¿A. In the

pendulum picture the latter case corresponds to the rotating pendulum. This behaviour has been observed experimentally (Webb et aI. Ig75b, Ig77), except for the region close' to the "frequency dip" at oy: e,', where the system is very sensitive to relaxation effects as well as to perturbations such as walls and heat currents (see Fig. 8.3). A second case allowing for an analytic solution is the precession of the magnetuation about a strong static magnetic field ^Ho G.e. yHs>> OJ after a large RF pulse perpendicular to ãs has tipped the magnetization by an angle 0. The dipole interaction may then'be considered as a perturbation. In the frame of reference rotating with frequency THo= ø¡ about the direction of the magnetic field, the spin is a constant vector ,s' : s(sin g, 0, cos g) as long as the dipole interaction is neglected. The vector â in the rotatins

8.3

Nonlinear NMR Phenomena in Uniþrm Textures

Figure

8.4

357

Precession of the spin S and corresponding motion of the vector a spin-tipping pulse with 0:30".

d after

frame precesses about this fixed vector,S' as described by

:

d'

: -:-y2^d' x S'. )(

(8.74)

Solving (3.63), one finds â in the laboratory frame as

ã1t¡:

/ 1t -

[ \

.or

cos o) sin

ø¡r

cos @Lt\

* cosz ayt I, sin 0 sin ø¡r /

0 sin2 a4t

L

(8.7s)

representing a motion where d traverses a "figure-of-eight" shaped orbit on the surface of the unit sphere (see Fig. 8.4). The projection of the two branches onto the (x,y)-plane is a circle of radius sinzl?, given by the equation (8.76) xt + (y - cost Le)' : sina |9, which touches the unit sphere at the point i. He.e we have taken the vector as pointing along the y ãxis, such ttrãt it is perpendicular to both the static field IIo un¿ ttr" RF held that tipped the ma[netization. Substituting ã1t¡ from (8.75) into the expression for the dipole torque, and neglecting higher harmonics of the Larmor frequency ø¡-, which are small by factors dl¡la\-, one finds

Ro(r)

:#(1

+ 3 cos g)Sr(r).û.

(8.77)

8

358

Spin Dynamics

r T/Tc = 0.754 ¡ o T/ Tc = 0.854 ì zrzs oe o T/Tç = 0.930 ì v T/T" = 0.952, 466.2 Oe

I

3

o 3

00

300 600 900 1200 1500 1900

2100

Tipping ongle 0

Figure8.5 Frequency shift A,a@):o)t--ú)t- in 'He-A as a function of the pulse-tipping angìe 0. (After Osheroff and Corrucini (1975a).) Solid

curve: theroetical result by Brinkman and Smith (1975;) as given Uy (8.78).

The dipole torque Ro(r) gives rise to a frequency shift of the free-induction signal (Brinkman and Smith I975a) that depends on the tipping angle and is given by @

t:

ú)r-

*

#-rr* 3 cos

g).

(8.78)

For small tipping angles, this result reduces to the transverse resonanr frequency in the linear regime (8.2L), expanded in the high-field limit. As the tipping angle increases, the frequency shift decreases and eventually becomes negative beyond a critical angle gc: cos-t (-å) - 109o. This

behaviour is due to the fact that for increasing g, the vector d oscillates with larger and larger amplitude, thereby going through configurations where the dipole torque is weaker or even reverses sign. The frequency shift given by (8.78) has been observed experimentally by Osheroff and Corrucini (1975a,b) (see Fig. 8.5). As shown by Poluéktov (1984), the Leggett equations may also be solved analytically in the case when the vector I is aligned with a constant external magnetic field. Such a configuration may be realized in a parallel-ptate geometry (see Section 7.9). An alternative and in a way more systematic approach to the nonlinear spin dynamics of superfluid 3He has been developed by Fomin (Lg76, L978a,b,c, L979a,b, L980, I984a,b,1985a,b, L986, L9B7a,b) (see also Fomin et al. (L982) and the review by Fomin (1981b)). Many of the results for the


Textures

359

A phase discussed so far,

as well as those for the B phase to be discussed below, may be derived using this approach. It is based on the observation that the motion of the spin variables may be decomposed into a fast precession and a slow motion of the axis (or the angle) of precession. One may then average over the fast motion to obtain a much simpler set of equations for the slow variables. The theory has also been generalized to include spatial variations and relaxation effects. In particular, it has been shown (Fomin 1984b, 1985b) that even a small inhomogeneity of the magnetic field can have a strong influence on the precession of magnetization in the B phase. It leads to a splitting into two homogeneous domains in the case of a linear magnetic field gradient. While in one domain the magnetuation is at rest, it precesses uniformly at a relative angle 0r:104" in the other. This effect has been observed by Bun'kov et al. (I9B4a) and Borovik-Romanov et al. (1984b, 1935). The energetic degeneracy of the two

possible orientations of ñ is lifted by the motion of the domain wall (Podd'yakova and Fomin 1983). Instability of homogeneous precession of the magnetization is also expected in the A phase (Fomin 1979b, I9B4a, Fomin and Shopova 19g9). The'predicted time dependence in the linear stage of the development of this instability has been observed by Borovik-Romanov et at. (I9g4a). The nonlinear spin dynamics in the A1 phase has been discussed in detail by de vegvar (1984), employing the adiabatic merhod of Fomin. Nonlinear driven longitudinal spin resonance has been investigated both numerically (Katayama et al. 1980a,b,c, 1983b, ooiwa et al. 19g1, yamaguchi et al. L982, Yamaguchi 19S3) and within an analytical stability analysis (Fishman and Folk 1984a). It has been found that behaviour familiar from other nonlinear systems, such as stable and unstable periodic motion as well as chaotic motion, may occur (ugulava 1984, Montgomery 1985).

8.3.4 Nonlinear spin dynamics in the B phase Let us now discuss the solution of (8.41), describing the spin dynamics in the B phase in a finite magnetic field. Again we concentrate on a few simple cases. As in the A phase, there is parallel ringing after a sudden change in the static magnetic ñeld ^[Io, i.e. a longitudinal oscillation of ,S driven by the dipole potential. If the magnetic field determines the direction of ñ (ñ ll H), one can find the equation of motion for the angle 0 by taking the time derivative of (8.41b) and eliminating S, with rhe aid of (8.41c): (i +Ofr sin 0 (1, +4cos 0) (8.7e)

-

- -yH,.

For small oscillations about the minimum of the dipole energy at 0:0t: cos-t (-å), (8.79) reduces to the linear longitudinal resonãnce equation (8.10a). In contrast with the A phase, the dipole energy in the BW state has two inequivalent maxima at 0 :0 and 0 : n (see Fig. 6.6). Consequently,

8

360

Spin Dynamics

there will be two values of the change in magnetic field Â.F/ for which the i'inging frequency approaches zero. These values are obtained by requiring that the kinetic energy E*,,,: âx(LH)' provided by the field change bã equal to the difference in dipole energy between the minimum and

maximum_ positions: LEr:& {*"Qly')G* cos g)2. with this, one finds y LH - (?),'' Os and y LH (ã)r,rd2, (Maki and Hu I975a). For values of

:

LH

large compared with dàrly, "rotating-pendulum" behaviour with frequency @,:2T ÂÉl occurs as in the A phase. These predictions are in qualitative agreement with experiment (webb et al. L97sb, 1977). As a second example, we consider the "wall-pinned configuration", where ñ is oriented perpendicular to H under the influence of, say, closely spaced walls (Brinkman L974, Maki and Hu I975b; for other configurations see Maki and Ebisawa L975a). The equation of motion (g.70) for e, obtained for the field-free case, allows discussion of this more complicated situation. In a field-turn-off experiment, with ñ. H:0 and 0 : 0r initially, the energy term involving Õr in (8.69) must be equal to the kineiic gained through the field turn-off, i.e. el:ã0 LH),. For small "n"rgy frelds, y LH (( 9", the dipole potential dominates in (g.70), and we may put 0(t) - 9¡ for all times. In this case ,S is stationary and ñ precesses in the p^la.n: perpendicular to the axis Cs with frequency Ea(îr): |erlsin, ä0r: (?)''ty Â1L Note that this frequency is independent of tft" Oipãt" "ouplingS owing to the special initial condition. The precession of fi couples backto if initially 0 is not precisely equal to 0t, giving rise to a traniverse ringing signal. Wall-pinned ringing of magnetization has been observed experimentally (Webb et al. L975c, 1977). obviously, the ringing mode discussed above is completely distinct from the oscillations of the vector â about the minimum of the dipole well. It may rather be compared to the free precession of S and fi, subject to the constraint ñ . S :0. It appears as if the dipole interaction is irrelevant in the latter case because the solution does not explicitly depend on it. This is not so, however: the dipole force is actually responsible for the constraint fi . S:0. Finally, we consider the precession of the magnetination tipped by a large angle in a strong magnetic field, @L>> fJs. Under these conditions, the total spin S is approximately a constant vector in the frame rotating with the Larmor frequency, which we take to lie in the (*, ,) plane. It is convenient to work in a coordinate system in which the ¡ and z axes are along S H and S + É respectively, *ú"t" S and Ê are unit vectors in the direction of S anrJ H. The equations of motion for the spin-orbit rotation angle g and the vector ñ then take the forms (Brinkman and Smith 1975b): â

:Zøt-sin ()Q)ñ*,

ñ,: ,rf- cos (+Ðñ, * cot (jg) sin GÐG - ñï1, ñ': r,tyfcos (|Q)ñ, - cot (j0) sin (Lþ)ñ,ûrl, ñ,: -ørcot(|0) sin (tO)ñ'û'.

(8.80a) (S.80b) (8.80c) (8.80d)

8.3

Nonlinear NMR phenomena in Uniþrm Textures

361

on a rong timescale, we may expêct the physical

system to be in a state of 0L, an¿ t á :b. 1s.aou¡, this implies ñ,:0, and .onr"qul n l! "n"" : :0 ñ, from (g.ggc,d). The .,ñ, requiremenr rhar the right-hand síde of'(gróou¡ íanish ¿eterÀinl s ñ, and ñ, as ñr:cotlettan|ç and, û,:(1.- n?,)*. Sínce the dipole energy is a miminum for this solution, no dipolé' torq* i, exerteã unJ- h"n"" no frequency shift occurs. There is a problem, however: the expression for û, becomes greater than unity for tipping ángtes > e 0r. Thus the above solution is acceptable

minimum dipore energy, i.e. 0

:

l""-ñiro

only for þ < e;. For larger tipping angles, one hãs to explicitly take into account the existence of relaxation processes, which tend io driíe the spin-oìúi, angle 0 towards its equilibrium value g¡_. This may be ]1.9rporated by adding a term -@- 9v)/t to the right-hand side"of (g.gOaj. Her" relaxation time whose significance and magniìude "'is a spin will be discussed in Section 8'6' with this modification, ttre quas'istationary ,olution of (g.70) discussed above remains meaningful even in trre regime þ > 0r: the vector i may now escape into the x direction, while staying p"rp"naiã"iuì ,o the z

axis.

A solution is then found with û.,-:0, ñ,_(0 _ ïr)/(2ø.rsinrr@) and ñr:cot)çtanle. rne stationary varue }s'of tìre spin-orbit angle has to satisfu 0o< þ) is obtãined trom the normalization (which condition O?,- 1. Tire angle 06 increases with ar.z from 0o- 0r for .¡,rt(( 1 to ** 0o- for avr >> 1'

It should be ¡oted that in general this þ consistent with the initial assumptions of a time-independentsolution is not tipping angle. However, in the limit of weak ielaxation, when o,y1ÐL, one ðan negrect the relaxation of z component of the magnetization, because fi and -the therefore the dþole toryue aie para[el to trre yäis and hence perpendicu_ lar to the magnetic field. In this case the dipole gives rise to a transverse resonant shift of magnitude (Brinkman andlorque Smirh ISZSA¡ (D

L - Q)L:

15

ffo+ 4 cos @).

(8.81)

Such a frequency shift has indeed been observed (osheroff and Corruccini, 1975b), although there are deviations from the behaviour predicted by (9.stl near @ -- ø {gee Fig. s.6). Further nonlinear effecrs have been discussed by Liu and Brinkman (1973). It has been suggested that in rongítudinal driven NMR in the B phase, nonlinear phenomena such as perioJdoubling, intermittent chaos and crisis

should occur (Katayama ,t ot. r.9g3a,b, iã-ãg""t i Lgg3, Monrgomery 1985). Note, however, that driven longitudinar ,þin ,"ronance has in fact been shown to be unstable with respect to transverse r'svesserv'ù fluctuations (Fishman \ and Folk 1984b,19S5).

8

362

13

Ï/T" = Q.7¿'3 a T/ T" = Q.$93 o

3.0

NM

to

G

t

tf)

:9

Spin Dynamics

o

o a

2.0

Oa

=J 3

o

L

J

a o

1.0

t000

t400

lg00

2200

Tipping c¡ngte e Figure

8.6

Ao : at_ - ú)L as a function of the pulse-tipping angle 3He-8. (After Osheroff and Corrucini (1975b).) Solid curve: in

Frequency shift

0

theroetical result by Brinkman and Smith (1975b) as given by (8.81).

8.4

TEXTURE.INDUCED MAGNETIC RESONANCE PHENOMENA

So far, we have concentrated on the spin dynamics of uniform textures, in particular of those textures where the orientation of the preferred directions was determined by an external magnetic field or the dipole interaction. As discussed in Chapter 7, one of the salient features of superfluid 3He is the existence of textures, both smoothly varying and strongly localized, of the order-parameter field. Nuclear magnetic resonance experiments provide a powerful tool for studying the textures related to the spin part of the order parameters. Some of the simpler results will be discussed in this section. In calculating the NMR shifts induced by a given texture, one may distinguish between two limiting situations. If the spatial variation of the texture occurs on a scale Ç¡ long compared with the dipole coherence length 6o 5o (ìee Fig. 7.30) is planar, i.e. parallel to the (.r, z) plane. The detailed form of the texture has been determined approximately by a variational method (Bruinsma and Maki 1979a), i.e. assuming the form

sinB:ksn(q, lk),

(8.12s)

where þ(z) is the angle made by I with the surface normal, sn is a Jacobian elliptic function (Abramowitz and Stegun 1965) with ft : sin B(0), and 4 is a variational parameter; B(0) and 4 are related by the condition K(k) : g, where K(k) is a complete elliptic integral. , To determine the NMR shifts caused by the texture, we have to obtain the operators A/ and z\8 in (8.107). This is done by expanding (7.183) in terms of fluctuations / and g of the vector

/ : (sin (90 + g) cos (rt,o + Ð, sin (gs + g) sin (rþo + f), cos (06 + S)) g.I26a) (8.126b) - ão- G(f'+ s'), f, -s) around the equilibrium texture âo: i, i.e. gs : âfr, rþo:0. The operators

zV and z\8 are thus found as

: -Eïifrt * As : -ELt[f t * N

sin2

B)#]

sin2 B)

.

#] -

sinz B,

(8.127a)

cos2þ.

(8.127b)

a

detailed treatment, the qualitative behaviour of the frequency eigenvalues îu¿ and Â., (8.108b), may easily be understood: for large plate separation (L >> 6o), the NMR frequencies approach their bulk values, i.e. )r¿- )tr: L. As L is decreased, the eigenvalues decrease, too. At L- E, the i texture resembles the so-called "bending i soliton" (Maki and Kumar 1977d), for which ]rc:0.LL6 and Lt:0.687. For even smaller L (L L0 E(T), the Mermin-Ho (MH) texture, paramerrized by (j.lgÐ, is srable. only for very narrow cylinders (R < 10f) is a disgyration-type táxture ;it; (7.197) or (7.199), energetically favoured. Their NMR signals have been calculated by Bruinsma and Maki (1978), as will be discussed below (see also Fishman 1978). In the following we restrict the discussion to the MH texture in the dipole-locked regime, Tle corresponding bending free energy, in terms of the angle 2¿madeby î:áwittr the cylindér axis, ias deriv"oîn (7.r92).The exact solution X@) is given by (7.193), where p_ erlR. Hère r is the distance from the cylinder axis and u:[2](3 - St,1lrîn weak coupling. The azimuthal angle is _denoted by g. The operators zV and ¡\s in 1's.ro1¡ to, this case are given by

ôl z a2ì * ,, ¡r_ : -o\ ^.r/Eo\'l 1 al,. * cos', 1¡)\., sin' x n ) tffirta, L'tr ðr)* ifu¡

zr'

(8.128a)

Ap: -*\,/€o\'l1al + cos". ôl'2fa2 rì * zxl] 2¡)rJ + t: (8.128b) plfæ*cos R ) t; ; lr(1 For symmetry reasons, the longitudinal fluctuation / must be radially symmetric, whereas the transverse fluctuation dependr on q according tó g(p, q) : E(p)e*tt Tlr" boundary conditions on / ãnd g at the surface of the cylinder, where x:T, may be obtained by integraiirrg (g.107) over an Âã

infinitesimal interval about the surface, with the result that

gll,:^:gll,:^ :0. ðp

(8.12e)

êp

The lowest eigenvalue of the equation for

hc:

/ is obtained for / :

const, i.e. (8.130a)

L.

The lowest eigenvalue of the operator Â8 may be determined variationally and is found as

h,:

!. r(*)',

(8.130b)

with 4 : 4.14. The gorresponding values for the radial disgyratio n (7 .I97) and the "double-half' disgyration (7.r99) are also given uv-(s.130a,b) with

:5.56

and r¡ :2.44 respectively. While the preceding derivation holds for arbitrary values of folR because of the scale invariance of the MH texture, the assumption of dþole-locking has to be abandoned for EolR < 1 or for a layer of thickn"r, near the T

io

wall of the cylinder. One may estimate the latter contribution for a wide cylinder (R > 6") by noting that the maximum angle between i and ã at the

8

374

Spin Dynamics

surface is of order EolR. Therefore the dipole energy density in this layer is of order (E"ln¡' and the contribution to the eigenvalue obtained by integrating over the cross-section is of order (6"/R)'. In the opposite limit (R Er), the ñ texture is uniformly oriented along H over most of the sample. In this situation the solution of (8.176) is characterized by periodic oscillations of the magnetization. The detection of excited states in an NMR experiment is then hardly possible since the integrated magnetization averages to zero. By contrast, for smaller plate sepaiation (L< Eò, ñ!: (ñ . H)'varies quadratically with z (see (7.176b)). If one now approximates lRt*,lt in (8.176)) bV a constant then (8.176) corresponds to the equation for a harmonic oscillator. The eigenvalues are therefore evenly spaced. The intensity of the excited modes given by (8.174) decreases as ft-l. A detailed numerical calculation of both the texture and the eigenstates is found to give excellent agreement with experiment (Osherotr L977, Osheroff et al. 1977; see also the earlier work of Osheroff et al. 1975). This also shows that the qualitative treatment discussed above already includes the essential physics. The agreement may 3He-8. be taken as a proof that spin waves exist in Cylindrical geometry As a second example, let us discuss the NMR signal from the B phase in a long circular cylinder of radius R with a magnetic field along the cylinder axis. For R ) å,, the equilibrium texture in this geometry (see (7.182)) may be written as (8.178a) ñ+: 2-''t(ñ* i. iñy)

- 2-tt2 Sin B a*i(u+v), ño: ñr: cos É.

(8.178b) (8.178c)

follows that the effective attractive potential V^: -L+ ln^lt is axisymmetric and given by V+: -L+ lsinz B and Vo: -sinz þ respectively.

It

8

388

Spin Dynamics

Eigenstates of (8.167) in an axisymmetric potential can be classified by their azimuthal angular dependence e'tq.In the usual case of an axisymmetric modes are excited. RF field only the For the transverse case (m: +), the eigenvalue equation (8.169) reduces to the following ordinary linear second-order differential equation:

/:0

(-R

+

R' tR'*,t')ELt:#

('#)] t,*,*-

(1

- å sin2 þ),t,*,o:

E*,t 1þ*,r, (8.17e)

where RL*.: û *Rtî (Hakonen and Volovik 1982, Maki and Nakahara L983, Fetter and Theodorakis L983, Jacobsen and Smith 1983). The potential -1 + ] sin2 B has a quadratic dependence on r near the centre, with very small curvature, and rises to its boundary value of -t only within a surface layer of thickness f¡7

u,,

*

8",¡Bì,+

6,v . s.), *

Dr(>BÌ,)'*

(D,

Eï(v . s"),

-D.) ). Bu.B?,. ß.34\

9

4L6

Hydrodynamic Theory

where B"¿þ

:å ) i

(n.,V,øÞ

t

RB¡Y,a).

Positivity of R requires

(u) the transport coefficients K, Ez, Es,

(b)

(.) 9.3

r¡,

v

and D, to be positive;

Ere'> E?; Dr> lDrl; Dz) -â(D, + D).

HYDRODYNAMIC EOUATIONS FOR THE

(e.3s) (e.36)

A PHASE

The hydrodynamics of the A phase has been a source of considerable controversy. It is only recently that a consistent picture has begun to emerge. Following the first derivation of the linearized hydrodynamic equations by straightforward application of the rules laid down in Section 9.1 (Graham 1974, Graham and Pleiner 1975, 1976), several amendments and refinements have been proposed (Liu 1975 , 1976, 1977, Lg7B, volovik and Mineev I976c, 1981, Liu and Cross L979, Mermin and Ho 1976). rn particular, the role of an intrinsic orbital angular-momentum density associated with the Cooper pairs has attracted much interest (see also Section 7.3). This is partly because microscopic derivations of the hydrodynamic equations revealed the existence of terms that could be interpreted as being due to an intrinsic angular momentum (Combescot and Dombre 1980, Combescot L978b,1980a, Mermin and Muzikar 1980, Nagai 1980, Ishikawa et al. 1980). While microscopic theories are of importance in providing quantitative information about the coefficients in the hydrodynamic equations, they are usually model-dependent. The microscopic approach consequently cannot be expected to provide the most general form of the hydrodynamic equations admitted by symmetry. Following Miyake et al. (1981), Liu (1985) and Hall and Hook (1986) (see also Hu and Saslow L977, Ho 1977, Lhuillier 1977, Nagai 1981a,b, Sonin 1,984b), we shall therefore attempt to derive the hydrodynamics of the A phase phenomenologically, assuming this state to possess an intrinsic orbital angular momentum of density t(r, t). Note that an intrinsic angular momentum is necessarily of quantum-mechanical origin. Alternative but similar derivations of the hydrodynamics of 3He-A have been considered by Khalatnikov and Lebedev (r977a, L977b, 1980), Lebedev and Khalatnikov (1977) and Gu (I982a, b). Early discussions of the orbital dynamics of 3He-A include those of Hall (L975, L976, 1977,1978), Hall'and Hook (1977), pleiner (r977b) and Brand and Pleiner (1981a). A hydrodynamic description of 3He-A in terms of quantum fields, where in addition to the ordinary (bosonic) variables anticommuting (fermionic) variables are also introduced ("superhydrodynamics"), has been formulated by Andreev and Kagan (1987).

9.3 Hydrodynamic

Equations

for

the

A

phase

4t

9.3.1 The concept of an intrinsic angular momentum What are the changes in the hydrodynamic equations if there is a finit, angular-momentum density r? As mentioned in -Section 9.1, in the of an intrinsic angular momentum the conservation of orbital absenc, angula momentum Z does not give rise to an additional independent conservatiol law. In fact, the angular-momentum conservation law can be derived fron the momentum conservation law by employing the definition of the orbita angular-momentum density L: r xg. Note that angular momentum is onl, conserved if the stress tensor is symmetric. In the presence of an intrinsii angular momentum, the angular-momentum density with respect to anr

given centre is given by

L:rxg+f.

(e.37"

Here r is defined to be the intrinsic part of L, i.e. r is independent of the size of the system, whereas the position-dependent part r x g scales with the size R of the system as R2. The thermodynamic equilibrium state of such a system is found by maximizing the entropy as given in (9.a). The resulting siate is characterized by a uniform temperature Z, chemical potential ¡t, angular velocity ll and uniform linear velocity t¿. However, because of the appearance of g in the expression for L, (9.37), the total velocity ttn obtained by taking the derivative of e with respect to g at fixed f is nonuniform. In fact, it is given by a superposition of linear and rotational motion:

un:u*l|xr.

(e.38)

Equation (9.38) may be used to express the angular velocity in terms of the velocity un as S2:TyXu,,. (e.3e) This relation carries over to a local-equilibrium situation, i.e. into the hydrodynamic regime. It implies that lt is not an independent variable. Furthermore, the thermodynamic force y,e¡ is a second-order spatial derivative of un¡, and may only give rise to dissipative terms of higher order in the gradients. Hence additional transport coèfficients connecteã with the internal angular momentum do not appear. There are, however, ne\ry reactive terms in the momentum conservation equation. Since r x g is no longer a conserved angular-momentum density, the stress tensor is no longer symmetric. Rather, a term matching the antisymmetric part of the stress tensor appears in the equation of motion for t, so that the conservation of L: r x g + t is guaianteed. It is no\ry convenient to define a new momentum density j (Martin et al. 1972) by

j-g+lyxt, (9.40) yillr the pr-operties (Ð Í dt, j : ! d3r g up ro surface rerms and (iÐ I dt, r xj : I dt, (, x g+ r). sin.e r xjõun u" seen to be identical with the

9

418

Hydrodynamíc Theory

angular-momentum density L, the stress tensor in the equation of motion for j must be symmetric. This property facilitates the task of collecting all possible terms in the stress tensor allowed by symmetry.

The intrinsic angular momentum

r partly originates from the orbital

angular momentum of a Cooper pair, which in the A phase is directed along the vector i. this part, f6, points along î, i.e.

ts:

tsl.

(e.41)

The angular-momentum density fe is a nonhydrodynamical variable, and may be viewed in analogy to the superfluid density p", for example. To understand the significance of fo, we consider a uniformly rotating container filled with 3He-4. In equitibrium the normal fluid will move rigidly with the walls of the container. There will be a change in the free energy proportional to the angular velocity là. For a large system, this change will be dominated by the macroscopic contribution to the angularmomentum density, , x g, which scales with the size of the system as R2 x volume. However, there will also be a change in free energy induced by the intrinsic angular momentum. It has not so far been possible to experimentally determine ús (but see Hall L985, Liu 1985). For finite vorticity g¿, there is also an induced intrinsic angular momentum

tin¿:0d2,

(e.42)

where 0 is a moment-of-inertia tensor, which has to be positive for stability reasons. The total intrinsic angular momentum is the sum of the two contributions (9.4L) and p.a\:

t:

to

*

f¡r,¿.

(e.43)

The induced angular momentum f¡,,¿ should be expected to exist even in the normal phase, but its value would be rather small u¡tr.- (ò"í.). k Here

u*:hklm*

(10.14)

is the normal-state quasiparticle velocity, and

ôn[:

ôn¿

* QLõeo,

(10.15a)

where

þL:*(h,r) t¡:

(10.1sb)

(E*l2k"T). Note that, owing to Galilean invariance, the momentum density is equal

and

tanh

to the bare mass times the current density:

g:

mj.

(10.16)

The equality of (10.16) and (10.11c) is guaranteed by the effecrive-mass relation (2.16), which continues to hold even in the superfluid state. In analogy, the spin current density jfl, the momentum current density II,, and the energy current density j" are found to be given by

ii,:

+k2 rofto (o* õnL),

(10.17a)

k

II,¡:Z hk,u¡,,,tr, (ôri),

(10.17b)

.Ps'r Je:-$t

(10.17c)

k

) Ç¡uptro (ô¿j.).

10 Transport

Properties

Evaluating the /s sum in the usual wây, namely by taking integrals over the energy f¿ and the angular variables, the spin current, for example, may be written as

ii,: änG + åFî) 2k ,o õn*,u.

(10.18a)

Here we have made use of the integrat

ll_-otreL: I__*r(H,r*

ffi#)

:,

(10 18b)

and introduced the notation

õn*,p: tto (ou ôn¿).

(10.18c)

The last term of (10.10) has in fact already been expressed in terms of the spin current via (10.18a).

10.1.3 Kinetic equation for Bogoliubov quasiparticles As already mentioned, the matrix kinetic equation may be simplified in the low-frequency macroscopic regime, where the pair condensate is in a local equilibrium state. In this regime transitions between the normal liquid and the condensate rarely occur. The Bogoliubov quasiparticles (BQP) may then be considered to be stable entities. Under these circumstances, a description in terms of a distribution function for BQP rather than by a matrix distribution function appears promising. This can be achieved by performing a particular Bogoliubov transformation on the matrix distribution function. In the present matrix notation the Bogoliubov transformation is defined as the unitary transformation diagonalizing the energy matrix: Qt'oeooQtf: E*ao,

(10.1e)

oll*: ( uo*' 'o'u),

(10.20)

where the components of otl¡,

\-u*nþ u*np/

have been calculated in (3.25). There we found

*knþ:

u"rl:çt*fi))''',

rrknþ:

,"*,1:(-*)l'"

(t0.ZLa) (10.21b)

The Bogoliubov transformation of the equilibrium distribution function is

10.1 Kinetic Equntions immediately obtained

443 as

oìrrltooolri

: f (E¡r^\o'-\ : ( f {nùt

o_

. ), o f(-Eùr)' _

(r0.22)

f @) is the Fermi function. We are now in a position to diagonalize the matrices multiplying ò.ff and ôe in the matrix kinetic equation. For this, we multiply (10.6) iro¡¡ the left by oVo* and from the right by oùIi_, which yields the result ot õ.f¡,- E**toô,in + õ.i¡,rsÛt _+ f (toÛo) õ8o- aÉo¡@o\o_) - aìr**õ.çkoui_. (10.23) Here ô.ri¿ and õ8e are the Bogoliubov-transformed matrix distribution where

function and energy change respectively. They are defined by õirk

:

õ8 k

:

k

* õt'k oùtl- = (y;;:, _u:Ë:_u),

$0.Zaa)

¡

* õe¡, etl - -

(::;:

(r0.24b)

ott

oìt

An expansion of (10.23) in small q leads new (diagonal!) distribution ôf¡:

_

3ft:'u)

to a Boltzmann equation of the

- (q . VeÊÉXôfÉ -/i ôEo) - (q õg 0lr*)**, (10.25a) where a suffix ++, +- etc. denotes components in particle-hole space; i.e. + + is the first diagonal element while + - is the upper off-diagonal ar ôf¿

element, etc. Furthermore,

rr:ß:9!o

: -!

êEo

ðto

(10.2sb)

zaÛk'

Y*E*:+ry. L*ffl'

(t0.25c)

In (10.25c) a contribution proportional to VoÂr has been neglected, since it is order T"lTi small. Note thatY¡E,.r, the group velocity of Bep, vanishes at thç Fermi energy E*:0. In (10.25a) the change in the Bep energy is

found

as

ôEo: tt/ ' t, 6o\ oÇ*' "'. - +(t+) -\ Ek/ :-ê *) lakl, ',

.

+,(ôÂo

^f

+

ò^f).

ôEre

^; (10.26)

ê As we shall see later, ôI9¿ is related to ôf¿ in a more complicated way than in normal Fermi-liquid theory, because ôÇ¿ and ôA¿ are defined in terms of ôn¿ and ôF¿ rather than ôf¿ and ôh¿ appearing in (I0.2a{. Indeed, the kinetic equation for the new off-diagonal distribution function ôh¿ is found to take quite a different form: (ot - 2E*) ôho - ,e ôf& : (A õg aU)*_. (10.27)

M4

10 Transport

Properties

For frequencies a¡ much less than the average gap frequency Alh (and also for collisi,irn rates r-r < Alh), (L0.27) may be solved for ôh¿ to give

ôho:

-hôro+

o(qr,r-r).

(10.28)

Equation (10.28) must be interpreted as the local equilibrium value of öho. The new off-diagonal energy change is calculated as

ôrÉ:

-(ôËo Â*

*

^*

ôEIft)

.E#ae(ô^;

h-#4(ô^¿

Âr

-r-

ê

- af

ôae)

a; ô^fr) + o(q').

(r0.2e)

In order to close the system of equations, one also needs to express the old distribution functions ôn¿ and òF¿, entering the expressions foi ôÇ¡ and ôA¿, in terms of the new ones. This is easily obtainèd from the reverse Bogoliubov transformation. One finds ôno

- å(r **)

u,,.

- +(t - Ðh^frôrre^';

- *-r(laol'ôEo

. ôFo

&E

+

orô^;

ðErfr

^É

+ ô^ft

:*zL*fuço * * L*ôflo) . +Li .

_

^f)

^;)

+ o(q),

(10.30a)

_Âo * Âr ôEro) - z,k 8,, uo-.

&,(ôEo

hft

(1o.3ob)

In principle, one would now have to eliminate ôh¿ via (10.28) in favour of ôf¿ and ôA¿ in all expressions. This is a rather cumbersome procedure, because it turns out that all the expressions have to be evaluated at least to order q. After all, we have to remember that the hydrodynamic variables associated with the order parameter are given by gradients of the angles specifying the rotation or the gauge of the gap parameter and not uy ttre

angles themselves.

Fortunately, there is a way to circumvent this laborious task. The clue is that the change in the order parameter in the hydrodynamic limit is describable in terms of gauge transformations of the ordei parameter. By performing the gauge transformations directly on the matrix distribution function, one is able to generate the terms coupling the condensate to the quasiparticle system in a natural way. The price that one has to pay for this is that one loses the equation of motion for the order parameter. This, however, is easily remedied: as we have seen befot", hydrodynamic "u"iy ôn the other variable of the condensate corresponds to a broken symmetry. hand, every one of these symmetries gives rise to a conservation law in which the particular hydrodynamic variable enters. Thus, by enforcing the conservation law, one is able to fix the value of the hydrodynamic variable.

10.1 Kinetic

Equations

Let us demonstrate this procedure for one particular example-the phase of the order parameter.

10.1.4 Gauge transformation of the kinetic equation A

change in the phase of the gap parameter

;

A*:

eiõaA1o

(10.31)

is accomplished by a unitary transformation of the energy matrix

t¡,:

aìlçeekoìli

:

ef,

+ ôef;,

(10.32a)

where

%o: ôef,:

exp (jit6 ò@),

(10.32b)

ô0 (roeå -

(10.32c)

åi

efuo).

Here the superscript r@ indicates the fixed gauge. Because a phase change ô@ in .Á¿ carries over to the nondiagonal distribution function F¿ as well, the matrix distribution function also transforms as

J{*: %otrl%i.

(10.33)

we may now choose the

phase change ô@(¿ r) at any point in space and time such that ee has constant phase. In other words, we may "gauge away" the phase. For spatially varying phase õþ(r, t), or equivalently for its Fourier transform õQ@, ø), this is accomplished by substituting õJ{o(q,

ø)

: õ$lþ"tîfllï) : ô,llry + åi ôø (.oJ{oo_- Jfoo*r"o)

(10.34)

into the matrix kinetic equation (10.6). After rearranging terms, the kinetic equation for the gauge-invariant distribution function ô"Áffl is found to have essentially the same form as before. The only difference is due to the appearance of a number of additional terms in the change in energy: ôe¿+ ôef;':

q:

ôeo

- iiha ôó to + +i õþ (r?*"o - toe?_). (10.35)

0, the last term is exactly equal to the change in õe under a change of phase by -ô4 as described by (I0.32c). (Note that (10.34) is equivalent to a phase transformation of ôJf* by -ô4.) We therefore define the gauge-invariant energy change by For

ôe[:

ôeo

- ôeão'

(10.36)

Furthermore, we introduce the notation

ôfr: )ika

õE,

'":ft4 õQ'

(10.37a) (10.37b)

446

n

Transport properties

which is borrowed from the phenomenological theory that relates the time derivative of the phase to the chemical potential ¡r and the gradient of the phase to the superfluid velocity u" (see (g.g) and (9.16)). Nõte that in this chapter ¡r is the chemical potentiat per particle. This identification can be verified later. Expansion of the last term in (10.35) to first order in q gives an expression for the change in the energy matrix in the gauge-invariant formulation:

ôef,"': ôrå - ôp ro +

+

h(k . u,)I-.

(10.38)

The degrees of freedom of the order parameter that relate to rotations in spin space or orbital space may be treated in an analogous way (see Combescot I974a). For example, the rotation of the order paiameter in spin space can be compensated by a non-Abelian gauge transformation ur: exp $i Eyo*ôji):.where ô0Ì, are the rotation ãngles in spin space and ou are the three Pauli matrices. This allows an interpretation of tire time and the space derivatives of ôg) as an effective -ugtt"ii" freld HE,

ttoHru: $iha

õ0"u,

(10.3ea)

and a spin superfluid velocity, u"p,r,

:

fi

õ0"u, - iq ou;r zmry

(10.3eb)

respectively. In this way, one generates additional gauge-dependent terms in the energy change, ôelni"

:)

p

[-p oH"*t* + + h(k . u"p,¡")trrsl,

(10.40)

which have to be added to the right-hand side of (10.3g). It remains to express the observable quantities, i.e. the densities and currents, in terms of the gauge-invariant distribution function ôJVï. For this, we first rewrite the diagonal elemenrs of ô"Áf* in terms of ô.fi 1seË 1to.s+¡;:'

õft*np:

õnek,p+

*lô*ur"+ ì

(or)

*ur,o,r].

a*þL.

(10.41)

Equation (10.41) is then inserted into the expressions for the conserved densities. one finds that in the equations (10. L'l.a,b,d) for ôr, and ôe the distribution function ôn¿ is simply replaced by ôn[. Í{o*"u"r,{, ?to.1Lc) for the momentum density acquires an additional term such that ks

(t0.42)

The momentum density g is henee seen to be given by the sum of the momentum density in the rest frame of the superfluid, àescribed by ònfl, and the contribution from a Galilean transformation to the system with

u"*

0.

L0.1 Kinetic

Equations

Since the energy change induced by the Fermi-liquid interaction involves the ungauged distribution function ôn¿, this part contains gauge-dependent terms too. Inserting (10.41) into the definition of ôe¿, (10.7), and combining all the gauge terms, one finds for the diagonal part of ôeli',

ôEil':

) [ttt, tr, (ônfl,) +2 fî,*,o*tro (o" o"t.)] -

ô¡r

l-

po}

n"*ou +

h(k. u,)l + yD [email protected],o,")ou.

r-"

ffi"

(10.43)

tt

The effect of replacing the ungauged distribution function by the gauged one in (10.43) is to remove the factor mlm* in front of /r . u, (in compliance with Galilean invariance) and to renormalize the prefactor of the spin supercurrent term (last term effective mass by

in (10.43)). Here we have defined a "spin"

'ffi,

_1 + åF'ì

m

(10.44)

I++Fî'

The Bogoliubov-transformed kinetic equation for the gauge-transformed distribution function ôffl still has the form (10.25a). However, the change in the BQP energy ôE[ is now given by (10.26) with ôÇ¿ replaced by ôÇf;'.

10.1.5 Gauge-invariant densities and currents

In order to

calculate the transport properties from the kinetic equation (I0.25a), we need to know how to express the currents in the hydrodynamic equations in terms of the gauge-invariant BQP distribution function ôffl. For this, we have to gauge-transform (10.30), which amounts to replacing òn¿ by ônfl and ôf¿ by ôffl as well as ôe¿ by ôeli'. Then we insert this gaugetransformed equation as well as (10.41) into the expressions (10.11)(10.18). In this wây, we find for the particle-number density ôn --

4r,"l|"r-;*tal'oet]

+(#)"rr, (10.45a)

where

(*9o)":à#.taot''

(10.4sb)

As is apparent from (10.45a), the quantity (ðnlðu)op relates a change in the chemical potential to a change in density at fixed BQP distribution (i.e. ôf¿ :0) and in the absence of Fermi-liquid interaction effects. Thus it only describes a part of the total density response and is therefore proportional to the so-called "bare partial compressibility" r$. An equivalent (and very

10 Transport Properties

448

useful) form of (10.a5) is given by ôr¿

:

(ôrfl

# [tr' }Etr,

-/; ôEå) * òr],

(10.46)

where ðnlðp, is the normal-state compressibility (see (2.19)). The momentum density g is obtained from its definition (10.11c) by using (10.18b), (10.30) and (10.41):

g:>

hk fto (ôfÐ + pu".

(10.47)

Similarly, the momentum current density is found to be given by

¡¡,,:\

kk,uo,""l*,(ôrå

-/; un;)1 * 6,¡n õ¡t,

(10.48)

where (10.18b) has again been used. The energy density and the energy current density are obtained ôe

:) k

E¡,tro (òfå)

* ¡t õn,

as

(10.4e)

: ) uo$¡ tro (òfå + QL õEò + I "1, k-m

1

:_

jo *

(10.50a)

ttg

1

*lrg.

Note that in a system with strict particle-hole symmetry the term involving ôf¿ is zero. The first term on the right-hand side of (10.50a) may be interpreted as the heat current. It consists of a sum of a convective and a diffusive term (10.sOb) Zsu,, * 2 roEotr, [òffl + f L(hk. u")1],

jo:

k

where u,, is the local drift velocity of BQP, i.e. the normal-fluid velocity. The corresponding expressions for the spin density S and the spin current jo are slightly more complicated because of the spin structure of the order parameter. By inserting (10.30) into the definition (10.1lb), one finds

^ S*

: h=-(

e.

*" õf o, " -

tk lAolt-

-

I

- ttoH"") ],

(10. sl ) ^p" where a notation analogous to (10.18c) has been employed. In the presence of an external magnetic freld Hs" is replacedby He"+ Hi'. Here s¿,"., is the BQP spin weight factor, which specifies the contribution to the spin component S, of a BQP with momentum /s and spin along the v direction. It is given by

ià

It

o,

r;t4

[ô",

(fr

)](ô E o,,

ç,9,

sk,t"u:

where

*.D¡,u*,* (r\ - *)n*"(o), Lp/

(I0.sza)

A*"(k) is the projecto¡

A*"(k):di(E')d"!k)

d*.d

'

(10.s2b)

10.1 Kinetíc Equations

449

In deriving (L0.52a), use has been made of the relation

:6ffi

o lo *L,e(o")'Âfl

:

6*u

-

zLpu.

(10.52c)

In analogy to the partial compressibility r$, we may define a Cooper-pair partial spin-susceptibility tensor by

)i"?:

r\\hffi[ô""

- A*"(k)|.

(10.s3)

Equation (10.51) allows one to calculate higher-order Fermi-liquid corrections to the thermodynamic spin susceptibility, i.e. those beyond the usual molecular-field correction considered in (10.10). To this end, õfo,* is replaced by its local-equilibrium form

õf*,u:

f

(10.s4)

LõEo,u,

where, using (I0.26), the BQP energy change is derived

õE*,p:

)

ro,,,,,(ô5o,"

-

as

(10.ss)

ltoH,).

Here Hu: H",:- + Hs", with.ðr'*t the apptied external magnetic field. In the B phase only the I :2 Fermi-liquid correction enters besides the / : 0 one, owing to the fact that only the weight factor s¿,"" and the projector A*"(k) have an angular dependence. The resulting spin susceptibility is given by (Czerwonko 1967)

)("

-*: rr

Í,o 1,

+ Fôio'

where

i,o:? + âyo+ &Fi!

¡t

:

"1,,2,3,

- yò'

(10.56a)

(1o.s6b)

and Y6 is the Yosida function defined in (3.101c). The spin current j'u, defrned in (10.17a), may be written in terms of the gauge-invariant distribution function as

ii::r-**lo

-,* õr L" -

htaot'

n*(ulo,"

+

.

+

n* ' ,,,.,)] (10.s7)

"rt'sou"p'¡''

In (10.57) we have defined pk,¡"u, the contribution of a BQP of spin component v and momentum /c to the spin current with spin orientation ¡,1 given by P*,p": Ur" -

(t -*)n.".o''

(10.s8)

450

1.0 Transport Properties

In local equilibrium the BQP distribution function is again given by (10.5a); however, this time only the part of õE*,p that is odd in k contributes, i.e.

õE*,t":) po,*u(ut*," + + n* . ,"o,,).

(10.59)

This may be substituted into (10.57). With the aid of (10.18b), it yields the local-equilibrium expression for the spin current:

i'í"' : #ì [t - år'r ,fror]"0åî,u"p,,, = )

pfl"u"o,".

(10.60)

Here the spin superfluid density in the absence of Fermi-liquid corrections is given by

o.

-

ys(Ê; r)l[ôu, A,"(Ê)l] (10.61) þ'Ð:1"":3so(Ê,Ê¡lr Note that j"*J* is equal to the spin supercurrentjlp defined in (7.16).

10.2 COLLISION

¡NTEGRAL

10.2.1 Matrix operator for binary collision processes The time rate of change of the matrix distribution function caused by binary quasiparticle collision processes is given by the golden-rule expression involving the scattering amplitude Z:

Ili*': - +: Ji L k2k3k4

"ri......1ri

,r,rt,y4:tr

x (ft3s1, kqs1lT lk,,s'r, kzsLl(2n)oô'(?'0,) U(?'

y,Eo,)

(

x{ô"",[Nli,i",NLi,t",(g'o:-#'o1,)"rt@i:,-Jüi)"*,^

- @ii- ttol),¡,@i,:-

" + (t,."í \s'ësJ)Ì)

JtlT)"*,NI1,,","rN1i,"4s¿l

$0.62)

To simplify the notation, the particle-hole index p and spin index ø have been combined into a single 4-component index s : (p, ø). Since the components of the distribution function belonging to the energy eigenvalues *Eeand -E¡have to be treâted separately, an additional index y: tl has been introduced in (10.62). Only in this way can we take energy conservation during collision into account, as expressed by the delta function with argument Ei y,Eo,: TtE*r* yzE*r- TsE*r- y¿E*o.This generalization leads to a further technical complication, since \rye now have to introduce a

10.2 Collision Integral quantity

451

9f,

which is the probability matrix for the state l/c) (with energy TE*) to be occupied or empty. This is not equal to ,rnity, since the quasiparticle may be in state lfr) energy +Ee or oi ,o,rrr", the gf, adds up to unity -Ek. sum of the spectral weight factors TF Erby: 7. The conservation of momentum is incorporated by it r O"ti" function with argument E'¡k¿:kr+kz-k=-ko.of course, the collisions also conserve

the total particle number and spin. This implies that the scattering amplitude (frrrr, kzszlT lk3s3, kqs+) is nonzero only for p1 * pz: ptlpq

p1u1* pzuz: pz&z* p¿uq, where p, is the partiôle-hoirirrá"" and ø¡ is the spin index of state |fr,s,). The scattering amplitude represents the sum of interaction processes involving momentum transfer over the entire Fermi surface. The pair correlations, on the other hand, are effective only in a small region of momentum space near the Fermi surface. Hence to order ntr; the scattering amplitude I is given by the scattering amplitude r in the normal state. More precisely, the components of z may be iãentified with and

: T----: t(I,2;3, 4), I T*-*-: T-*-+ : t(1, T****

-4;3, -Z), I T*--*: T-++-: t(L, _3; -2, 4), )

where the subscripts *, - refer to the particle-hole indices p,, and we have used the abbreviations L =(kr, ut), etc.'The factor 1/3! -2-(-kr, -ez), in (L0.62) accounrs for the fact that all 3! peimutaiions of (2,3,4) in rhe sums in (10.62) are equivalent and should be-counted only on"". Finally, the matrix structure of Jft introduces an asymmetry, for exámple into the first term in the curly brackets in (10.62), which is removed bv taking half of the symmetric combination (for a derivation from microscópic theory, see Wölfle L976a)- The collision integral has been investigated, in particular in the limits T-7. and T+0, by perhick et al. (Ig75,lgle, Ig77i. Next, we derive explicit expressions for the lineanzed colliiion integral

!-ø in (10.6) using (10.62). The equilibrium distribution function *16, describing the distribution of quasiparticles with energy rE*, must be proportional to f(yEù. It is then easy to decompose the complete equilibrium distribution function tîoe, (10.3a), into componenrs

J{2:

Q0.6aa)

Y:*T

with

rüo:)(t.#)r*uu. The linearization of

(10.64b)

Jf[ is done about the equilibrium value J{f,o:

JtI:

Jt"oo

+

õJtI.

on the other hand, the distribution of

(10.65)

quasiholes with energy yE¡, is

452


proportional to

L-ffuEo) s:o

and hence

- rt': +(r .#),

- r@Eùt.

(10.66a)

From (10.64b) and (10.66a), the equilibrium spectral weight function gf,o is found as

eto

: +(t *#),

r: lrt.

(10.66b)

a projector, i.e. (glo)r:g|o, and Øfo and g¡o are ît orthogonal, 9[o9¿o:0. Upo{ substituting the equilibrium distribution functions ,Á[f,o, (10.64b), and 9f , (10.66b), the collision integral (10.62) is found to be zero. This follows from the fact that, owing to energy conservation, the combinations of Fermi factors in the out- and in-scattering parts, frfr(r-.ÊXl - fo) and, (1 -,txl - fr)rtfo, have exactly the same value. cãñs"q"enuy, the two parts of the collision integral are found to cancel each othet. ihir is true not only in global equilibrium, but also for any local equilibrium distribution functions, i.e. distribution functions of the form 1rb.o+u¡ and (10.66a), where eo and E¿ depend on space and time. The linearized collision integral takes the form Note that 9Io

ô1¿1"",

: - +: 3!2

kTkok^

x (krsä,

""i;r, ,,,r,,y4:rt

kqs,al

T lk6,1, krsl)(zn)ao3(), t) U,(?, y,E n,)

xf î,if ",¿(r-f î))e-f i,:l{rii?¡,p\1f"*,p"o1!,"f x [ôóIl"i"

,+ õe?i"t",- õþ8"*t-

l^^?^,t

)],

-F I ".*"í õþIi","il '4ù4t \s, *sr/J'

(10.67)

where ô@ is a combination of linearized distribution functions,

ôøL,

: lflT - fÐPL,,l-'[ôNL

f[:f(yE*)

,

- fi"ôpL.

-

pk,,f Ly ôE*1, (10.68)

is the Fermi funcrion. The last term in (10.6g), which is proportional to the change in ener9y õE*, comes from the linearization of the energy variables in the energy delta function in (L0.62). Because the collision integral is zero in local equilibriuffi, â change in energy òE¿ in the delta function is equivalent to a . change -ôE¿ in the õrresponding equilibrium distribution function. Tþiq leads to the additional changè in thã distribution function given by rhe lä$fi,terms of (10.6g). The linearued collision integral (10.67) is seen to act on four distribution functions, õJü and õ9'[,, T : x.L, instead of the one function ô"Áf¿ we started from originally. However, these functions are not independent. ihis may be and

L0.2 Collision Integral

453

seen by looking at the- kinetic equations for these four functions, which are obtained by decomposing the kinetic equatiol(ro.-o) for ô,Áf¡ irrto it, parts. To do this, one simply repraces $ooby rnepdua,ior* for

ï: i1 ô.fl , are then given ttr". kinetic equations for ô,Áf¿ *itn ¡rl ,"pr"."¿ or-*f !v. and the full collision integral trfff

replaced by the cõmpo.r"rri- Oy?. m order to derive the equation for õgl, one starts from the mãtrix kinetiË equation for the quasihole distribution function ô gi- õJti. This -equatiol is gü"r, by the normal kinetic equation with replaced by yf,o-¡1o. TÃe collision 'f[ the one in integral has the opposite sign from ttre quasiiarticle equation because adding a quasiparticle to a quasiparticle siate'i. eqrriualent to removing a quasihole from the same state. The sum of the two kinetic equations for quasiparticles and quasiholes with energy yE¡ yierds the kinetic equation for õ9!,:

a õ9f,-

rfr*

õØ[+ õØf,e¡_:

ô€*

gil- g[lõeo.

(10.69)

Note that here the collision integral has dropped out. On inserting Øf,o into (10.69), the right-hand side of (10.69) is seêñ ro be proportional

to

hence

õgI:

T

õ9*.

y,

and,

(10.70)

From the kinetic equations for õ.lt|, ô"Ál¿ and õgl, it is straightforward to derive õJf[in terms of ô"Áf¿ and õg¡ alone:

õ"ifI:

+(g|_ôJfo + õ"r.fk gl,_) + ty(õg¡ J{** * .if k_ õgk). (10.71) It follows that only two of the four distribution functions,.q.rutìparticles ô"â and õg¡, arc independent. While ô"Ál* describes the distribution of and cooper pairs in momentum space, corresponding to both branches xE¡, of the excitation spectrum, the additional function ôØo determines the relative spectral weight of the two excitation branches. When substituting Q0.7L) into the collision integral (10.67) the small momentum changes *lq due to the external field b" neglecíed; as they only introduce correction terms of order elke into "un the collisi,on integral.

10.2.2 Gonservation properties of the collision integral The conservation of mass, momentum, energy and spin is reflected in corresponding integral properties of ô9. one finds that for any distribution

function ôJA,

k,v--+l k,v--+.1

(10.72a)

(r0.72b) (10.72c)

k,v:+L k'v:tL

(10.72d)


454

10.2.3 Gollision integral for Bogoliubov quas¡particles In the macroscopic regime, i.e. when the condensate is in local equilibrium, the matrix collision integral (10.67) assumes a simple form. Let us first perform the Bogoliubov transformation (10.20) on the distribution function På",ô0L, appearing in the linearized collision integral (10.67). The result is

pä", ôoä"

,-vi!

- f'o)l-':2 ((t:.]]u*\ ôU -(1

un1

- ), - y)õt-¡l'

(10.73)

where òfr,

:

ôfk

-/;

-

:

ôho

+;:

ðho

ôEÉ, tr-

LL*

òfe.

$0.7aa) (10.74b)

However, according to the discussion below (L0.27), ôh¿ may be put equal to zero in the macroscopic regime, so that the transformed matrix distribution function PL",ô@l becomes diagonal. Also, only one of the diagonal elements is nonzero for given y. In deriving (10.73), small terms of order q and ø have been neglected and the solution of (10.69) for ôØf (more precisely its Bogoliubov transform), i.e.

õØk:å('lt has been employed.

u;-)

.o(ry,i)

(10.7s)

equation (10.67) for the distribution function of Bogoliubov quasiparticles may now be written as the sum of an out-scattering and an in-scattering contribution (Einzel and Wölfle 1978, F;inzel 1984):

The collision integral

of the scalar kinetic

(10.76)

Here the Bogoliubov quasiparticle relaxation rate is obtained from the Bogoliubov transformation of the term proportional to ôQ¿ in (10.67). Employing the fact that at low T all momenta lie on the Fermi surface, the energy and angular dependences may be separated (see the discussion for the normal state after (2,35)). Owing to the anisotropy of the condensate, the relaxation rate is also anisotropic in general, i.e. it depends on the vector kt. Furthermore, the orientation of the momentum vectors is specified by three angles instead of two, as in the normal phase: (i) the angle 9 enclosed by ft1 and k2, (ii) the angle @ between the planes spanned by (ftr, k) and (kr, ko), and (iii) the angle @2 describing rotations about lc1.

10.2 Collísion Integral

455

The result of these reformulations ir (6, = E*,, E¡= E*,, f,=fo,)

Inf* ;,: t*"(l x

+ e-E*/k"\Jo dErdEroío

*,à,*. :*1

(ô(E' * pzEz- þEEs-

fr! - f)G - fòA(kr, kz, kz, kùl ".

þ¿Eq)

(10.77a)

Here we have defined a generalized transition probability by Q(kr, kr, kr, ko) : W - Wo"[Di . Dt* Di . Dol - W@i . DÐ(Dr. Dq) + zw@i . Dr)(Di . Dò + 2w4(Di . Dò@i . Dr), (10.77b)

where the quantities ry are combinations of normal-state quasiparticle scattering amplitudes given by (2.39) and (2.49) and Wou(o, e)

:

(Ao

+ A')(Bo + B t) + 4At Br .

(10.78)

The quantities At(e, @) and Bt(e, Q) are the singler (S:0) and triplet (S:1) components of the quasiparticle scattering amplitude introduced in chapter 2 (see oot!l'*r"(krar,

kquo),

kzeú kz&', Npa(ktut, -k4- ugk3d3,

B:

-kz- uz),

(10.79a)

(10.79b)

where e¡: LI are spin indices. The vector quantity D¡ is the normalized equilibrium gap parameter,

Dj:

þ¡d(k¡)lE*,, and the angular brackets in (I0.77a) denote the average,

(r2)"-/'o{"o,;øf

*"

Xf

Dj in

(10.80)

(10.81)

(10.77b) come from processes in which a quasiparticle transforms into a hole by condensing into a Cooper pair (or vice versa). The quasiparticle or quasihole character of a momentum label k¡ may therefore be different for either of the two scattering amplitudes, which leads to the asymmetric combinations in (10.78). The in-scattering part ôÍi" has a very similar structure. We only give the result for the spin-symmetric part:

The terms involving

ôi:1,'"

: iå*xf xrrfr(L

f* Jo

-

otrd|rdEo fr)(L

r,2,*,(õ(Er* :*1

pzEz- þzEz- pzEs-

trqEq)

- rò[a(kt, kz, h, k4)(õQ¡ * ô0o - ô4r)]),, (10.82a)

where fJ is defined in (I0.77b). The distribution function appears in the combination

ôó, - r! :

L

(L

z

+

p¡)

õf*,ro,(L

(1'

- p) õf -*

- r*,)

'' þ¡: tL'

(10.82b)

10

Trønsport Properties

Equation (10.82a) is valid in the case that ò/¿ is an even function of f¡. In the opposite case the quantity in square brackets has to be replaced by Eo

u'1,ÍW(õV¡

with

lrtz,

* ô?o -

õtþr)

-

Wou(Di . Dò

ôr/rl

(10.83a)

Wo" and D, deñned in (2.39) and (10.78) and

s

¡rrl.

¡

1.

Ç*, -,

(l\ + 'F) ò/o, + (t -

't zgkj --

p¡) õl-.o,

fo,(L-fo,)

(10.83b)

The complete collision integral for the spin-antisymmetric case has been derived by Einzel (1984). The collision integral (10.76) does not conserve Bogoliubov quasiparticle number and spin. This is due to the fact that Bep may be gènerated or annihilated during a collision process. Ffowever, momentum and energy are still conserved. This gives rise to the following integral properties of ôi:

)k ft tt, (ôio¡ :

g,

(10.8aa)

2k notr, (ôio¡ :

g.

(10.84b)

At the same time, the collision integral functions of the form

õf'fåB: f L(anr - hk.

un

vanishes

for any distribution

- urf)u,u,

(10.8s)

where un and ôI must be interpreted as the local velocity and a change in local temperature of the BQP. This implies that collisions do not alter a distribution of the form (10.35). Consequently, the distribution function (10.85) must correspond to a local-equilibrium state of the Bep system.

10.2.4 Relaxation-time approximation The collision integral for Bogoliubov quasiparticles derived in Section L0.2.3 is complicated enough to defy any general solution. It is therefore necessary

to introduce an approximation that simplifies the collision operator sufficiently to allow an exact solution for a large class of transport problems. At the same time, this approximation must be able to describe the essential physical properties of the system under consideration. This task is achieved by the following relaxation-time approximation (RTA): (10.86a)

10.2 Collision Integral Here the sum on cr: s, â extends 9ver, spin-symmetric and -antisymmetric components, and the sum on r--f(-1)': f 1 runs over even (f :+1) and odd (r - -1) functions in 6. For the spin-symmetric case the projectors p are defined by

pi^,(k,

k,):

OiG)

|f

x Qi(E')

,rr,* ro.u. yrL]ll> rLf]'"

(t0.eza)

460

L0

The mean collision time relaxation rate Í

^"

2...,

:{

Transport Properties

will be defined by the inverse of the

average

ll }r, r] ll? rt] }-',

(10.e2b)

so that the mean free path is given by

l:

(10.e2c)

uuurû.

This yietds a low-temperature limiting behaviour

-,_(zn)''' (J

12

3 \^ö)

1

(10.e3)

*[u'zR'(r,,)1""'''

The normalized mean free path is seen to increase exponentially for T+0 (see Fig. 10.1). It should be noted that this normalized mean free path l(T)l\D depends only weakly on pressure. It is of the order of centimetres for T :0.2T.In this temperature regime, and even more so at still lower T, the collisions of quasiparticles with the wall of the container dominate the collisions between the quasiparticles. This is the so-called "Knudsen r(r)

m 101

0.2 Figure 10.1. Mean free path temperature.

0.t, /(f)

0.6

0.9 T/Tc

10

of Bogoliubov quasiparticles in the B phase versus

10.3

Transport

Cofficienß

46I

regime", or "ballistic quasiparticle regime", some aspects of which will be considered in Section 1-0.4. The A phase

of the BQP

in the A

phase is more not separate in this case. At low tempÞratures, where the A phase may exist in the presence of a magnetic field, only quasiparticles with momenta in the neighbourhood of the two nodes of the gap, i.e. along tÎ, where Ak^ o*po,*"k(+Fî#so'pir

+

+ a"^t)ø(ff1^,,t.Ì

: ôÍ{ôri},

(10.100a)

where the tensor Vis defined by (see (9.65)) V¡

:

äi(q¿uni

*

Qiun¡

-

36,¡9' u").

(10.100b)

Equation (10.100a) is an inhomogeneous integral equation for ôfi, which cannot be solved in general terms because of the complexity of the collision operator in a superfluid Fermi liquid. Exact results can be found for some of the transport coefficients close to T. and in the limit of low temperatures (Pethick et al. 1975, L976, 1977, Bhattacharrya et al. 1977b, Einzel L984). This is due to the fact that for some of the transport quantities the leading correction term to the normal-state result just below 4 is of kinematic origin and may be calculated from the normal-state kinetic equation with the BQP group velocity inserted in place of the normal QP velocity. At low T the back-scattering part of the collision integral, i.e. the one involving integrations, vanishes in some cases more rapidly than the out-scattering part, so that the simple relaxation time approximation becomes exact. In general, the integral equation (10.100) can only be solved

L0


approximately, for example by assuming the kernel to be separable (Wölfle and Einzel L978, E,irløel1984) or by employing a variational method (Ono e/ al. 1977, 1982, Hara et al. L980, Ono and Hara 198L, Hara 1981). For a somewhat different formulation based on the Green's-function technique, see Houghton and Maki (1977) and Dörfle et al. (1980).

10.3.1 Transport coefficients of the B phase

In the B

phase the task of solving (10.100a) is greatly simplified by the isotropy of the energy spectrum. This allows one to separate the dependences on energy and angles, as in the normal state. The mathematical problem may then be reduced to the solution of integral equations in the energy variable. The angular dependence is easily obtained from the left-hand side in the kinetic equation (10.100a) for each particular case.

Shear viscosity

In response to shear flow, characterized by q¿un¡*O, i+j, the spinsymmetric distribution function, hereinafter denoted by ô/i, assumes an angular dependence (Y eE)rkr. Separating out this behaviour as well as the main dependence on energy, ô/i may be written in terms of a dimensionless function ,lrr(E): õf

L: i )ij "R/i(Y ¡E¡)¡k¡q,u,¡úr(E*).

(10.101)

substituting this into the collision integral (10.76), (10.82a), (10.87a), Vn is found to satisfy the integral equation

ú,(8,): r(ç,)[1 .

I: dE,&(8,, Ð(Z)',Þ,G,)],

(10.102a)

i(E): t*ldu

(10.102b)

; is the BQP relaxation time (10.87a) divided by the normal-state relaxation i';'tate z$, (2.1S). The integral operator t in (10.102a) is a special case of a more general operator S, (i.e.the one for l:2):

ü(å" E):

)'ì*Jo(Er,

E)

- üz(&, E),

(10.103)

which is determined by the functions.Is and J2 defined in (10.87b) and by the parameters Â|* and y¡ introduced in (2.51)-(2.53) and (10.88a) respectivety. This follows from specializing (10.83a) to the B phase. To proceed, we equate the off-diagonal components of the stress tensor (10.48) and their hydrodynamic expression (9.28): ¡7,i

:2 k

F onò,k¡

õf

L: -iîle¡uni,

i + j.

(10.104)

10.i Transport Substituting

Cofficien*

469

for ô/í. from

(10.101), the following expression for th

shear-viscosity coefficient is found: T

:

rfl,

2 F onn)?k'kiçf L)rp,Gù: ,r(#rþ,)

,.

4o: !nm*u!"R,

(10.105a

(10.10sb

and we have deûned the energy average

(# r,),= l-

d'o Gr Ð

f;r,re-t'

(10.105c

Note that (10.105a) does not depend on i or j for symmetry reasons. In tht immediate vicinity of 7: the temperature variation of 4 is dominated by thr change in the quasiparticle excitation spectrum, which introduces a factor EIE into the expression for th_e group velocity (10.25c) (Shumeiko 1972) The ensuing factor of (Ela¡', for example in the energy average ir (10.105a), gives rise to a square-root dependence of the change of in thr 4 superfluid state: qQ) - qQ) * - (?: - T)'''. This may be seen from the limiting behaviour of the f integral over the expression

IxÉ):Hà

t#*)

:r -naõ(Ç)

(10.106)

if we recall that Á *(7.- T¡rrz for r+ Zi. The integral equation (10.102a) for rþr(5) -ay be solved exactly to first order in A, using the eigenfunctions of the transport problem in the normal state, as discussed in Section 2.3 (Pethick et al. 1976). In fact, the only correction linear in A (and hence

nonanalytic in the "natural variable" Az) comes from replacing the singular term ETln3in (10.102a) by (10.106). The relaxarion time t(Ç2)ãs welt as rhe kernel sr(Er, Sr) possess analytic expansions in a', as may be seen by

inspection, and

to order A may therefore be approximated by their

normal-state expressions. Introducing a function

þ(x) where

x:

(n'+

:

)

V,(Ç)

sechz

(10.107)

)x,

ElksT, the integral equation (10.102a) takes the form x')Q@)

- )tr[ dx' K(x, x')e@,) :

sech2

]x

- nt 21*r*,0)O(0) (10.108)

(here and in the following we write )u2as shorthand for Â)*). To first order in a, the quantity @(0) is approximated well by its Á :0 value. using the eigenfunction expansion (2.6r) of the integral kernel K(x, x'), (10.10g) is easily solved. The result is conveniently expressed as

ô(x'l:

ón@\ + ôó(x).

110.109a)

L0

466


where Qou)

¿odd

J

is the solution in the normal state and

: l_ ut1', dx

f v-

xrz

(10.10ec)

eo(o)loo(o).

The shear-viscosity coefñcient is obtained by inserting (10.109) into (10.105a), using the relation (10.106):

rt: r¡.t-!

nrÍoo(o)l'

#.

o(a'),

(10.110a)

where TN is the normal-state result (2.66). The quantity @s(0) may be obtained as a rapidly converging series whose first few terms are

oo(o):L*(r.i&.äh;

)

(10.110b)

Turning now to the limit of low temperatures, we observe that the integral term in (10.102a) vanishes somewhat faster than the term on the left-hand side owing to the extra factor of EïlEïæTlA in the limit T+0. For T lT.(( 1, we may therefore neglect the integral term. The function r/ is then given by

,l'r$):

t(5)[1 + oQ lr)1.

(10.111)

Substituting this into (10.105a) and using the low-temperature result for t given by (10.90), one finds a remarkable result: the viscosity coefficient tends to a finite value as I+ 0 (Pethick et al. L975, Einzel 1980, 1984):

fot rtrl

: cnl(rl)(t * o,T),

(I0.LLza)

where

2n I ks\12 1 c': T L^(o)l wof,(hy

(10.112b)

un:4u"

(I0.ILzc)

W

and Ws, fr(Lr) and u" have been defined by (10.90b), (2.68) and (10.90c) respectively. Estimates of the coefficient cn using realistic values of Á(0), I4zs and h2 yield values of 0.1-0.2, depending on pressure. The behaviour of q at intermediate temperatures may be calculated either by solving the integral equätion (10.102a) approximately using a variational method, or by approximating the integral kernel in (10. L02a) by a separable

10.3

Transport Coeficients

467

kernel as given by (10.86), which then allows an exact solution of the integral equation. In the present problem the symmetry is specifred by I 2 and r: *L (even function in 5), and the problem is symmetric in spin space. The integral kernel in (10.I02a) may therefore be approximated by

slto(6r, Er.): lr,L t(Er) (1,1-r(Ë))e -t(E)'

(10.113)

where Z(f) is the normalized relaxation time given by (10.102b) and the energy average ( )e ir defrned by (10.105c). The integral equation (10. L}Za) with the separable kernel (10.113) is readily solved. The viscosity coefficient calculated from (10.105) using this solution is then obtained as

,:

rt?z

(E'lE')'e

\

r,L\þ-r(Ð)

(10.114) -r(Ð) E+ ^z (U e - ),r((Ez l E1l i> El' Note that the result (10.114) becomes exact in the low-temperature limit and yields the correct behaviour near 7" as given by (10.110), with @s(0) approximated by the first two terms in the parentheses on the right-hand side of (10.110b). A numerical evaluation of (10.114) with parameter values appropriate for pressures of 0, I0,20 and 30 bar but neglecting surface effects is shown in Fig. 10.2. The initial drop in 4 below 7" is well described theoretically, whereas the low-temperature plateau is not borne out by experiment. Qualitatively, the reason for this can easily be understood: it is due to the deviations from the hydrodynamic behaviour associated with the increasing mean free path at low 7- The latter may eventually become larger than the characteristic dimension of the sample used in the experiment. Consequently, scattering processes of the quasiparticles at the wall become more important than those between the quasiparticles. This effect is not included in the theory described above and will be further discussed at the end of this chapter.

Bulk viscosity The hydrodynamic equations for the B phase derived in Chapter 9 contain three coefficients of the bulk (or "second") viscosity: Er, Er. and (3. These are dissipative coefficients multiplying V . u,, and V . g" in the expressions for the stress tensor and the dissipative part of the chemical potential (9.28) and (9.23). On the other hand, the trace of the dissipative part of the stress tensor is obtained from (10.48) by inserting the deviation from local equilibrium of the distribution function: uf r* tDi n3: å >ka k. uo* E¡,

:

-e ty. g, -

nm õ¡P

Ery . u,.

(10.11s)

L0

468


0¿ rl (T) rl(Tc)

03

o9o fo$

oor

0.1 0.2 Figure

0.4

0

6

0.8 l trc 1.0

10.2 Shear-viscosity coefficient q(T) for tHe-B versus temperature at pressures P:0, 10, 20,30 bar as calculated from (10.114) (after

Einzel 1984); neglecting surface effects. Experimental values are from vibrating-wire measurements of Carless et al. (L983a,b): V, 0.L bar; A, 9.89 bar; O, 19.89 bar; E, 29.34bar.

The dissipative part of the chemical potential, i.e. ô¡,lo: ôpt"" - ôpto', where ô,,rt"" is the local equilibrium value, can be determined with the aid of (10.46), which expresses the density õn in terms of ô¡r and ô/*. By definition, the local-equilibrium value of the density change ôF¿ro'is equal to the exact value ôn, which implies that the dissipative part on the right-hand side of (10.46) has to vanish, and hence öpo

: - 1¡r.'^ >+ õf L: - 6rV . un - 6rV .g.. m ãE*

(10.116)

Here the factor Ilm accounts for the fact that ô¡rD is the chemical potential per unit mass. It is clear that the two terms in (L0.115) may be combined into a single expression:

+> nÌ i

: ! nZ (k . uo ku

keue)

*L¡, uf'r

(10.117)

10.3 Transport Cofficients Now the quantity in parentheses vanishes at the Fermi energy. Therefore only particle-hole-asymmetric terms contribute to the viscosity coefficients 6r and Ç2. Since each particle-hole-asymmetric term is order-ft/ft,small and since one needs two of these to get a nonvanishing result for the energy integration in (10.1-Ll), e, and (2 are at least order-([ l|*)'smaller than, for example, the shear viscosity,. The coefficient Ç2 multiplying V'u' acquires yet another factor of (?:/f.)'from the driving term in (10.100a), which is of the same structure as the term multiplying ô/i in (10.117) (Shumeiko 1972). The remaining coefficient Ez describes the friction arising from a normal-superfluid counterflow with nonvanishing divergence, which exerts a force on the superflow. It is again convenient to split off the dependences contained in the driving term by defining

ò/l

:

*ff

tr*,q'

(10.118)

s"t'e(E*).

The dimensionless function ?E is found to satisfy the integral equation

,þeG): ?(å,)[1

.I

d5, so(6,

,t'rfrrÞeG,)f,

(10.11e)

which is rather similar to the corresponding equation (10.I02a) in the case of the shear viscosity except that the kernel is now given by ,S0 (see (10.103)). From the experience with the shear-viscosity calculation, we expect to be able to calculate (3 exactly near T" and for T+0. The general result for Ët is obtained by substituting (10.119) into (10.116):

E,: es(#úr(6))u

(10.120a)

Here

tS:#urz$:l#r,

(10.120b)

sets the scale and the angular brackets denote the energy average defined by

(10.105c).

In the limit Á+0 there is again a nonanalytic contribution from

the factor of ETIEZ in (10.119). Replacing 7 and So by their normal-state limits and introducing a new function

Q@): where

(n'

x:

Elk"T,

jrr<e)

sechz

(r0.Lzt)

)x,

one arrives at the following integral equation:

+ x')e@) - a.' K(x, x)lr [

n

#

urr'rlOf''l

:

sechz

lx.

(10-122)


4t0

However, this time one is not allowed to treat the term proportional to Á in perturbation theory, because the coefficient Âð* appearing in the definition (10.103) of 56 is unity and is thus equal to the eigenvalue kt- L of the first of the eigenfunctions tþ"(x) introduced in (2.56). This simply expresses the fact that particle number is conserved in the normal state (Âå* : L !) but ceases to be conserved in the superfluid state, where the number of quasiparticles is no longer equal to the number of atoms in the system. As one approaches 7: from below, the coefficient of the eigenfunction It r(x)

:#

sech2

lx

(10.123)

in the expansion of @(x) in terms of eigenfunctions ú"(x) diverges as .Á-1. The leading term in the bulk viscosity (3 is easily found by employing the eigenfunction representation (2.6I) of the kernel K(x, x'); it is given by Wölfle I977b)

E,: ESI:*T."o,,,t]

as

r+

r"-

(r0.124)

In other words, (3 diverges as (1 -f lf)-t/z as T+ [. The corresponding term in the Josephson equation (9.22), -6rV'g", still vanishes as it should,

:

-

because g, pr(u. u") approaches zero as A2, and therefore overcompensates the divergence of (3. In the opposite limit of low temperatures the integral term in (10.119)

again may be neglected compared with the out-scattering terrh. For the same reason as discussed in the context of shear viscosity, Ç, is found to tend to a constant:

E,(D:

ceeS(r")(r

*

"rT

as

r+ o,

(10.125a)

where

,e

:

2n 1. /k"T"\'

T

\^(0)/ yo &E:4U, - 1I4r,

,

w,

(10.12sb)

(10.125c)

At intermediate temperatures we may approximate the integral kernel in (10.119) by the relaxation-time expression (10.113) with L3*:\. The resulting expression for (3 is found to be

s,:ce[(

#rrt):ffi]

(10.126)

Equation (10.126) tends to the correct limiting values for T+ 7" and T + 0. In Fig. L0.3 the result of a numerical evaluation of Ç3 according to (10.126) is shown versus temperature.

10.3 Transport Cofficients

47t

.=t(¡l

î qt

800

1¡l

E

l¿

(v)

lJr

600

/.00

o oo o

200

o

"

0

Figure

0.8

o.?

o

o o

0.9 1 t Îc

l.o

10.3 Second viscosity coefficient CtQ) for 'He-B versus temperature

at as calculated from (10.126) (after Einzel 1984). Experimental values (O) are from the vibrating-wire measurements at "1..28 bar of Carless et aI. (1983a,b).

P:0

The coefficient Ç3 is not easily accessible experimentally. In principle, it determines the attenuation of fourth sound, as will be shown in Section tL.L, but in practice this contribution is very small compared with the one from mean-free-path effects. It has recently become possible to extract f3 from vibrating-wire experiments (Carless et al. 1983a,b). A comparison of the data with theory is shown in Fig. 10.3. Thermal conductivity We begin by splitting off the ft dependence of the driving term in (10.100a) from the distribution function: õf

L:izfl./Í.

Z+

@o-

q)ôr,/.(5).

(10.127)

It can be seen that ô/i is odd in f as in the case of 4 and (3, but is also odd in k, in contrast with the previous cases. The function qr"(Ð is found to satisfy the integral equation

ú-(E): r(F,)[1 .#,Il o;,sf-)(6,,

tU#,,Þ.G)f,

(10.128)

where one factor of E-L has now been shifted outside the integral, owing to

1.0 Transport Properties the extra factor of E in (10.127). T.aking into account the oddness of ò/¿ with respect to k+ -k, the kernel sÍ-) is obtained from (10.s2) and (10.g3) AS

sÍ-)(5,,

E):

Âì-/5-)(6 r, Er)

- yrrs)Gr, Er).

(10.129)

Here /f-) is defined by (10.87b) with, however, an additional factor of I,tr. in the Ír2 sum. The parameters Âi- and yr are given by (2.52) and (10.8ga) respectively. Inserting (L0.127) into the expression (10.50b) for the heat current and comparing with (9.33), one obtains the thermal conductivity as

r:

rs(r)

(10.130a)

@h(E'rt,*)e,

with

ro(z)

: åCN(r)ufr$(r).

(10.130b)

The thermal conductivity varies smoothly near [, because the factor E7ln3 (giving rise to the nonanalytic behaviour of 4 and fr) has been replaced by the nonsingular term E3lnr. Nevertheless, the slope of r changes discontinuously at T". In the limit of low temperature-s the integral term in (10.128) is again negligible owing to the factor 87. The various factors of T from the

relaxation time 7 and the energy averages (accidentally) happen to combine to a T-r law as in the normal state. The complete result is

¡r(r) :c*xs(rl(r *

""T)

as z->Q,

(10.131a)

where

21 n

ü*:4ut

(10.131b)

W6'

1"__Zy,

-l -'Hu,

(10.131c)

with l4ro given by (10.90b) The relaxation-time approximation for the kernel SÍ-) should take into account the fact that {-l1Er, E) vanishes for S1 : 0 and Ez:0 in the normal state. This property is readily incorporated by introducing factors E, and E, such that

sÍ-)(Er, 6r) l*ro: åÂì-

h

E2

(Ez liG))

e

¡G)'

(10.132)

With this approximation, excellent results are obtained in the normal state.

10.3

Transport Coefficients

473

1.2

l-lro l-l¡-'

t'r-=

o.s

Ylx 0.8

0.1

o'ob

Figure

'l l'lc l0

3He-B versus temperature at P :20 bar according to (10.133) using Âi- values from 0.5 to 2.O (after Einzel 1984). Experimental data points (x) are from heat-pulse measurements at 20bar by Wellard (1982) (see Hall and Hook 1936).

10.4 The scaled thermal conductivity r(T) of

For the B phase, a simple calculation yields

K(r) : *,

@f.ul<e'rfe>)e

+

åÂì-

l (10.133)

At low temperatures

(10.L33) reduces to the exact behaviour (10.131). A numerical evaluation of (10.133) shows that Trc(T) is roughly constant (see Fig. 10.4). This is essentially borne out by experiment. However, at low temperatures the experimental curve (Wellard 1982, Wellard et al. 1982) falls below the theoretical results. This may again be caused by finite-meanfree-path effects. Additional scattering processes involving phonons and, at high pressure, fluctuations of the solid phase have also been considered but have been found to be small (Geilikman and Chechetkin 1975). Spin diffusion

The spin-diffusion coeffrcient describes the dissipative spin current, which is driven by a gradient in @p, the deviation of the spin density from its equilibrium value (see (9.29) and (9.30)). Therefore the last term in (10.100a) is the relevant driving term. For simplicity, we assume the Landau parameter Fi to be equal to zero. When we discussed the spin density and spin current density in (10.51) and (10.57), we already noticed that the coherence factors here are rather complicated. Indeed, the spinantisymmetric component of the collision integral is so complex that we

10

474


it here. A complete treatment has been given by Einzel (1980, 1981). Here we only consider the simplest relaxation-time approximation, where the in-scattering part is completely neglected. The shall not attempt to discuss

distribution function is then given by (see combescot 1975c) s

. Í¿oI r¡l L,p- i'--:! L qÍL@o. q)(o*)*Bp*,¡"u{D"n.

,.,

õf

(10.134)

Inserting this into the definition of the dissipative spin current,

(i"-t)"

: ä nZ u*rp*,¡"utr, (o" ôfi) kv

(10.13s) vl

one finds the spin-diffusion coefficient )

D,,;u; : fr 2 \. t - f L), ¡,u -trtvt

x-x,

¡¡(pz¡,)

*ú *.

(10.136)

u¡ing the definition (10.58) of p*,¡"u and the projector property of A2: A, one has (p,o)

*"

:

õ *u

-

4 #R*,Ê,R,¡Ê¡,

A¡,u,

(10.137)

whgre &, is the rotation matrix characterizing the B phase. As anticipated in (9.31), it follows that the diffusion tensor consists of three independent components: D,"ru¡: D1õ,"u6i¡

+

DzRpiR"¡

*

D3Ru¡Ru¡,

(10.138a)

with (10.138b)

Dz: Dz:

-+oo(ifetfi)r,

Do:åu!"|n Z& x

.

(10.138c)

(10.138d)

we already know that the factors E'ln'produce nonanalytic dependences proportional to A near [. Hence D.r3 vanishes as Á for T - 7., whereas D, tends to the normal-state coefficient Ð ". Just below T, D, falls, which is similar to the behaviour of the viscosity r¡, but the effect is smaller here, owing to the factor of ! in front of E'/8,. In the limit of low temperatures the spin-diffusion coefficients of (10.136)

10.3 Transport Cofficienß

475

are found as

Dt:

-4Dr,:#D"#;ffi

(10.13e)

since Do(T) is proportional to r$, 1z.l_s), whose low-temperature dependence is given by r$(f): rR(?:XT.lT¡', the spin-diffusion coefficients are seen to diverge as T-1 for T+ 0. The qualitative behaviour near I and as T+0 shown by (10.136) is in agreement with a more refined calculation (Einzel 1981). The spin-diffusion coefficient is difficult to measure quantitatively, because the dipole torque and spin supercurrents tend to mask the effects of spin diffusion (see the discussion of spin-echo experiments at the end of Section 8.6).

10.3.2 lntrinsic spin relaxation In this subsection we discuss situations in which, by some means, normal and superfluid components are driven out of equilibrium with respect to each other (see Section 8.6). For example, we might envisage increasing the density of the normal component (instantaneously) by injecting hot quasi-

particles into the system. This type of experiment has been carried out on superconductors, where a suitable device is available-the tunnel junction. One expects the system to relax towards equilibrium within some intrinsic relaxation time. In superfluid 3He the equivalent of a tunnel junction is not available, but there is none the less a way to prepare a nonequilibrium state of this kind for the spin densities, as discussed phenomenologically in Section 8.6. There we argued that the dipole torque acts only on the Cooper-pair spins, which adjust to this field within a time of order klA. The quasiparticle spins, on the other hand, relax towards the mutual equilibrium state via quasiparticle-number-nonconserving collisions in which quasiparticles are converted into Cooper pairs and vice versa. The corresponding intrinsic spin-relaxation rate is typically of the same order as the quasiparticle collision time and hence much longer than hlA, except very close to 7i. In the expression (10.51) for the spin density S" one can identify the quasiparticle contribution as

Sf:Lh2to,*"õfo,u,

(10.140)

kv

where s¿,u., is the BQP spin weight factor defined in (10.52a) and a notation as introduced in (10.18c) is used. The remaining terms in (10.51) describe the polarization of the Cooper pairs by the gauge field Hs", the external field and the Fermi-liquid molecular field. In the approximation where the Landau parameters Fi with

Hi'


476

l>-2 are neglected, the Cooper-pair spin density is given SË: y-t

Z

by

(10.1ala)

Xf"lt""u,

where lPo is the cooper-pair partial susceptibility defined in (10.53) and

HïÏ: Hï, + HE"-,ä ott

(10.141b)

is the total effective magnetic field acting on the Cooper-pair spins. As

usual, this field is screened by the total spin polarization S, via the Landau parameter Fô. The effective field Hin may be expressed in terms of the spin densities by inverting (10.141a):

Hi"u:

yZ 0oo);,1(s" - sT),

(10.1a1c)

where we have defined the quasiparticle spin density S! - Sr, SË. The quasiparticle spin density in a local-equilibrium state, described by the distribution function

ô/fi, : f L(õe*,*- ttoHï'_ _f LttoH"*u, is found by substituting

ttoV"*)

(10.142)

(I0.I42) into (10.140a): S¡'t".

:

y-rZ

(10.1a3a)

x?"o"Uîu.

Here we have defined the quasiparticle partial spin susceptibility by xf"ou:

p3> Gf k

ÐG?)*",

(10.143b)

where ,i',:

(s?)u,:

(*)'u*"

*ff

ru",

(10.1a3c)

wrth Apu given by (10.52b). Of course, the two partial susceptibilities add up to the usual spin susceptibility 10, discussed in (3.102)-(r.t-o+): xo*u:ï¿Xo"+

xf,.

In order to calculate the relaxation time

(10.144)

zçB introduced in (8.188) from the kinetic equation, we need an approximation for the relevant part of the collision integral. To neglect the in-scattering part completely, aJ we did in the case of the spin-diffusion co,efficient, is not allowed here, because spin is a conserved quantity in the normal state. As one approaches ?: from bélow, the Cooper-pair density S! vanishes and Sl- S". However, in the absence of spin-conserving interactions, the spin-relaxation time is infinite in the normal state, and one therefore expects tçB to diverge for T-+ ?i. This

L0.3

Transport Cofficients

behaviour is brought about by a cancellation of the in-scattering and out-scattering parts of the collision integral. A simple relaxation-time approximation along the lines of (10.68a), which keeps much of the structure of the exact collision integral, is given by

ô/r,r,

---;['"-?t'"-

E s*,,u¡,õfL,.^lro

k'L

E

.

f L"lro

(10.14s)

The sum over k' in the last term of (10.1a5) may be related to Sl. Multiplying the kinetic equation (10.25a) in the limit Q:0 (we consider spatially homogeneous relaxation processes here) by E*s*,*ntr,(op. -.) and summing over k, one finds o,*"õf 1,") : iøsr ]!!----"E,{r-to)fL,lto, f / r¡, = ã\

(10.146)

s.

In deriving (10.1a6), *e have replaced (s?)r" by

-:#) f s,*)*u = ôpusz*: õ*u' "l('-#,1 (t

(

(B phase), (10.147)

(A

phase).

This is an exact result for the B phase. In the case of the A phase we assume that Sfl is approximately parallel to S[, i.e. perpendicular to d. Then the term propottønal to A*u in (10.143) does not contribute to s2¡,. The equation of motion for Sä is obtained by substitution of the local-equiiibrium solution (L0'r42) into ôfi: ôft - ôff" in the first term of the collision integral, then solving for ôf¿ and finally integrating as before. The resulting expression is so simple that we need not specialize to the hydrodynamic limit øz o)

which ensures quasiparticle conservation. For the normal Fermi tiquid as well as for an isotropic superfluid (B phase), the slip length E may be calculated exactly by assuming diffuse boundary scattering and employing a relaxation-time approximation of the collision integral that is consistent with momentum conservation (F,inzel et al. 1983). Foi the normal Fermi liquid, one finds E :0.579/, where I : upt. For more general types of scattering, variational methods aÍe useful, allowing one to obtain bounds on the slip length. For example, a lower bound on the slip length, which is within one per cent of the exact solution for diffuse scattering, is given by

,r(Lz-

Kz) + (L'+ K")(Lz/\-r --'1" . zLz(Lr* Krr)

Kz)(L'-

K..-)

(10'r77)

where

Ln

: k'>' ef þkf;vo,,@vo,')-' , k

K^n: tP>' 2' ef L)k"k''vk,,"(rv*,")^-'(rvo',")- t(tl n ak'), kk' and the prime on the summation sign indicates Vo,") 0

ot V¡,',,> 0.

10.4.2 Andreev reflection

At a boundary the order parameter is distorted within a layer of thickness

EQ), the coherence length. This leads to a modification of the scattering of BQP off a boundary. In conventional superconductors this effect may occur at a normal-superconducting interface and will lead to important changes in

L0.4 Flow in Restricted Geometries

487

the transport properties, as frrst discussed by Andreev (1964). Indeed, the thermal conductivity of a superconducting metal in the intermediate state is dominated by the scattering of quasiparticles off the interfaces between normal and superconducting regions, at which the gap parameter tends to zero within a distance EQ). A similar effect occurs at a boundary in superfluid 3He, where the gap parameter varies in space. The gap distortion induces an interaction of the BQP with the boundary. This effect, called Andreev reflection, acts in addition to the microscopic scattering process right at the wall. It is similar to the reflection of a quantum-mechanical particle by a potential threshold in the case of a particle approaching the threshold from the top of the step. The Bogoliubov quasiparticle is turned back by means of an elastic transition from the particle-like excitation branch with Eo> 0 to the hole-like excitation branch with fo'L., only a thin layer at the walls of the cavity is disturbed by the boundary scattering, and the momentum transfer to the wall is described by the transverse surface impedance Zr:Il,,lul"", as described by (11.3). The surface impedance is perhaps the physical quantity that reflects the properties of the boundary surface most directly. The transverse surface impedance has been calculated from an exact solution of the model kinetic phase by applying equation for the diffuse-scattering boundary condition (Einzel et al. 1983; see also Ebisawa 1980) and using a variational approach for a general scattering law (Einzel et al. 1987). Within the slip

B

a

approximation (valid for at 11I), the transverse surface impedance is readily found from the hydrodynamic equations as

Z,=#:(1 -i)3(t-?r),

(10.182)

where 5 : (2r¡ I pn )t'' is the viscous penetration depth and Ç is the slip length. The surface impedance in the A phase has been discussed by Hook and Zimmermann (1981). Another flow situation in which boundary effects are very important occurs in a so-called U-tube oscillation experiment. There the two vertical columns of a U-shaped vessel are connected by a narrow channel, such that only superflow connects the two parts. When the level of liquid in one of the columns is raised by means of an electric field applied to the liquid, and if the field is then switched off suddenly, one might expect oscillations to occur about the equilibrium position of the liquid levels in both columns. However, experimentally, only a slow relaxation back to equilibrium is observed (Eisenstein and Packard 1982, Eisenstein 1983). This finding may be explained by taking into account superfluid-to-normal-fluid interconversion near the free surfaces and the friction of the normal flow at the walls of the vertical tubes (Brand and Cross L982). From the hydrodynamic equations supplemented by the usual boundary conditions on the normalfluid velocity field, u': usu.ra.. at the free surface and u,,:0 at the walls of the channel (plate separation d), one flnds a pressure change associated with the velocity i of the moving free surface:

ôP: p(l2rtEr)'''L d' At lower temperatures,

(10.183)

7., the mean free path is no longer small in comparison with the diame{er of the tube. In this ôase the viscosity 4 must be replaced by the effective viscosity coefficient r¡"o relevant for Poiseuille ,T,> A, all the

11.i

Time-Dependent Mean-Field Theory in the Collisionless Regime (¡)

525

breoking

2L

squoshing

(El',','.L

\ l-/

(8)"',A

)

Zeemon sptitting

reoI squoshing zero sound

112

collisionless spin waves

l/E

q

Figure 1L.1. Collective modes in the B phase (schematic). The transition from the hydrodynamic to the collisionless regime is indicated by a broadening of the excitation branch. The dashed lines indicate the splitting in a magnetic field. The collective-mode frequencies at q:0 indicated in the figure are the weak-coupling results.

modes possess linear dispersion. The teading qt correction for small q is listed in Table Ll.L. The numerical coeffrcient in some cases involves the quantity Â1, which follows from expanding )'(r) to first order í, q', and is defined by

)'r[r(q:

0)]

4az 1481,,_(kr),^***ll ::l!^,or,^[._ oço14EodE) lo:o' Luh rnry J

(11.6s)

@æL4ET- ØæooThe frequency in the argument of )', is the rnode frequency at q :0. A schematic plot of the collective mode frequencies as functions of q is shown

in Fig. LI.L. Effect of a magnetic field

An external magnetic field may be incorporated into the description by allowing for the Zeeman splitting in the normal quasiparticle energies: E*": Ç¡,Xs¡tsH,

s: tl,

where f¿o is the nuclear magnetic moment. The

LL Collectiue Modes equations then become quite complicated, so w9 do not reproduce them hére (see Tewordr and Sóhopohl 1979b, Schopohl and Tewordt 1981). The magnètic fietd induces a spin-singlet component of the order-parameter fluctuations, which is responsible for a linear rather than quadratic splitting of the mode. Such a splitting has already been calculated for the spin-wave spectrum, i.e. the 1* mode, in (11.37). -" :2 collective-mode For small magnetic frelds the linear splitting of the J frequencies is given bY (11'66) @*ù(T, H): o2*(T) + mrgz*(T)av(T, H),

where ffit:Q, t1, t2. The g factors may be expressed in terms of the yosida function Ys defined in (3.101c) and the function Â as

8,,(r):ílr*#], and

at(T, H):

yH

r+Fi(3+å%)

(t1..67a)

(11.67b)

real is the renormalized Fermi-liquid Larmor frequency. The splitting of the aI. squashing mode (I :2+) hai been observed experimentally (Avenel et rôso) urrã ir in full agreement with the theoretical prediction by Tewordt and Schopohl (1971Ui. T.ne splitting of the squashing mode is harder to observe b""u.rr" of the stronger coupling of the sound to the collective mode (Meisel et al. 1983c). It has nevertheless been detected in measurements of the group velocity (Movshovích et aI. 1938) by employing the theoretical anaiysis of this experimental situation developed by varoquaux et al. (19g6). Ri trigtrer fields the splitting deviates substantially from linear behaviour (Shivaram et aI. 1,983, 1986) because the equilibrium gap iò itself distorted by the magnetic field (Schopohl et al' 1983' parameter 'Brurou and Popov 1984a,b,c, Fishman and Sauls L986, 1988b)' The possibility of inducing transitions between the sublevels by an RF field has L""n disóussed by Dombre and Combescot (1982b). The collective-mode similar to freq,uencies are also affected by superflow. The effect is somewhat of a magnetic field (Kleinert 1980a, Sauls and Serene 1984, Brusov

thai

1985a,b).

11.3.2 Gollective modes in the A phase In the axial state the equilibrium gap parameter is given by t dou(it) :,fr\n)r' A o âo*Yr rçt¡ .

(11.68)

Y11, We shall take âo*to lie along the z axis. Since do*çft¡ is proportional to

is convenient to

expand the order-parameter fluctuations

it

in terms of

1-1.3 Time-Dependent Mean-Field rheory in the collisionless Regime

52i

spherical harmonics Y1*:

õdp(c):

_ì*,

õdr"* Yr^(Ê).

(11.6e)

Furthermore, we introduce real and imaginary parts_of õd¡, , d,*. and d.,i,^, in analogy with (11.63). Subsrituting (11.69) inro (11.57), weïnd in îti limit q+ 0 the following set of equations:

l(X)' ^- - z^^ + +a^fa; + 2ô^,^*(o;^ - zâ".4 â"¿'"^) ^ ' \-\ r*^â?(ôåo,^)¿, :-Ziha¡ ô^rtrà (11.70a)

f

l/hat\'. L U" / ^^

.-l-..

2h^ + +a^]a';^

/

- 26*t)"^(o';^ - ,¿",4 â"¿i^)

:zry ô^t)Ltâot"(o6*)¿.

(11.70b)

ao

Here we have defined

L^:

I oo, lYr^(k)l' t (k), tr^: oo, lv,^(Ê)l'ffi I E*: oor lYr^G)lzE@).

^rÌ

(11.70c)

I

In

deriving we have exploited ^(11.70a,b), functions )'(Ê) and E(k) abour î for q->9.

the axial symmetry of

In contrast with the B phase, different angular

the

averages of t () are involved in (11.70a,b,c), so that Âo does not simply cancel as in (11.64a,b). A further difficulty arises from the frequency dependence of )r(c), as defined by (11.55). The eigenvalue equations, obtained by putting the right-hand sides of (11.70a,b) equal to zero, are now nonlinear in ar2 and may have more than one solution. As usual, a nontrivial solution of a system of linear homogeneous equations exists if the determinant of coefficients vanishes. A numerical analysis of the eigenvalue problem for (11.70) yields well-defined zeros of the real part of the determinant at real frequencies (WöIfl e I976c, !977a, Tewordt and schopohl r979a). However, the imaginary part of the determinant does not vanish at these frequencies (in contrasì with what happens in the B phase). This is a consequence of pair-breaking processes, which may occur at any frequency owing to the two nodes in the energy gap in the axial state. Consequently, one might expect a zero of the determinant to exist somewhere in the complex frequency plane. But this is not the case,

LL

Collectiue Modes

as will be discussed below. In fact, it may be shown that-strictly speaking-the determinant does not have any zeros in the complex frequency plane, except for ø :0 (Dombre and Combescot 1981). Hence in the A phase there are no well-defined modes at all-there are only "resonances". Nevertheless. we shall call these resonances "collective modes".

The limit løl

:0

Following Table 1I.2, we group the collective modes according to the label ffi:0, t1 of the order-parameter fluctuations di- and d'i"^ (at least in weak-coupling theory).

(i) m : L. For m: tl,

the angular average of .E'(É) defined in (11.70c) vanishes, i.e. ,E*t :0. It turns out that in this case there is at most one collective mode per variable. The order-parameter variables d',r, d'r, and d')r, are associated with rotations in spin space about the ¡ and y axes and the gauge-orbit symmetry respectively. Their eigenfrequencies are zero (they are Goldstone modes) and the corresponding dispersion laws are linear in q, as will be discussed below. The remaining variables d',r, d'], and dlr do not even show any resonance because the real part of the eigenvalue equation

(T)'^,-

4)",:s

(Lt.7L)

does not have a solution for real a (at least in the absence of Fermi-liquid corrections). Note the simplicity of (11.71), which is a consequence of

E^:+l:0'

(ii) m--1. The six variables dL,_r, d'i,_, (p:*,y,2) are found to oscillate at a frequency

. hrr"t:

12 Re Â,1r/2

\ ú'/

Ao:

L.23A0(T).

(1r.72)

Hence they correspond to pair vibration modes. Since the motion of the order-parameter structure, i.e. the motion of the unit vectors râ and á, resembles the clapping of hands, as explained in the beginning of Section r1.2, this mode has been called the "clapping mode" (see Fig. ll.2).lt is remarkable that ø.¡ scales with the energy gap A¡(T) at all temperatures within one per cent accuracy (see Fig. 11.3) in spite of the presence of the Â factors in (I1.72). Among the six modes, only d'i,-, co,uples to sound waves (neglecting the coupling due to particle-hole-asymmetric terms), whereas d',,_¡ and d'r,_, couple to spin waves.

11.3 Time-Dependent Mean-Field Theory in the Collisionless

I

Figure

Regime

529

qx¡ol stote

11.2 A pictorial representation of the order-parameter modes in the A phase.

The case m:0 plays a special role because here there exist three resonances for each of the variables d'*,s and d'i",o. This is due to the nonzero value of go, given by

(iä)

m:0.

Eo

: -)

[ ao, vß(îù ^(fr)f I "r;#

(r1.73)

(*)'^-2)'o*

(1t.74)

(here a partial integration with respect to cos 0 has been performed to bring Eo into the form (11.73)). A small admixture to the gap parameter proportional to Yro(k), as given by these variables, causes the preferred directions ñ and ñ to oscillate out of plane. This motion may be visualized as a flapping of wings (see Fig. lL.Z)-hence the name "flapping mode" (Wölfle I975b). The m:0 eigenvalue equation is given by

4Eo:

s

More-or-less stable oscillations are found to occur at two different nonzero eigenfrequencies. In addition, there is a solution at zero frequency corresponding to a diffusive mode (see below). Let us discuss the solution of (1I.74) in nlore detail. There is one solution with a strongly temperature-dependent eigenfrequency: ha¡nn

_T

(tt.7s)

Ao r"' More precisely, in the limits hann ao

: (t)'''

T-T.

and

i.l, - #

hr,¡no: (!)t/znk, r(rn

f

T+0

Ee)

*

the analytic results are

#l

"on,

,)-'''

", r

-+ r.,

as T-+

0.

(IL.76a) (11.76b)

L1 Collectiue

530

Modes

rlrc 064 084

*r99P

normol flopping

0.8 06 0.4 0.2 lttl?vz (a)

Figure

0

1

0.8 06 0.¿ il-T/Tc)

0.2

1t2

(b)

11.3 Collective-mode frequencies of the A phase versus temperature in

units (a) of the temperature-dependent gap An(T) and (b) of the gap at T :0, ^n(0).

The complete numerical result is shown in Fig. 11.3. The physical origin of this mode may easily be visualized. It arises from the fact that the flapping of the vectors ñ and ñ is equivalent to an oscillation of the vector i about its equilibrium direction (remembe^r that î: rh x ñ). Since i specifies the axis of the energy gap, a motion of J will change the quasiparticle spectrum and hence the equilibrium quasiparticle distribution fi(Ð. The quasiparticles then have to adjust to the new equilibrium state by means of collision processes. Hence the relaxation to equilibrium will take a characteristic time r¡, which is of the order of the quasiparticle collision time z. As one may expect, r¿ is closely rela.ted to the orbital-viscosity coefficient Âo calculated in 1fô.fZO¡. Now, if the oseillation of i is too rapid for the quasiparticles to follow, the quasiparticle dïstribution will not be changed much. In this case there is a wêll-defined direction Î* given by the time averaged axis of the energy gap, about which the superfluid component of the system oscillates. The restoring force that maintains this oscillation is provided by the potential well formed by the energy of the quasiparticle system. Therefore

11.3 Time-Dependent Mean-Field rheory in the collisionless

Regime

531

the oscillation frequency will decrease with temperature and will eventuallr tend to zero as l--+0 in the same way as the density of quasiparticles goes to zero in this limit. Because of the important role played by the normal-fluid component in the mechanism of this mode, it has been callec the "normal-flapping mode". It is interesting to note that this mode ir exceptional since it owes its existence to the thermal excitations in the superfluid and hence is not a property of the superfluid condensate itself. In contrast with the "normal-flapping mode", the frequency of the other flapping mode scales approximately with the gap, similarly to the clapping mode. It may therefore be viewed as a ground-state property of the superfluid as in the case of the squashing modes in the B phase. Hence we shall refer to this mode as the "super-flapping" mode. The ratio of its eigenfrequency to as(T), i.e. ha"olao(T), is found to decrease from z at T, to approximately I.56 at T:0 (see Fig. 11.3). The leading correction for T - T. is given by ht lru

ao(T)

:rl'-:#ee)#l ", r-r"

(rr.77)

Pair-breaking effects The anisotropy of the energy gap in the A phase, l\&)l: Aosin g, implies that the minimum energy required to break a Cooper pair into two quasiparticles varies from zero to ZAs, depending on the position on the Fermi surface. Hence pair breaking via absorption of collective-mode quanta may occur at arbitrarily low frequencies. However, at low frequencies (Ð this effect is reduced because of the limited phase space available. Mathematically, the pair-breaking process is described by the vanishing of the energy denominator in the expression for ),e given by (11.55), which gives rise to an imaginary part of ).(*), as evident from (LL.62a,c).

To determine the frequency eigenvalues, we have so far omitted the imaginary part, because it is relatively small as long as ha 1ZA. and has

little influence on the real part of the eigenfrequency. However, one has to keep in mind that the collective modes discussed above are damped by pair-creation processes. Clearly, the higher the frequency, the more damping will occur. We defer a discussion of the damping until later, because in cases of practical interest (e.g. coupling of collective modes to sound waves) the broadening of the absorption peaks caused by the collective modes is also influenced by the imaginary part of the coupling function. Therefore it is more sensible to discuss directly the physically observable quantity (".g. sound absorption) rather than collective-mode frequencies themselves.

There is yet another problem associated with pair breaking, which has already been briefly mentioned. Even though the small imaginary part in

11

532

Collectiue Modes

(1I.72) and (11.74) induced by pair-breaking processes may not be of great quantitative importance, it is conceptually relevant. Namely, it implies that, strictly speaking, there are no collective poles in the complex frequency plane corresponding to the clapping and flapping modes. These modes are only peculiar resonances on the real frequency axis, and are not related to a complex pole. The argument runs as follows: In the absence of collision effeóts, the left-hand sides of (11.71) and (11.74\ are functions of ar2, expressing the fact that time-reversal invariance is strictly valid in this limit. Then, for any given zero of, sâY, (11.71), at frequency @+:@ct -if (assuming f >0), there is a corresponding zero at o-- -@ct +ill which lies in the upper half-plane. However, any collective pole in the upper plane violates causality and causes an instability of the system. Hence there can be no complex zeros of (11.71) and (11.74) other than a¡ :0. An investigation of the functions on the left-hand sides of (11.71) and (1L.74) (Dombre and Combescot 1981) has shown that this is indeed true. Finite wave vector, lql + 0 Even at lql + 0 the different spin components of the order parameter decouple. This is a consequence of the fact that the spin and orbital degrees of freedom separate. On the other hand, the respective orbital components become mixed. The eigenvalue problem thus reduces to a set of six 3 x 3 matrix equations. In thã following we only consider the special case q ll î for simplicity, where the 3 x 3 matrices assume diagonal form. The dispersion of the Goldstone (i.". gapless) modes with m:l associated with the variables d'!, and d',t, d'rt is given by the single expression

a),: ("rq)'lri\

((k . Ð'^(¡el )¡ ( r(re )

(11.78)

)¿

In the limit T--+0 this reduces to

ú):

(rt.7e)

*.3-t/zurQ.

gapless mode, with ffi:O, is obtained for the "flapping variables" d'u,uandd'i,.o as a solution of (11.74) in the limit a¡, q--0. The Â functions in (11.74) are highly singular in this limit and have to be analysed with great care. One finds (11.80) o¡ : a(urq)' as T ---> 7.,

:,

A further

where

a-h(r.-rl##-u*]

,

while in the low-temperatiire :iimit

a:-.i.j-(*r)'"(#) (,.q)' as r-+0.

(11.81)

11.i

rime-Dependent Mean-Field rheory in the Collisionless Regime

533

This mode is the so-called "orbital wave", i.e. the Goldstone mode associated with the breakdown of rotational symmetry in orbital space. It can be seen that this mode is overdamped at all temperatures because of its strong coupling to the quasipar(icle system (see the discussion on orbital relaxation in Section 10.3). The dispersion of the normal-flapping mode follows from (11.74) as

oÅ"(q): a?n(q:0) + u?@ .ï'.

(11.82)

where (11.s2) now holds for arbitrary directions of f and i. nt T:0, where a^n (g - 0) : 0, the dispersion of the normal-flapping mode comes out to be linear. This behaviour is changed into a quadratic behaviour owing to particle-hole-asymmetric terms in the very-low-frequency regime ho¡ en)Aå;#

(11.83b)

In the A phase the collision-induced broadening adds to the pair-breakinginduced broadening (rWölfle and Koch 1978). A strong-coupling theory of the collective modes has not yet been worked out, although it is in principle known how to calculate the leading correction in T.lf, (Serene and Rainer L978, 1983); see, however, Wartak (1978, 1980a,b, 1981, 1982). Judging from the smallness of the strong-coupling corrections left in the thermodynamic properties once the strong-coupling gap parameter is inserted in the weak-coupling theory, one may anticipate that genuine strong-coupling corrections will be small.

11.3.4 Observability of collective modes Among the set of collective modes found in each of the phases, only the Goldstone mode associated with broken gauge symmetry (which is essentially a sound wave) may be excited and detected directly by experiment (Wheatley 1978). Besides that, it has also been possible to observe standing

11.4 Collisionless Sound

s35

spin waves using a gradient magnetic field (Masuhara et al. 1934). These Goldstone modes are in turn found to couple to some of the orderparameter pair-vibration modes. This allows for an indirect detection of the latter modes via absorption and dispersion effects on the Goldstone modes. In particular, sound waves are very well suited for this purpose since the sound velocity is high compared with the Fermi velocity, implying that the sound-excitation branch will intersect the pair-vibration branch (see Fig. I 1.1).

This is not necessarily the case with the spin-wave branch. So far,

collective modes have only been seen in sound-propagation experiments. In fact, sound waves may be used as a spectroscopic probe to investigate the pair-vibration modes in superfluid 3He, much as electromagnetic waves are employed to study the electronic-excitation spectra of atoms and molecules. In the next section we shall therefore give a relatively detailed account of the theory of sound propagation in the collisionless regime. A general discussion of the symmetry aspects of the coupling of collective modes to observable quantities has been given by Serene (1983).

11.4

COLLTSTONLESS SOUND

The detailed discussion of order-parameter modes in the preceding section has equipped us to calculate the physically observable effeôts of p-wave pair correlations on the high-frequency dynamics of 3He. Our starting point will be the matrix kinetic equation (10.6) and the conservation laws for particle number and momentum (2.2I) and (2.22), together with the definirion of the momentum current density Ir,, given in (10.17b). Combining (2.2r) and 2.22), we find at once the Fourier-transformed equation of motion for the mass-density fluctuations õp(q, ,) in terms of the momentum current density: @' õp

*

iøqlI.q

:

(11.84)

Q.

The second term in (11.84) may be rewritten by substituting the expression for II, (10.17b); this yields

qfiq

çr2

:

*4

@ . q)'(õn*

+

eLô60).

( 1 1.8s)

Using (r. Ð':?P2((. Ð + {, where P2 is the l:2 Legendre polynomial, and employing the integral property (10.lSb) of QL, this may be cast into the form

qnq::#

@,ù'Le+

Fi,) ôp + zm(t +

*Få)> P,(Ê-4> a"rf. (n.86) k

It follows from (11.84) that the dispersion law for sound waves is (,)2:c1q'[r+22Ç(a)1.

(r 1.87)

11

536

Collectiue Modes

where

.E )e

P,(k- Q) õro

E@¡): 2

Z_

Eônn k

(11.88)

'

2t + +F) 51+Fi)

(11.89a)

ln fact, if small terms of order (urlr,)a are neglected, the coefficient Z

is

given by the relative difference between the velocities of zero sound c,, and first sound c, in the normal state: ct

7:co-

(11.8eb)

C1

The response function f(ø) may be expressed in terms of ôf¿ and ôÁ¿ by taking the l:2 component of the diagonal mean-field equation (11.53a):

4

,,',

Q'(k)

(Â1É¡f Takagi S 1975a J. Phys. C8 1507 Takagi S 1975b J. Low Temp. Phys. lE 309 Takagi S 1977 J. Phys. C10 4045 Takagi S 1978 Prog. Theor. Phys. 60 934

Smith

References

590

Tankersley LL L979 J. Low Temp- Phys.37 L41 Tesanovich Z t984 Phys. Lett- 100Ã' 158 Tesanovich Z and Valls O T 1986a Phys. Reu- B'311374 Tesanovich Z and Valls O T 1986b Phys. Reu. 833 3139 Tesanovich Z and Valls O T 1986c Phys. Reu. B,34 7610 Tewordt L 1974a J. Low Temp. Phys- 15 349 Tewordt L L974b Z. PhYs. zffi207 Tewordt L and Einzel D 1976 Phys. Lett- 56A' 97 Tewordt L and Schopohl N 1979a J. Low Temp- Phys. Y 489 Tewordt L and Schopohl N 1979b J. Low Temp- Phys- 37 421 Tewordt L, Fay D, Dörre P and Einzel D 1975a Phys. Reu. B1I l9l4 Tewordt L, Fay D, Dörre P and Einzel D 1975b J. Low Temp. Phys. 21645 Tewordt L, Scúopohl N and vollhardt D L977 J. Low Temp. Phys.29 l\9 Tewordt L, Fay D and Schopohl N 1978 J. Low Temp. P4lt',30 177 Theodorakis S and Fetter A L 1983 J. Low Temp- Phys. 52 559 Thouless D J 1960 Ann. Phys. (Ny) 10 553 Thuneberg E V 1986a Phys. Reu. Lett.56 359 Thuneberg E V 1986b Phys. Reu- B'33 5124 Thuneberg E V 1987a Phys- R¿u. 836 3583 Thuneberg E V 1987b Europhys. Lett- 3 lLI Thuneberg E V and Kurkijärvi J 1981 Phys. Leu- ffiÃ' 35 Thuneber! E V, Kurkijarvi J and Rainer D 1981a Physica L07B + C 43 (Proceedings of thã 16th Internãtional Conference on Low Temperature Physics, LT-16) Thuneberg E v, Kurkijärvi J and Rainer D 1981b I. Phys. c14 5615 1853 Thuneber! E V, Kurkijarvi f and Rainer D 1982 _Phys. R¿u. Lett. edn (Adam Superconductiuity,2nd and Superfluidity tqse and'Tilley D ñ. Tilley

f

ß

Hilger, Bristol) Tinkharñ M 1975 Introduction to Superconductiuity (McGraw-Hill, New York) Jensen H 1984 J. Low Temp. Phys.55 469 Topsoe F and Højgaard "1þ65 Zh. Eksp. Teor. Fiz. 49 1875 lSou. Phys. JETP 22 L282 Tkàchenko V K (1e66)l Tougi J T-lgBZ in Progress in Low Temperature Physics, Vol. VIII, ed. D F Brewer (North-Holland, Amsterdam), P. 133 Toulouse G and Kléman ML976 J- Physique Lett' 37 149 Truscott W S 1979 PhYs. Leu 74A' 80 Tsuneto T 1960 PhYs. Reu. l1.8 1029 Tsuneto T 1977 Prog. Theor- Phys- 58 719 Tsuneto T, Ohmi T and Fujita T 1978 in Physics at Ultralow Temperatures, ed- T Suguwara, S Nakajima, T Ohtsuka and T Usui (The Physical Society of Japan, Tokyo), p. 153 Ueda K and Rice T M 1985 Phys. Reu. B3L7LI4 ugulava A I 1984 Zh. Eksp. Teor. Fiz. ffi 497 lSou. Phys. JETP 59 290 (1984)l -?Ullah S 1988 PhYs. Reu. 837 5010 Valatin J G 1959 Nuouo Cim- 7 843 Varma C M and Werthamer N R L974 Phys. Reu- Ã9 1465 varoquaux E, Williams G A and Avenel o 1986 Phys. Reu. B34 7.617 Vdovin Y A 1963 in Application of Methods of Quantum Field Theory to Problems of Many Particles,^ ed. A I Alekseyeva (Moscow: GOS ATOM ISDAT)' p. 94

(in Russian) Vinen W F 1957a Proc. R- Soc- Lond128 Vinen W F 1957b Proc. R- Soc. Lond. ^2/;0ll4 Vinen W F 1957c Proc. R. Soc. Lond- ^?A0 493 ^242 Physics, Vol. Vinen W F 1961 in Progress in Low Temperature (North-Holland, Amsterdam), P. 1

[II, ed. C J Gorter

References

s91

Vollhardt D 1979 Doctoral Thesis, Universität Hamburg Vollhardt D 1982a Phys. Bltitter 39 4L Vollhardt D 1982b Phys. Blötter 39 L20 Vollhardt D 1982c Phys. Bltitter 39 151 Vollhardt D 1984a Reu. Mod. Phys.56 99 Vollhardt D 1984b in Proceedings of the 17th Internøtional Conference on Low Temperature Physics, LT-17, Part I ed. U Eckern, A Schmid, W Weber and H Wühl (North-Holland, Amsterdam), p. 5 Vollhardt D and Kumar P L978 J. Physique 39 Colloq. C-6, Vol. I, p. 43 (Proceedings of the 15th International Conference on Low Temperature Physics, LT-15) Vollhardt D and Maki K L978 J. Low Temp. Phys. 31457 Vollhardt D and Maki K L979a Phys. Lett.72!¡21 Vollhardt D and Maki K 1979b Phys. Reu. B.20 963; ibid. 823 E1489 Vollhardt D and Wölfle P 1981a Phys. Reu. Lett.47 L90 Vollhardt D and Wölfle P 1981b Physica 1088 + C 1055 (Proceedings of the 16th International Conference on Low Temperature Physics, LT-16) Vollhardt D, Lin-Liu Y R and Maki K L979 J. Low Temp. Phys. 37 627 Vollhardt D, Maki K and Schopohl N 1980 J. Low Temp. Phys. 39 79 Vollhardt D, Lin-Liu Y R and Maki K 1981 /. Low Temp. Phys.43 189 Vollhardt D, Wölfle P and Anderson P W 1987 Phys. Reu. B.35 6703 Volovik G E 1975a Pis'ma Zh. Eksp. Teor. Fiz.22234lLJEfP Lett.22108 (1975)l Volovik G E 1975b Pis'ma Zh. Eksp. Teor. Fiz. T¿ 4l2IJETP Lett. 22198 (1975)l Volovik G E 1978a Pis'ma Zh. Eksp. Teor Fiz. 27 605IJEZP Lett. 27 573 (197S)l Volovik G E 1978b Pis'ma Zh. Eksp. Teor. Fiz. ?Å 65 ÍJETP Lett. ?.8 59 (197S)l Volovik G E 1979 in Souiet Scientffic Reuiews, Section A, Physics Reuiews, Vol. 1 (Harwood Academic Publishers, Chur), p. 23 Volovik c E 1980 Zh. Eksp. Teor. Fiz.79 309 fSou. Phys. JETP 52 L56 (1980)I Volovik G E 1984a Pis'ma Zh. Elcsp. Teor. Fiz.39 169 IJETP Lett.39 200 (1984)l Volovik G E 1984b Usp. Fiz. Nauk. 143 73lSou. Phys. Usp. 27 363 (1984)l Volovik G E 1984c Pß'ma Zh. Eksp. Teor. Fiz.39 304 ÍJETP Lett. 39 365 (19S4)l Volovik G E 1985 Pis'ma Zh. Eksp. Teor. Fiz. 42294ÍJETP Lett.42 363 (19S5)l Volovik G E 1986a Pis'ma Zh. Eksp. Teor. Fiz. 43 428ÍJETP Lett. 43 551 (1986)l volovik G E 1986b Pis'mø Zh. Eksp. Teor. Fiz. 43 535 VETP Lett. 43 693 (1e86)l

Volovik Volovik Volovik Volovik Volovik

G E 1986c Pß'ma Zh. Eksp. Teor. Fiz. 4/ I44 ÍJETP Lett. 44185 (1986)l G E 1986d Pis'ma Zh. Elcsp. Teor. Fiz. 4388TLIETP Lett. 4 498 (1986)l G E 1987 J. Low Temp. Phys. trl 30L G E and Balatskii A V 1985 I. Low Temp. Phys. 58I G E and Gorkov L P 1984 Pis'ma Zh. Elcsp. Teor. Fiz.39 550 IETP Lett.

3e 674 (1e84)l Volovik G E and Gorkov L P 1985 Zh. Eksp. Teor. Fiz. 88 L4L2lSou. Phys. JETP 6t 843 (1e8s)l Volovik G E and Hakonen P J 1981 J. Low Temp. Phys. 42 503 Volovik G E and Khazan M V 1982 Zh. Eksp. Teor. Fiz. 82 t498 fSou. Phys. JETP ss 8e8 (1e82)l Volovik G E and Khazan MV 1983 Zh. Eksp. Teor. Fiz.85 948 lSou. Phys. JETP s8 ssl (1e83)l Volovik G E and Khazan MV L984 Zh. Eksp. Teor. Fiz.87 583 fSou. Phys. JETP 60 276 (1e84)l Volovik G E and Konyshev V A 1988 Pis'ma Zh. Eksp. Teor. Fiz. 47 207 IJETP Lett. 47 250 (1988)l Volovik G E and Kopnin N B 1977 Pß'ma Zh. EI<sp. Teor. Fiz. ?ß 26 PETP Lett. 2s 22 (re77)l

References

Volovik G E and Mineev Y P I976a Pis'ma Zh. Elcsp. Teor. Fiz.23 647 IETP Lett. 23 se3 (1e76)l Volovik G E and Mineev V P 1976b Pis'ma Zh. Eksp. Teor. Fiz. A 605IJETP Len. s6r (1e76)l Volovik G E and Mineev V P 1976c Zh. El<sp. Teor. Fiz. 7l ll29 lSou. Phys. JETP set (re76)l Volovik G E and Mineev Y P I977a Zh. Eksp. Teor. Fiz. 722256lSou. Phys. JETP 4s 1186 (re77)l Volovik G E and Mineev V P L977b Zh. El<sp. Teor. Fi2.73767 [Sou. Phys. JETP

a 4 6

401,

(1977)l

Volovik G E and Mineev V P 1981 Zh. Eksp. Teor. Fiz.81 989 fSou. Phys. IETP 54 s24 (1e81)l

Volovik G E and Mineev V P 1982 Zh. Eksp. Teor. Fiz.83 1025 [Sou. Phys. JETP s6 s7e (1e82)l Volovik G E and Mineev V P 1983 Pis'ma Zh. Eksp. Teor. Fiz.37 103 IJETP Leu. 37 r27 (1e83)l Volovik G E and Mineev V P 1984 Zh. Eksp. Teor. Fiz. K 1667 fSou. Phys. JETP se e72 (1e84)l Volovik G E and Salomaa M M 1985a Pis'ma Zh. Eksp. Teor. Fiz. 42 42I VETP Lett.42 521 (1985)l Volovik G E and Salomaa M M 1985b Zh. Eksp. Teor. Fiz. 88 1656 lSou. Phys. JETP 61 e86 (1e8s)l Volovik G E and Salomaa M M 1987 Pis'ma Zh. El<sp. Teor. Fiz. 45 44 IETP Lett. 4s s6 (1e87)l Volovik G E and Yakovenko V M 1989 J. Phys.: Condens. Matter 15263 Vulovic V Z, Stein D L and Fetter A L 1984 Phys. Reu.B.29 6090 Vuorio M 1974 J. Phys. C7 L5 Vuorio M L976 J. Phys. Cg L267 Wartak M S 1978 I. Physique 39 Colloq. C-6, Vol. I, p. 11 (Proceedings of the 15th International Conference on Low Temperature Physics, LT-15) Wartak M S 1980a Phys. Lett. 794' 435 99 Wartak M S 1980b Acta Phys. Polon. Wartak M S 1981 I. Phys. C14 L155 ^57 Wartak M S 1982 Phys. Lett. ßÃ' 84 Webb R A 1977a Phys. Reu. Lett.38 1151 Webb R A 1977b Phys. Reu. Lett.39 1008 Webb R A 1978a Phys. Reu. Lett.40 883 Webb R A 1978b J. Physique 39 Colloq. C-6, Vol. I, p. 31 (Proceedings of the 15th International Conference on Low Temperature Physics, LT-15) Webb R A 1978c Phys. Lett. 67A, t97 Webb R A, Greytak T J, Johnson R T and Wheatley J C L973 Phys. Reu. Lett. 30 210 Webb R A, Kleinberg R L and Wheatley J C L974a Phys. Lett. ß4,421 Webb R A, Kleinberg R L and Wheatley I C L974b Phys. Reu' Lett.33 145 Webb R A, Sager R E and Wheatley J C 1975a Phys. Reu. Lett.35 615 Webb R A, Sager R E and Wheatley J C 1975b Phys. Reu. Lett.35 1010 Webb R A, Sager R E and Wheatley J C I975c Phys. Reu. Lett.35 1164 ttrebb R A, Sager R E and Wheatley J C L977 I. Low Temp. Phys. 26 439 Wellard N V 1982 Doctoral Thesisi University of Manchester rilellard N V, Alexander P W, HaIt H E and Hook J R 1982 Physica 109/1108+C 2096 Wheatley J C 1966 in Quantum Fluids, ed. D F Brewer (North-Holland, Amsterdam), p. 183 Wheatley J C 1970 in Progress in Low Temperature Physics, Vol. VI, ed. J C Gorter (North-Holland, Amsterdam), p. 77

References

Wheatley J C 1973 Physica 69 2L8 Wheatley J C 1975 Reu. Mod. Phys. 47 41'5 Wheatley J C 1976 Phys. Today 29 32 Wheatley J C 1978 in Progress in Low Temperature Physics, Vol. VIIA, ed. D F Brewer (North-Holland, Amsterdam), p. 77 \Wheatley J C, Rapp R E and Johnson R T 1971 J. Low Temp. Phys. 4l Whitham G B 1974 Linear and Nonlinear Waues (Wiley, New York) Wilczek F and Zee A 1983 Phys. Reu. Lett. 512250 Wilks J 1967 The Properties of Liquid and Solid Helium (Clarendon Press, Oxford) Wilks J and Betts D S 1987 An Introduction to Liquid Helium, 2nd edn (Clarendon Press, Oxford) Williams G A and Packard R E 1980 J. Low Temp. Phys.39 553 Williams M R and Fetter A L 1979 Phys. Reu. B.20 169 Witten E 1979 Nucl. Phys. B156 269 Wojtanowski W 1986 J. Low Temp. Phys. g 309 Wojtanowski W and Wölfle P 1986a Phys. Reu. Lett.56 488 Wojtanowski W and rWölfle P 1986b Phys. Lett. ll5Ã,49 Wojtanowski W and Wölfle P 1987 Phys. Reu. Lett. 59 C2387 Wölfle P 1973a Phys. Reu. Lett.30 1169 Wölfle P 1973b Phys. Reu. Lett. 311437 Wölfle P L974 Phys. Lett. 47A,224 Wölfle P 1975a Phys. Reu. Lett. Y l2l0 Wölfle P 1975b in Quantum Statistics and the Many-Body Problem, ed. S B Trickey, W P Kirk and J W Dufty (Plenum, Ne'v York), p. 9 Wölfle P I976a J. Low Temp. Phys. 22 152 Wölfle P 1976b Phys. Reu. B1.| 89 Wölfle P L976c Phys. Reu. Leu.37 1279 Wölfle P 1977a Physica 908 96 Wölfle P 1977b J. Low Temp. Phys. 26 659 Wölfle P 1978a in Progress in Low Temperature Physics, Vol. VIIA, ed. D F Brewer (North-Holland, Amsterdam), p. 191 Wölfle P 1978b in Physics at Ultralow Temperatures, ed. T Sugawara, S Nakajima, T Ohtsuka and T Usui (The Physical Society of Ja.pan, Tokyo), p. 87 Wölfle P 1978c J. Physique 39 Colloq . C-6, Vol. III, p. 1278 (Proceedings of the 15th International Conference on Low Temperature Physics, L T-15) tWölfle P 1979 Rep. Prog. Phys. A 269 Wölfle P 1980 in Proceedings of the XVI Karpacz Winter School of Theoretical Physics (Lecture Notes in Physics, Vol. 115), ed. A Pekalski and J Przyslawa (Springer, Berlin), p. 47 Wölfle P and Einzel D 1978 J. Low Temp. Phys. t2 39 Wölfle P and Koch V E 1978 J. Low Temp. Phys.30 61 Wölfle P, Götze W, Kurkijärvi J and Smith H 1980 J. Phys. Cl}246l Woo C W L976 in The Physics of Liquid and Solid Helium, Part I, ed. K H Bennemann and J B Ketterson (Wiley, New York), p.349 Wu M K, Ashburn J R, Torng C J, Hor P H, Meng R L, Gao L, Huang Z J, Wang Y Q and Chu C W 1987 Phys. Reu. Lett.58 908 Yamaguchi Y 1983 Prog. Theor. Phys.69 1377 Yamaguchi Y, Katayama T and Ishii C L982 Phys. Leu. 9LA,299 Yang C N 1962 Reu. Mod. Phys. Y 694 Yanof A W and R"ppy J D 1974 Phys. Reu. Lett.33 631 Yarmchuk E J and Packard R E 1982 J. Low Temp. Phys. 46 479 Yip S 1985 Phys. Reu. B.32 291,5 Yip S 1986l. Phys. Clg l49L Yip S and Leggett A J 1986 Phys. ReÚ. Lett. 57 345 Yosida K 1958 Phys. Reu. tl:O 769

594

References

Zhang W and Kurkijärvi J 1988 J. Low Temp. Phys.73 483 Zhang W, Kurkijärvi J and Thuneberg E V 1985 Phys. Leu. Lt9A' 238 ZhangW, Kurkijärvi J and Thuneberg E V 1987 Phys. Reu. 83ó 1987 Zhang W, Kurkijärvi J, Rainer D and Thuneberg E V 1988 Phys. Reu.B.37 3336 Zinov'eva K N and Boldarev T S 1969 Zh. Eksp. Teor. Fiz. 56 1089 lSou. Phys. IErP 2e s8s (1e6e)l Zotos X and Maki K 1984 Phys. Reu. 830 145 Zotos X and Maki K 1985 Påvs. Reu. B3l7120

Author lndex

Note: page numbers in bold refer to reference section Aarts, J. 589: see also Steglich, F. 22 Abel, W. R. 28,45,561 Abragam, A. 23,394, 403, 561

Abraham, B. M. 576,586: see also Ketterson, J. B. 45. Roach, P. R. 104 Abramowitz, M. 278, 372, 561

Abrikosov, A. A. 34, 36, 59, 92,561 Abud, M. 165,561 Adams, E. D. 571: see also Flint, E. B. 365 Adenwalla, S. 542,561 Aharonov, Y.245,561 Ahonen, A. I. 94-6, 101-3, 188, 345, 363-4, 495, 497-8,561-2,572: see a/so Giannetta, R. W. 105

Ainsworth, T. L. 49,562,

5g

Akimoto, T. 575: see also Ichikawa, K. 488 Aldrich, C. H. 46,48,49,51, 55,57,562 Alexander, P. W. 592: see a/so Wellard, N. V. 473 Allen, J.F.2,562 Alonso, B. 521,562 Alpar, A.22,171, 5E5 Alvarez, L. W. I, 562 Alvesalo, T. A. 98, 106, 491, 561,562, 563,574: see ¿/.ço Ahonen, A. I. 363, 364.495: Archie, C. N. 106,430; Halperin, W. P. 98. 100 Amar, A. 56E: see also Davis, J. C. 435

Ambegaokar,Y. 14,117, 131,165, 185-6,193, 203,281 ,289, 395, 487 , 494,562 Amend. B. 570: see also Eska, G. 403 Amit, D. J.36,562 Anderson, A. C. 561: see also Abel, W. R. 28,45 Anderson, P. W. 6, 7,ll-12, 22-3 , 52, 54 , 66 , 73 , 82, 92, ll4, r23,136,150, 172,228-9,250,263, 265,266,280,289,293, 294, 320 , 337 , 339 , 346, 347 , 403, 483, 487 , 494, stt,524,550, 562-3, 565-6, 568-9, 5E4, 591: see also Blount, E. I. r95,196,203; Brinkman, W.F.127; Brueckner, K. A. 5; Engelsberg, S. 181, 183; Vollhardt, D.52,54 Andreev, A. F. 416,487, 488,495,563 Andres, K. 94,563 Andronikashvili, E. L. 109,

257,5ß'

574: see also

Hakonen, P. J. 109 Aniola-Jedrzejek, L. 5t6: a/so Roach. P. R. 98

.re¿

Anufriyev, Yu. D. 94,562, 563: see a/so Alvesalo,

T. A. 106 Arai, T. 291, 330, 563, 5E9 Archie, C. N. 106,490, 563, 574: see a/so Halperin,

w.

P. 38,95, 100 Ashburn, J. R. 593: see also

Wu. M. K.22 Ashida, M. 135, 495,533,

s44. ir63

Avenel, O. 105, 108, 380, 399, 494, 526, 542, 546, 563, 5E2,5E5,587, 590: see also Bloyet, D. 398;

Meisel, M. W.542; Piché,

L.

544; Salmelin,

R. H. 542; Varoquaux, 8.526,542 Babu, S. 49,563 Bagley, M. 335, 561,572: see also Gay, R. 107, 133, 335

Bagnuls, C. 478, 563, 569: see also Dombre, T. 478 Bailin, D. 128, 263, 280, 299, 563

Bais, F. A. 155,5E8 Balachandran, A. P. 251,563

Balatskii, A. V. 205,208, 563,591 Balian, R. 6, 7, 11, 80, 117,

128,165,167,59 Balinskii, A. A.234,5U Band. W. T. 580: see also

Main, P. C. 106,335 Baramidze, G. A. 431, 564 Baranger, M. 550,564 Bardeen, J.4,61,U,5U Bartolac, T. J. 381,564,565, 573 see also Bozler, H.

M.495; Gould, C. M. 377,381 Barton, G. ll4, 117-18,

120-1,157, r59, 165, 170,326, 494,59 Bashkin, 8. P.21.,59 Batenburg, P. 208,sEt Bates, D. M. 311,5(A Baym, G.26,36,59,497, 564.5E7: see also

596 Baym,

Author Index

G.-{ont'd.

Bowley, R. M. 479,565 Bozler, H. M. 102, 103,399,

Salomaa, M. M.497,

495,59,565, 573, 576,

498

Béal-Monod, M. T. 53,495,

5g

Bedell, K. S. 49, 51,59,123, 131,562,564, 5E6: see ¿/so Ainsworth,

T. L.

49

Bednorz, I. G.22,5g Berezinskii, V. L. I23, 495,

5& Berg, R. F.544,564, 569, 575: see a/so Engel, B. N. 546; Ihas, G. G. 541 Berglund, P. M. 565, 587: see also Bloyet, D. 398;

Salmelin, R. H.542 Berk, N. F.7,36,53,Sffi Bernier, M. E. 563, 565: see abo Avenel, O. 380, 399; Bozler, H. M. 102, 103, 399

Berthold, J. E. 106, 107,5(A, 584: see also Parpia, J. M.490 Bertsch, G. F. 46,564 Betbeder-Matibet, O. 438,

4gg,5g Betts, D. S. 24,56E,575,

579,5t7, 593: see also Dahm, A. J. 330,336; Hutchins, J. D. 330; Ling, R.-2. 330,336; Saunders, J.374 Beyer, J.491,5(A Bhattacharyya, P. 196, 298, 300-2,395,463, 479-9, 564,5E5: see also

Pethick, C. J.451,463, 465,466 Bialek, W. 559,564 Black, W. C. 561: see also

Abel, W. R. 28,45 Blaha, 5.292,5U Blatt, J. M. 550,564

Blount, E. I. 150, 195, 196, see qlso Brinkman, W. F. 186, 187, 364, 388

203,íffi,565:

Bloyet, D. 398,565 Bogoliubov, N. N. 68,524, 56s

Bohm, D.245,561 Boks, J. D. S. 495,594 Boldarev, T.^.2,576 Borisov, N. 5E3: see also Neganov, B. S. 93

Borovik-Romanov. A. S. 331,359, 402, 494, 565 Bouchiat, C. 557,565

57t: see a/so Bartolac, T. J. 381; Bates, D. M. 311; Gould, C.M.377, 381; Israelsson, U. E.

98,253; Lawson, D. T. 104

487-8,56é Buhrman, R. A. 574:

see also

Halperin, W. P. 95 Bullough, R. K. 577: see also Kitchenside, P. W. 384 Bun'kov, Yu. M. 359,392, 402, 565, 566, 574, 576:

Brand, H. 198, 416,431,46I, 492,510,565, 569, 5E6: see also

Buchholtz, L. J. 185, 289, 290,291,313, 369,394,

Dörfle, M. 461,

464

see also

Borovik-

Romanov, A. S. 331, 359,494; Hakonen, P. J. 254, 391,392, 394; Ikkala, O. T. 109,391 Burckhardt, T. W. 46,56ó Butcher, K. J. 578: see also Lea, M. J. 45

Bredl, C. D. 589: see aßo Steglich, F. 22 Brenig, W. 7,36,565 Brewer, D. F. 331,568,575, 579, 5E7: see also Dahm, A. J. 330,336; Hutchins, J. D. 330, 365; Ling, R.-2. 330,

Calder, I. D. 542,566, 5t2: see also Mast, D. B. 105,

336, Sachrajda, A. S. 490; Saunders, J. 374

Campbell, C. E. 47,59, 566,

Brinkman, W. F. 7, 11,

542 578

22-3,36,52-4,82,92, 102,114,123,126, l2g,

Candela, D. 582: see also Masuhara, M. 45,535,

736, L82,184, 186-8, 200,204,223,229,231, 263-6,289,337 ,348, 354, 358, 360-2,364, 369,384,385, 388, 400,

Capel, H. W. 135,5E3: see c/so Nijhoff, F. W. 135 Carless, D. C. 468, 471,491,

403, 473, 434, 482, 499, 505, 510, 5ll, 547, 562, 565, 56E-9,5E0, 5t4, 589: see a/so Blount, E. I. 195, 196,203; Engelsberg, S. 181, 183, 364; Osheroff ,D. D.32, 102, 103,283,345,364,

H. l8ó, 282,284,384,510 Bromley, D. J. 369, 374,375, 498,565 Brooker, G. A. 38,40,565, 387,510; Smith,

5E9

Brown, G.8.49,59,562, 563,565: see also Ainsworth, T. L.49 Bruder, C. l2l, 150, 161, 163, 167 , l7t, 5({t Brueckner, K. A. 5, 46,Sffi Bruinsma, R. 278, 287,3I3, 373,374,380-1, 56ó, s81

Brusov, P. N.495, 521,524, 526.547,56(t Buchal, C. 5E4: see also Owers-Bradley, J. R. 22 Buchanan, D. S. 489,56ó Bucher, E. 104,563

547

56ó

Carlson,

J.46,5U

Carneiro, G. M. 36,56ó,5E5 Carney, J. P.491,56ó Carton, J. P. 135,567 Castaing, B. 54,567

Castelijns, C. A. M. 491, 494,567

Caudrey, P. J - 577: see also Kitchenside, P. W. 384 Ceperley, D. M. 46,567 Chainer, T. 488, 504,567 Chapellier, M.572:- see also

Godfrin, H.495 Chechetkin, V . R. 228, 473, 523,567, 572 Cheng, 'f . P. 172, 567 Chervonko, E. 534,567 Chester, G. V. 579: see also

Lee,

M.4.46

Chocolacs, H. 5E4: see also

Owers-Bradley, J. R. 22 Choi, C. H.206,567 Chu, C. W. 593: see also'V,lu,

M. K.22 Clark. W. G. 572: see also Godfrin, H. 495 Clarke, G. R. 580: see also London, H. 93 Coates, K. F. 491, 567: see

Author Index a/so Castelijns, C. 434

597

A. M.

326,330 den Breems, A. 56é,5t3:.çee

Coffey, D.37,567 Coleman, S. 129,567 Collan, H. K. 562: see also Alvesalo, T. A. 106,491

alsoCapel, H. W. 135; Nijhoff, F. W. 135 de Bruyn Ouboter, R. 56t:

Combescot, M.511,533, 546,567 Combescot, R. 86, 195,

de Gennes, P. G. 15, 92, 182, 189, 196,227,231, 286, 289 , 298 , 319 , 337 , 511, 562, íffi: see also

205-6,208,251,395, 396, 398-9, 416,420, 428. 431,433,474, 4779,480,493,499, 503, 505,517,524,526,528, 532-3, 546, 563, 565, 567-9, Stl:. see also Avenel, O.542:' Bagnuls, C. 478; Bloyet, D. 398; Dombre, ^f . 478 Coombs, G. J. 58E: see also Scherm, R. 45 Cooper, L. N. 4, 5, 61, 564, 567,56t: see al.so Bardeen, J. 64 Cornog, R. 1,562 Corruccini, L. R. 31, 102, 105, 358, 361-2,399400,403, 509,56E,584: see also Osheroff. D. D. 102, 103,345,364,387 Cowley, R. A. 5E7-9: s¿e ¿/.ço Scherm, R. 45; Stirling, W. G.45 Crooker, B. C. 330,56t, 576: see a/so Israelsson,

. u. E.98,253

Cross, M. C. 10, 23-4,128, 146-7, 154, 157, tgz, 184, 188, 196-7,2(n, 203-6,210,223,231,

263-6,299,300-l,320, 327,337 ,354,369,394-

5, 401, 403, 407, 4lt-12, 414, 416, 420, 434, 4923, 492, 495, 4gg, 571, 533, 548,565, 56E, 580,

5E4,5E9: see abo Paalanen, M. A. 182 Czerwonko, J. 90, 449, 542,

5(ú

Dahl, D. A. 182,184, 56t Dahm, A. J. 330, 336,568, 575: see a/so Hutchins, J. D. 330 Daniels, M. 8. 542, 56t, SEE: see abo Saunders, J.542 Das, P. 93,568 Davis, J. C. 495,56E, 5t5: see also Pekola. J. P.

see also

Das. P. 93

Ambegaokar, V. 14, 185,186, 193,203,299, 487,494 de Vaard, A. 565: see also Borovik-Romanov. A.

s. 494 de Vegvar, P. G. N. 359, 402,546,56t, 587: see ¿/so Polturak, E. 546; Sagan, D. C. 98 Delrieu, J. M. 182,568 Dianoux, A. J. StE: see also Scherm, R. 45 Dmitriev, V. V. 565, 566: see ¿/so Borovik-Romanov.

A.

S. 331 ,359, 494; Bun'kov, Yu. M. 359, 402

Dobbs, E. R. 24,51ú, 57E-E0,

5ß:

see also

Daniels, M. E. 542; Lea, M. J. 45; Ling, R.-2. 542,544:' Saunders, J. 542

Dombre, T. 205-6, 208 ; 251,

416, 428, 43t, 478, 526, 528, 532, 563, 567-9: see a/so Bagnuls, C. 478 Doniach, S. 7, 36, 53,495,

564,569

Donnelly, R. J. 257,498, 569,572 Dörfle, M. 198, 461,464, 565, 569: see also Brand, H.46r Dörre, P. 590: see also Tewordt, L. 127,523 Dow, R. C. M. 321, 569 Duru, I. H. 311,569 Dy, K. S. 56,569 Dyugaev, A. M. 55,569 Dzyaloshinskii, I. E. 92,561 Eastop, A. D. 575: see also Hook, J. R. 287 Ebisawa, H. 272, 278, 344,

353-4,360,376-7 ,3956, 398, 401,477,492,

505,521, 546,563,567, 569,5E1: see also

Avenel, O. 105.526 Eckern, U. 438,569 Eckstein, Y. 586: see also Roach, P. R. 98

Edwards, D. O. 495,569, 570,575, 5E2: see also Feder, J. D. 98; Hoyt, R. F. 98; Masuhara, M. 45,535,547

Efetov, K. B. 128.569 Eilenberger, G. 72, I 18, 569 Einzel, D. 403, 454,456, 459, 462-4, 466, 469,

471,474-5,479_9,496, 489,490, 492-3,523, 544,590, 593: see also Tewordt, L. 127,523 Eisenstein, J. P. 98, 326,

330-1, 490,491,492, 569, 5E9:.ç¿e ¿/.ço Swift,

G. W. r82 Emery, V. J. 5, 6, 545,569 Engel, B. N. 32, 542,546, 569,575: see alsolhas,

G. G. s4l Engelsberg, S. 7, 36, 53,

tgt-3, 364,565, 569, 5t4, 5t9: see also Osheroff, D. D. 102, 103, 345, 364,387; Smith, H. 186,282,284, 384, 510 G. 102, 105, 403,479, 503.570-l: see also

Eska,

Einzel, D. 403; Fomin,

I. A.

403

Essmann,

U.

Fak. B. 58t:

109, 570

see also

Scherm, R. 45

Fal'ko, V. I. 488,570 Fantoni, S. 5t1: see also Manousakis, E. 47 Faraj, E. 575: see also Hook, J. R. 287, 289,492 Fay, D. '7, 127,570, 57E, 590: see also Tewordt, L. 127,523 Feder, J. D. 98, 569,570: see ¿Iço Edwards, D. O. 495 Feenberg, E. 47,53, 59, 570 Fetter, A- L. 23, 92, 135, 180, 184, 197-8,233, 239,246,248,256,259, 262-3,286-91,301-4, 308, 311-13,319-21, 326, 337 , 363-5, 369 , 371-2,393, 385,388-9, 403,497,499, 56ó, 570, 590,592-3: see also

Author Index

598 Fetter, A.

L.--{ont'd.

Freeman, M. R.364,

Vulovic, V.2.233,243, 389, 390 Feynman, R. P. 257, 327 262,

37'.1.,

-8,

337,570 Fisher, M. E. 329,57E Fishman, F. 290, 359, 361,

402,570-l Fishman, R. S. 45, 526,534, 532,571 Flachbart, K. 565: see also Borovik-Romanov. A.

s. 359 Flint, E. B. 365,571 Folk, R. 359, 361,570 Fomin, l. A. 23, 182, 186-7, 205,283,331, 358-9, 390,402-3,570-1, 573, s86

Fomin, N. V. 244, 262,589:-

Golo, V. L. 402 Ford, P. J. 574: see alsoF{all, H. E.93 Forster, D. 188, 434,57t Franz, W. 589: see also Steglich, F. 22 Frautschi, S. 165,571 Fredkin. D. R. 564: see also Béal-Monod, M. T. 53 Freeman, M. R. 3(A,495, see also

571

Friedman, L. 5E7: see also Sagan, D. C. 98 Frossati, G.572: see also

Godfrin. H. 495 Fujita, T. 238, 251, 259, 261, 495,571,5t3, 590: see ¿/so Nakahara, M.26L; Ohmi, T.237,245,254, 292;

Tsuneto,T.26l

Fulco, J. R. 575: see also Ho,

T. L. 208 Fulde, P.22,571 Furukawa, F. 576: see also Katayama, '1. 361, 4Oz

Z. 5,571 Gammel, P. L.46,107, 108, 190, 330, 332,334,566, 571.574: see also Hall, H. E.331 Gao, L. 593: see ø/so Wu, M. K.22 Garg, A. 206, 251, 572, 5t9: see also Stone, M. 205 Garibashvili, D. I. 574: see a/so Hakonen, P. J. 109 Gay, R. 107, 333, 335, 572 Geilikman, B. T. 473, 572 Germain, R. S. 571: see also

Galasiewicz ,

Giannetta, R. W. 102, 105, 381, 403, 54r,564,572: see also Berthold, J. E.

A. 8. 298, 572,

575: see ø/so Hoffberg,

M.22 Glyde, H. R. 46,572 Godfrin, H. 495,572,

SEE:

R. 45 Goldman, M.23,403,561 Goldstein, L. 55,572 Goldstein, S. 57t: see also Lawson, D. T. 103, 104 Goldstone, J. 501, 572 Golo, V. L. 150,291,353, 402, 403, 508,572,573 see also Scherm.

Gongadze,

A. D.

183,252,

253,254,277 ,288,363, 507,573 Goodkind, J. M. 104, 105,

584,sEt Gorkov, L. P. 22, 92, 150, 561,573,591 Götze, W. 593: see also Wölfle, P.497 Gould, C. M. 374,377 380-1, 564'573,576: see abo Bartolac, T. J. 381; Bates, D. M. 311; Israelsson, U. E. 98,253 Gould, S. G. 575: see also Gully, W. J. 182,399; Hook, J. R. 287,288, 482

Grabinski, M. 403, 493,573 Graham, R. 412, 414, 416, 431, 505, 565, 569, 573, 586:. see

43r Guernsey, R. W. 106,573 Gully, W. J. 96, 182,399, 565, 570, 573,57t, 584:

106, 107

Ginzburg, V. L. 127,572 Glaberson, W. I. 257, 572 Glassgold,

Castelijns, C. A. M.

49I,494; Coates, K. F.

495

¿/so Brand. H.

461; Dörfle, M. 461,464 Greaves, N. A. 183, 347,

498,573 Greytak, T. J. 96, 573,592: see also Webb, R. A. 98

Greywall, D. S. 28, 29, 31-2, 36,54,55, 59, 95-9, 573 Grilly, E. R. 589: see also Sydoriak, S.,"ç. 1 Gros, C. 5ß: seelalsp Seiler, K. 37, 55Gu, G.-Q. 416,573 Guckelsberger, K. StE: .see ¿Iso Scherm, R. 45 Guénault, A. M. 491,sffi, 567, 57t: see also Carney, J. P.491;

see

abo Bozler, H. M.

102, 103,399; Feder, J.

D. 98; Lawson, D. T. 103, 104; Osheroff, D. D. 18,99, 1(n, 101 Gurgenishvili, G. E. 431, 507,56/.,566,571: see alsoBaramidze, G. A. 431; Bun'kov, Yu. M. 392;Gongadze, A. D. 183,252,253,254,277, 288,363, 507 Gutzwiller, M. C. 52,53,573 Haavasoja, T. 561,

562:. see

A. I. 363, 495; Alvesalo, T. A. 98 Hahn, K. D. 57E: see also Kotsubo, V.326 ¿/so Ahonen,

Haikala, M. T. 561: see also Ahonen, A.f..94,96, 363,364 Hakonen, P. J.15,24, 109-11 ,129,233,242, 252,254,2&,285,335, 371, 388, 390-4,5(6, 574,576,5t5, StE, 591: see also Ahonen, A. I. 94,96; Bun'kov, Yu. M. 392; tkkala, O. T. 109,

391; Pekola, J. P. 330, 391,393; Seppälä, H. K.

241,243,390 Hall, H. E. 23, 93, ll1, 134, 190, 196, 205, 2l'1., 298, 320,327,331, 333-5, 337,416,478,427, 434, 473,49r,499, 563, 566, s7l., 572, 574-5, 577 , 5t0,592: see also Bagley, M. 335; Carless, D. C. 468, 471,491; Gammel, P. L. 107, 190, 330; Gay, R. 107, 133, 335; Hook, J.R.287, 288,482; Kiewiet, C. W. 106; Main, P. C. 106, 335: Wellard, N. V. 473 Halperin, W. P. 95,98, 100, 542, 547 , 5(ú, 574, 577 , 5t2,SEE: see also Calder, l. B. 542; Ketterson, J. B. 538, 542; Mast, D. B. 105,

Author Index 542; Meisel, M. W. 526, 541,544; Shivaram, B.

s. 526,542 Ham, T. 8.312,574,575: see also

Hu, C. R. 294

Hammel, E. F. 5t9: see also Sydoriak, S. G. 1 Hara, J. 4(4,499,574, 5t4: see abo Ono Y. A.4U

Harris, C. G.299,574 Hasegawa, Y. 247, 347, 403, 523,574 Hebral, B. 568: see also Crooker, B. C. 330 Hernadi, S.I. 46,572 Herring, C. 53,574 Higgs, P. W. 172, 574 Hilton, O. A. 589: see also Stirling, W. G.45 Hirai, A. 576, SEE: see also Ishikawa, O.401; Simola, J.T.1I0,263, 498

Hirashima, D. S. 534,574 Hirayoshi, Y. 569: see also

Einzel,D.492 Hirschfeld, P. J. 294, 569, 575: see also E,inzel, D. 492

Ho, T. L.17,196,210, 229-30,261,298-90, 321,331,333,416,4r920, 423, 434,5&, 572, 575,582: see also Bhattacharyya, P. 196, 298,300,301,302; Gammel, P. L. 107, 108, 130,322,334 Hoffberg, M.22,575 Højgaard Jensen, H. 38,40, 489,491-2,505, 569, 575, 590: see also Einzel, D. 486, 488,492 Hong, D. K. 129,575 Hook, J. R. 23, 111,134, 190 , 196, 205 , 211. , 287 8, 298,312, 327, 327 , 333-5,337, 416, 427, 434, 473, 482, 492, 499, 563,566, 569,572,5745,5E0, 592: see also Bagley, M. 335; Carless, D. C. 468, 471,491; Gay, R. 107,133,335; Main, P. C. 106,335; Wellard, N. V. 473 Hor, P- H. 593: see also Wu, M. K.22 Houghton, A. 464,575 Hoyt, R. F. 98,575, SE2: see

599 a/so Masuhara,

M. 45,

535,547

Hsu, W. 22,575 Hu, C. R.292-4,301, 304, 3r2-r3,331,360,416, 43r,434,574-5, 5t1, 5E7

Huang, K. 66, 172, 188,575:

Wu, M. K. 22 Huang, Z. J.593 Hubbard, J. 52,575 Hutchins, J. D. 330, 365, 56t, 57E: see also Dahm, A. J. 330, 336 see also

lchikawa, K. 488,575 Ihas, G. G.32,285,541, 542,544,564, 569, 575, 589: se¿ a/so Engel, B.

N. 546 Ikkala, O. T. 109, 391,574, 576: see ¿/so Hakonen. P. J. 109, 110, 129,399, 391,392,393 Ishii, C. 576,ífd/'593: see also Katayama, T. 359, 361,402; Ooiwa, K. 359; Yamaguchi, Y. 359 Ishikawa, M. 205, 416,576 Ishikawa, O. 401,576 fslander, S. T. 574, 576: see ¿/so Hakonen, P. J. 109, ll0, 129,388,391,392, 393; Ikkala, O. T. 109, 391

Israelsson, U. E. 98, 253,576 Jacak, L. 135,523,576 Jacobsen, K. W. 285,326,

388,576 Janke, W.326,576 Jarió, M. V. 150, 165,576 Jiang, Q. 580: see abo Lu, S.

T.

403

Johansson, B. 137,188, 582 Johnson, D. L. 508,576 Johnson, R. T. 106, 573,576, 585.593: see also

Greytak, T. J. 96; Paulson, D. N. 96, 103, 540,541:' Webb, R. A. 98; Wheatley, J. C. 93 Jones, D. R. T. 128,576 Jones, R. B. 117, 12L,165, 168,576 Josephson, B. D. 108, 494, 495,576 Kagan, M. Yu. 416,563

Kalos, M.

H. 46,567,579,

5E8: see also Lee,

M. A.

46

Kamenskii, V. G. 390, S71 Kamerlingh Onnes, H. 2, 4, 576

Kanamori, J. 52, 576 Kane, J. W. 562: see also

Amit, D. J.36 L.2, 4, 576 Katayama, T. 359, 36I,402, Kapitza, P.

576,584, 593: see also Ooiwa, K.359; Yamaguchi, Y. 359 Kats, E. I.402,508, 564, 572:. see

¿/so Balinskii.

A.4.234

Kaul, R. 128,576 Kawamura, K. 574,584: see also Hara, J. 464; Ono, Y. A.4& Keen, B. E. 45,576 Keesom, W. H. I,2,576 Keith, V. 573: see aßo Guénault, A. M.491 Keller, J. 571: see also Fulde. P.22 Keller, W. E. 59,576 Kennedy, C. J.573: see aßo Guénault, A. M. 491 Kesaev, V. I. 402,576 Ketterson, J. B. 45, 104-5, 538, 542, 547,561, íffi, 576-7,5t2, 5t6-8: se¿ a/so Adenwalla, S. 542; Calder,I. D. 542; Mast, D. B. 105,542; Meisel, M. W. 526,541,544; Roach, P. D. 497,498; Roach, P. R. 104; Shivaram, B. S. 526, 542

Khalatnikov, I. M. 34, 36, 59,327,407,416, 425, 434, S6t,577-E Khanna, F. C.46,572 Kharadze, G. A. 379,38L, 431,507, S&,sffi,573, 576-7: see also

Baramidze, G. A. 431;

Bun'kov, Yu- M.392; Gongadze,

A. D.

183,

252,253,254,277 ,288, 363,507; Ikkala, O. T. 109, 391

Khazan, M. V. 245,512,5I5, 518, 542,577,591 Khokhlov, R. V. 546,586 Khriplovich ,I. B. 559,577

Kieselmann , G. 488,577

Kiewiet, C. W. 706,577,

Author Index

600 Kiewiet, C. W.-{ont'd. 5t0: serr a/so Main, P. C. 106,335

Kim, J. S. 150, 165,571,577 Kim, N. SE3: see also Movshovich, R. 526 Kinsey, B. B. 583: see also Oliphant, M. L. E. 1 Kitchenside, P. W. 384,577 Kivelson, S. A. 550,5E6 Kjäldman, L. H. 326, 494, 577

Kleb, R. 585: se¿ a/so Sköld, K.45 Kleinberg, R. L. 99, 311, 575,576,577, 5t5, 5E7, 592: see a/so Johnson,

R. T. 106; Paulson, D. N. 103; Sager, R. E. I02,399; Webb, R. A. 101,102,103 Kleinert, H. 128, 301-4, 321, 324,326, 521, 526, 547 , 576-7 Kléman, M. 213, 216, 337, s90

Koch, V. E. 524, 533, 534, 54'1.,

544, 545, 577, 593

Kodama, T. 503, 561, 575, 577: see ¿/so lchikawa, K. 488 Kojima, H. 105,430, 503,

567,575,577, 579, 585, 586: s¿e ¿Iso Chainer. T. 488, 504, 505; Ichikawa, K. 488; Lu, S. T.403; Paulson, D. N. 96, 101 Kokko, J. 380, 497, 561, 562, 577:. see

ø/so Ahonen.

A. I. 101, 103,363,495, 497,498 Kondo, J.497,577 Konyshev, V. A. 208,5Á1, 591: see a/so Balatskii, A. V. 205.208

Kopnin, N. E. 261, 335,489, 495,577,591 Kop, T. 546,569,577: see alsoE,inzel, D.403 Korhonen, J. S. 5E3,SEt: s¿e ¿/so Nummila. K. K. 498; Simola, J. T. 110, 263,390,498 Korshunov, S. E. 495,577 Kosterlitz, J. M. 495,57E Kotsubo, V.325,578 Krotscheck, E. 47, 52, 57, 59,57E

Kruppa, D. 575: see also Hutchins, J. D. 365

Krusius, M. 336,561, 56ó, 574,577,5t5,SEE: see ¿/so Ahonen, A. I. 94, 95,96, 102, 188,345,

363,364; Bun'kov, Yu. M.392; Hakonen, P. J. 129,254,390,391,392, 394; Paulson, D. N. 103, 104, 336, 448,544,545; Pekola, J. P. 330, 391, 393; Seppälà,H. K.241, 243,390 Kubota, M. 584: see also Owers-Bradley, J. R. 22

Kuchnir, M. 5E6: see also Roach, P. R. 104 Kumar, P. 195, 272-3, 275, 277 -80, 365, 372, 37 6-8, 390-1, 575, 57E, stl, 591: see ¿/so Hu, C. R. 292,293,294 Kurkijärvi, J. 336, 377,488, 4W, 497, 499,570, 5778,590,593-4: see also Kjäldman, L,H.326, 494; Thuneberg, E. V. 498; WöIfle, P.497; Zhang, W.487 Kuroda, Y. 123, 127, 135, 57t Kurten, K. E. 47,57t Kurti, N. 94,578 Kyynäräinen, J. M. 98,57t, 587: see ¿/so Salmelin. R. H.542 Landau, L. D. 4, 25, 27, 28,

32,33,43, 50, 59, 78, l\3, 162,321, 577 , 578 Langer, J. S. 257, 329, 57E Lawson, D. T. 103, 104,573, 57E: see ø/so Gully, W. J. 96 Layzer, A.7,I27,57t Lea, M. J. 45,578 Lebedev, V. V. 416, 577,578 Lee, D. M. 7, 23-4, 92, 95,

lll,136,188, 374, 380, 39% 403,565, 568, 5724 ; SV9, 582-4, 586-7 : see

95,96,99, 100, 101; Polturak, E. 54ó; Sagan, D. C.98 Lee, M. A. 46,579 Lee, P. A.22,579 Leggett, A. J. 6, 10, 13-14, 18-19, 22,24,59,96, 90,92, 102, 123-4, 1289, 133, 136,146,154, 174-9, r82,188,244, 247 ,298,339-43,347 , 355,395-7 ,399, 401, 403,439, 479, 493, 4879,499,511, 546, 550-7, 560, 573, 578-9, 593 Lekner, 1.496,576 Leman, A. A. 402,403, 573: see also Golo, V. L.402 Levin, K. 37,53, 56, 59, 127, 131,135,136,579 Levy, M. 395,562 Lhuillier, D. 416,579 Li, L. F. t72,567 Liburg, M. 5t3: see also Neganov, B. S. 93 Lieke, W. 5t9: see also Steglich, F. 22 Lifshitz, E. M. 32, 33, 78, 92, 113,162,57E, 579 Ling, R-Z. 330,336, 542,

544, 579,5E0

Lin-Liu, Y. R. 188,277-8, 283,304,309,

31 1, 314, 376,387 ,577, 5E0, s8l, 591: see ¿/so Kleinert, H.302,303, 304;

Vollhardt, D. 309,310, 365,367,374 Liu, M. 10,144-7,154, I57, 159, 188,210,298,3(n2, 361, 407, 409, 4lr-r2, 414, 416, 418, 420, 431, 434,493,507, 509, 5ó8,

573, 5E0,589

London, F. 4,5t0 London, H. 93,5E0 Loponen, M. T. 562: see also Alvesalo, T. A. 106,491 Lounasmaa, O.V. 15,24, 104,337 , 494,561,562,

'aliöBozler, H. M. 102, l03l 399; Corruccini, L. R. 31; de Vegvar, P. G. N. 546; Giannetta, R. W. 102, 105,381; Gully,

574,5t0,5t3,5t5, 587: see also Ahonen, A. I.

w.

P. J. 109, 110, 129,399, 391, 392, 393; Pekola, J.

J. 96, 182,399;

Halperin, W. P.95; Lawson, D. T. 103, 104; Movshovich, R. 526; Osheroff, D. D. 7, 18,

, 96, 36? , 364 , 495 , 497,498:' Alvesalo, T.

94

A.

106,491; Hakonen,

P. 107,110, 190, 330, 391,393; Salmelin, R. H.542

Author Index

601

Love, A. 128, 217, 263, 280, 299,563,576, 5E0: s¿e

D. 128; Jones, D. R. T. 128 Lu, S. T. 105,403,504,5E0 Luey, K. 565; see also ¿Iso Bailin,

Bozler. H. M. 495 Lyden, J. K. 573: see also Guernsey, R. W. l0ó Ma, S. K. 564: see also Béal-Monod, M. T. 53 Maack Bisgaard, T. 575: see also Højgaard Jensen, H. 489.491

McClure, M. G. 205,579 McCoy, R. J. 573: see also Guernsey, R.

W. l0ó

Mclnerney, M. F. 135,5E0 McKenzie, R. H. 546,580 Magradze, O. V. 5E7: see ¿/so Salmelin, R. H. 542 Main, P. C. 106, 335,563, 577,St{l: see also Bagley, 335; Kiewiet, C.

w.

5E2

Markkula. T. K. 574: see also Hakonen, P. J. 109 Marquardt, W. 5E8: see also Schopohl, N. 533 Martin, P. C. 417,ítz Mast, D. B. 105,542,íffi, 582,SEE: see also

Calder, L D.542¡' Meisel, M. W.526; Shivaram, B. S. 542 Masuhara, M. 45, 535,547, 582

Mathews, P. W. 576: see also Keen. B. E. 45

Mattuck, R. D. 137, 188,5t2 Meisel, M. W. 526,541,542, 544,547,577, 5t2, 5E6, 588: se¿ ¿/so Ketterson. J. 8.538,542; Roach, P. R.98; Shivaram, B. s. s26,542 Mendelssohn,

106

Maki, K. 15,23,182, 188, 195, 208, 230,233,2434,252,263-4, 267 ,2723, 27 5, 277 -80, 283, 285, 287, 289, 291-3, 3 13- ls, 318,326,337 ,344,3536, 360, 365,372-4, 37683, 387, 388-90, 391, 395, 399, 401,403,464, 505, 507, 5t3, 521, 523, 546, 563, 566,569, 575, 577,5E0-1, 5E3, 591, 594: see ¿/so Hu, C. R. 292, 293, 294; Kleinert, H. 302, 303, 304; Lin-

Liu, Y. R. 304, 309, 3l I, 376; Vollhardt, D. 309, 370, 324, 325 , 365 , 367 , 374

Makrotsieva, V. 565: see also Borovik-Romanov, A.

s. 494 Mamaladze, Yu.

Salmelin, R. H.542 Manousakis, E. 47,581 Markelov, A. V. 489,494,

G.

109,

257,563 Mammiashvili. G. 5t5: see also Pekola, J. P. 330, 391,393 Manninen, A. J. 578: see also Kyynäräinen, J. M. 98 Manninen, M.'1. 326, 330, 336,491,562, 578, 5t0, 5E1,587: see also

Alvesalo, T. A. 98; Lounasmaa, O.

Y.

494;

K. 1,5t2

Mendoza, E. 580: see also

London, H. 93 Meng, R. L. 593: see also

Wu, M. K.22 Mermin, N. D. 17, 24,ll4, 117-18, 129, t3t,165, 169,

l7l-2,

192, 196,

205,2t0,213-14,220, 224,229,250,263,266, 281,289-90,292,293, 295,33r-2,337, 416, 419-20, 428-9, 434, 550-1, 562, 564, 575, 5t2: see also Bhattacharyya, P. 196,

298,300,301,302 Meschede. D. 59t: see also Steglich, F. 22

Metzner, W. 53,5t2 Meyerovich, A. E. 21,59

Michel, L.

l4l,

166-8, 188,

2t3,5t2 Mikeska, H. J.7 ,565:

se¿

akoBrenig, W.7,36 Millis, A. J.74,5E2 Mills, D. L.495,Sil Mills, R. L. 56t: see also Cooper, L. N. 5 Mineev, V. P.23,24, l4l, 150, 188,204-6,208, 213,217,220,223,2301,242,250-1,254,263, 265-8,270,278,280, 292,335,337,416, 433,

434,499,542, 550, 5ú3, 574, 5t2_3,597, 5gl_2: see also

Hakonen, p. J.

254,39r,392,394; Mermin, N. D. 220; Salmelin, R. H. 498.542

Misener, A.D.2,562 Mishra, S. G. 36,5t3 Miyake, K. 205, 416,420, 434,495,576, 5E3: see a/so Ishikawa, M. 416 Mizusaki, T. 576: see also Ishikawa, O. 401 Monastyrsky, M. I. 150,572: see also

Golo, V. L.291

Monien, H. 346,494,5E3 Montgomery, R. 359, 361, 5E3 Moody, J. 564: see also

Bialek, W. 559 Moore, M. A. ll4, ll7-18, 120-1,157, l59, 165, 170,326, 494, 563-4, 576: see ø/so Bailin, D. 128; Jones, D. R. T. 128 Morel, P. 6, 7, 12, 73, 82, 562,Sffi: see also Brueckner. K. A. 5 Morii, Y. 567: see also Chainer, T. 488, 504, 505

Movshovich, R. 526,583

Mueller, R. M. 571,SfM-: see a/so Flint, E. B. 365; Owers-Bradley, J. R. 22 Muething, K. A. 570: see ¿/so Feder, J. D. 98 Mühlschlegel, B. 75i 82,5t3 Mukharskiy Yu. M. 565,566: see also BorovikRomanov, A. S. 331, 359, 494; Bun'kov, Yu. M.359,402 Müller, K. A.22,5ø Mussett, S. G. 567, 573: see a/so Castelijns, C. A. M. 494; Coates, K. F. 491; Guénault, A. M.491 Muzikar, P. 22, 204-6, 208, 242, 416, 428-9, 434, 550, 567, 5t2-3, 5E9: see ¿/so Stone, M. 205

Nagai, K. 123, 127,135,

205-6, 416, 420, 428, 482, 499,503, 508, 521, 523,533,544, 563, 575, 583-4: see also Hara' J. 4641-Højgaard Jensen,

H. 489,491; Ono' Y. A.

4&

Author Index

602 Nagi, A. D. S. 57E Nair. V. P. 572: see also Garg, A. 251 Nakahara, M. 252, 285, 261, 291,381,388, 391, 495, 571,

stl, 5ß; see also

Fujita, T. 259,261, 495 Nakajima, S. 7,5E3 Namaizawa ,H. 523,574 Nambu,Y.437,524,5t3, 5E4

Nayak, V. S. 569: see also Edwards, D. O. 495 Neganov, B. S. 93,5E4 Nenonen, S. A. 580: see also Lounasmaa, O. V. 494

Neumaier, K. 569, a/so Eska,

570:. see

G. 102, 105,

479,503

Nijhoff, F. W. 135, 584: see alsoCapel, H. W. 135 Novikov, S. P. 572: see also Golo. V. L.291 Nozières, P. 26, 28, 33, 51, 54, 59, 438, 499, 550,

56/-.567.5t4, 5E5 Nummila, K. K. 498,5t3, 585,5E8: see also Pekola, J. D. 107, 110, 190, 330, 391, 393; Simola, J. T. I10,263,

390.498

Ohmi.'r. n7,

245, 254, 263,

291,292,571, 5t3, sEE,

Fujita, T. 251,259,261,495; Nakahara, M.261; Seppälä, H. K. 241 ,243, 59n: see c/so

Tsuneto,T.26l Oliphant, M. L. E. 1,5E3 Ono, Y. ^.464,574,5t4: Hara, J. 464 see also Onsager, L. 140,585 Ooiwa, K. 402,576,5E4: see also Katayama, T. 359, 390;

402

Osgood, E. B. 104,5t4 Osheroff, D. D. 7, 18,23,

32,95-6,99-103, 105, lll, 128,283,336,3456, 358, 361-2,364,387 , 399-400,403, 508,510, 565,56t, 569,573, 5E4: see also Brinkman, W. F. 186, 187,364,388;

Corruccini, L. R. 3l; Gully, W. J.96; Paalanen, M. A. 182 Østgard, E. 46,59, 5E4

Ostrowski, G. E. SEt:.çee a/so Sköld. K. 45

Ott, H. R.22,5U Owen, J. C. 47,5U Owers-Bradley, J. R. 22, 566,5E2, 5t4: see also Calder, I. D. 542; Mast,

D. B. 105,542

A. 182,336, 561.562, 577, 583, 5t4:

Paalanen, M.

Ahonen, A. [. 95, 101, 102, 103,345, 363,3(A, 495, 497 , 498 Packard, R. E. 98, 109, 331, 492,5(8,569, 585, 5E9, 593: see a/so Davis, J. C. 495: Eisenstein, J. P. 98, 326,49O:. Kokko, J.380, 497; Pekola, J. P. 107, ilO, 190, 326,330,333, 391,393; Swift, G. W. see also

499,562, 56/,-7,569, 571,5E7: see also

Aldrich, C. H. 46,48; Baym, G.497; BhattacharYYa, P. 196, 395, 463, 478, 479;

Brown, G. E.59; Fomin, I. A. 182; Salomaa, M. 497,498 Pfitzner, M. 22, 49, 51, 52, 56,57,58, 5t5 Piché, L.544,563, 5t5: see ¿/so Avenel , O. 542,546 Pickett, G. R. 566,567,573: see aßo Carney, J. P. 491; Castelijns, C. A. M.494; Coates, K. F. 491; Guénault, A. M. 491

182

Palmer, R. G. 280,292-4, 562

Palmeri, J. 489, 5t4 Pandharipande, V. R. 47,

581,5M: see also Manousakis.E.4T Panoff. R. M. 46,5U Parodi, P. 582: see also

Martin, P. C.417 Parpia, J. M. 106, 108, 330, 336.490, 493, 494, 569, 57E,5M: see also Kotsubo, V - 326 Passvogel, 'f . n7 , 245 , 246, 247,262,584, 5t5 Pati. J. C. 173,5E5 Patton, B. R. 58, 128,5t5 Paulson, D. N. 14,96, l0l, 103-4, 182,336,484, 540-1, 544-6,555,573, 577-t,585:. see also Greytak, T. J. 96; Kleinberg, R. L. 99; Kojima, H. 105,504; Krusius, M. 336 Pekola, J. P. 107, 110, 190, 326,330,336, 391, 393, 491, 56t, 57E, 5t0, sEl, 5t5, 587: see also Davis, J. C.495; Kyynäräinen, J. M. 98; Lounasmaa, O. Y. 494; Manninen, M.T.326: Salmelin, R. H.542 Pelizzari, C. A. 45,46, 58E: see also Sköld. K. 45 Penrose,

Pershan. P. S. 5E2: see also Martin, P. C.417 Pethick, C. J. 26, 36-7, 56, 59, 451, 463, 465-6, 483,

O.

140,5E5

Pines.

D.22,26,28,46,

48-9,51 ,55,57,59, t7l,562,564, 575, 5t5: see also Aldrich, C. H. 46.48 Pitaevskii, L. P. 5, 92,579, 5E6

H. 412, 414, 416, 431, 505, 510, 565, 573,

Pleiner,

5tó Pobell, F. 5M: see also Owers-Bradley, J. R- 22 Podd'yakova, E. V. 359,586 Polturak. E. 546, 56t, 572, 587: see also de Vegvar, P. G. N. 546; Giannetta,

R. W. 105; Sagan, D. C. 98

Poluéktov, Yu. M. 319,358, 506.5Eó

Pomeranchuk, I. 104, 5t6 Popov, V. N. 495, 521,524,

526,547,562,56 Price, P. B. 5E5: see aLso

Pekola, J.P.326 Privorotskii, I. A. 290,488, 571,5Eó Pushkina, N. I. 546,5tó Putterman, S. J. 434,5t6

Quader, K. F.49, 131,562, 564,58ó: see also

Ainsworth,

^f

. L.49

Rainer, D. 23, 74, 124, 136, 182, 185, 198,206,208,

286-7,298, 300, 438, 487-8. 495, 498, 499,

Author Index

603

52r,534,562, 566, 56E, 577-E,583,5Eó, 588, 590,594: see also Ambegaokar, V. 14, 185,186, 193,203,289, 487,494; Kjäldman, L. H.326,494; Thuneberg, E. V.498, Zhang, W. 487

Ramakrishnan, T. V. 36, 583 Ramsey, N. 559,586 Rapp, R. E. 593: see also

Wheatley, J. C. 93 Rasmussen, F.

8.

574: see

alsoHalperin, W. P.95, 98. 100 Reppy, J. D. 105, 108,257, 329-30, 336, 504, 563-4,

568,571,574,578, 5E4, 593: see ¿/so Archie, C. N. 106,490; Berthold, J. E. 106, 107; Crooker, B. C. 330; Gammel, P. L. 107, 108, 190,330,332, 334; Hall, H. E. 331; Parpia, J. M. 106,490 Rhodes, T. L. 490,584 Rice, M. J. 7,5E6 Rice, T. M. 53, 150,565,

579,stt, 59O: see also Lee, P. A.22; Seiler, K. 37,55 Richardson, R. C. 7, 23, 92,

95,102, ll1, 136, 188, 399,403,561-3, 565,

5ß,572-4,577-9,5E4, 5t6:

see ø/so

A. I.

Ahonen,

101, 103,363,495,

497,498; Archie, C. N. 106,490; Bozler, H. M.

102,103,399; Corruccini, L. R. 3l; Freeman, M. R. 364, 495; Giannetta, R. W. 105; Gully, W. J. 96, 182,3991' Halperin, W. P. 95,98, lfi); Kokko, J. 380; Lawson, D. T. 103, 104; Osheroff, D. D. 7, l8ì 95, 96,99,100, 101 Richardson, R. W. 575: see ¿/so Hoffberg, M.22 Rickayzen, G. 92,5Eó Riedel, 8.7 ,36,565, 5tó: see also Brenig, W. 7, 36 Ripka, G. 47,5U Roach, P. D. 497-8,576, 5E6: see ø/.ço Ketterson, J. B. 45, 104; Roach, P. R. 104

Roach, P. R. 45,98, 104-5, 547,576,586: see also Ketterson, J. B. 45, 104; Roach, P. D. 497,498 Robinson. F. N. 57E: see abo

Kurti, N.94 Rokhsar, D. S. 550,5t6 Rosen, G. 63,586 Roubeau, P. 574: see also Hakonen, P. J. 109 Rozhkov, S. S. 208,5t6 Ruderman, M. 575: see also

Hoffberg, M.22 Ruel, R. 430,5E6

Ruoff, M. 546, 563, 5E5, 5E7: see also Avenel, O. 542; Piché, L.544 Rutherford, E. 583: see also Oliphant, M. L. E. 1 Sachdev, S.582: see also

Millis, A. J.74 Sachrajda, A. S. 490,5E7 Sagan, D. C. 98,56t,5t7: see also de Vegvar, P.

G. N.546 Sager, R. E. 102,399,587, 592: see ølço Webb. R. A. 102,283,345,356, 360,479 Salam, A. 173,sts Salmelin, R. H. 263,498, 542,5t7 Saloheimo, K. M. 574: see ¿lço Hakonen. P. J. 109 Salomaa, M. M. 23,24,128, 150, 155, 234-7, 239-41, 243-8,250-4,256,259, 263,337 ,389,393-4, 403,499, 497-9, 542, 564, 569,573,5t2-3, 5E6-t, 592: see also Baym, G.497; Hakonen, P. J.129,254, 391, 392, 394; Salmelin, R. H. 498; Seppälä, H .K.241,243,390 Samalam, V. K. 545,5E7 Sanchez-Castro, C. 131, 564 Sandiford, D. J. 563,572, 5E0,584: see also Bagley, M. 335; Gay, R.

107,133,335; Main, P. C. 106, 335; Parpia, J.

M.490 Sarma, B. K. 543,561,56ó, Sn,5E2,5ß: see also

Adenwalla, S. 542; Calder, l.8.542: Ketterson, J. B. 538,

542; Mast, D. B. 105. 542; Meisel, M. W. 526. 541,544; Shivaram, B. s. 526, s42 Sartori, G. 166,561 Sasaki, Y. 576: see aßo

Ishikawa, O.

401

Saslow, W. M. 195, 301,304, 416, 431,434,493, 503, 511, 563,567, 575, 5E7:

Avenel, O. 542; Hu, C. R. 294 Sauls, J. A. 22, 45, 56, 123, 127 , 171,253, 526, 534, 542,546,570-1, SE0, 5E3,587: see also Fetter, A. L. 197,799,262; Muzikar,P.22 Saunders, J. 374, 541, 542, 5û,572,579, 580, StE: see also Dahm, A. J. 330,336; Daniels, M. E. 542; Giannetta, R. W. 105; Ling, R.-2.542, see also

544

Schäfer. J. 5E9: see also Steglich, F. 22 Schakel, A. M. J. 155, 208,

217.sEt Scherm, R. 45, 5t7-9: see aÍso Stirling, W. G. 45

Schertler, R. 571:

Fomin,I. A.

see also

358

Schmidt, K. E. 46,579,StE: see also Lee, M. A. 46 Schmitt-Rink, S. 550, 5E3 Schoepe, W. 561, 562,570,

571,577,5ß: see also Ahonen, A. I. 101, 103, 363, 495, 497, 498; Eska,

G. 102,105, 479, 503;

Fomin, I.4.358; Kokko, J.497; Simola, J. T. 110, 263,498 Scholz, H. N. 570,575,5t2: see also Feder, J. D. 98; Hoyt, R. F.98; Masuhara, M. 45, 535, 547

Schopohl, N. 180, 262,

ll8,

128,135,

3ll , 526-7 ,

533,542, 546,569, 5t4, 5t5, 58E, 59O,591: see a/so Passvogel,T.237, 245,246,247,262; Tewordt, L. l2l,533; Vollhardt, D.324,325 Schrieffer, J. R. 4, 7,36,53, 61,92,550, 564, 575, SEE: s¿¿ also Bardeen, J -

Author Index

604 Schrieffer, J. R.--Cont'd. 64; Ho, T. L. 208 Schwarz, K. W. 329,sEt

Seiler, K.37,55, sEt Seppälä, H. K. 233,241,243, 262,389,390, 574, 58E: see also Hakonen, P. J. 390

Serene, J. W. 23, 56, 124, 127, 136, 171, 198, 300,

438,495, 499,521, 523, 526, 534-5, 542, 546, 547,565, 571, 579, 5&3,

5t6-E: see also Brinkman, W. F. 127; Fomin, [. A. 182; Lee, P. A.22; Muzikar, P. 22; Sauls, J. A.22,253 Sergatskov, D. A. 565: se¿ ¿lso Borovik-Romanov, A. S. 331,494 Sessler, A. M. 5, 298,56E, 569,572:. see also

Cooper, L. N. Shahzamanian, M.

5

A.

481,

s88

Sham. L. J.579:. see also

Lee,P.4.22 Sharma R. G. 5E0,5t1: see ¿lro Lounasmaa, O. V. 494; Manninen, M. T. 326 Sherill, D. S. 5E2: see also

Masuhara, M. 45,535, 547

Shields, S. E. 105,58E Shigi, T. 575: see also

Ichikawa, K.488 547,577,582, 5EE: see a/so Ketterson, J. B.

538,542; Meisel, M. W. 526,541,544 Shopova, D. V. 359,571 Shumeiko, V. S. 465,469,

5tt

T.

107

, 110, 190,

263,330,390, 498, 574, 583,5E5, 5ß: see also Hakonen, P. J.129,254, 391,392,394; Nummila, K. K. 498; Pekola, J. P. 107, 110, 190,330,391, 393; Seppälä,H.

102. 381

H. 186,200,282, 284-5,326,358,360-1,

Smith,

362,379,384, 386, 388, 400,413, 483,499, 505,

510,564-5, 569, 575-7, 584-5, 5E9, 593: see also Bhattacharyya, P. 196, 395, 463, 478, 479;

Brinkman, W. F. l8ó, 787,3&,388; Einzel, D. 486, 488, 492; Højgaard, Jensen, H. 38,40, 489, 491, 492, 505; Osheroff,' D. D.32,283,387,510; Pethick, C. J. 451,463, 465,466; WöIfle, P. 497 Soda, T. 291, 330, 40I, 497, 563,56ó, 577,5t9: see ¿/so Brueckner, K. A. 5 Soinne, A. T. 562: see also Alvesalo. T. A. 98 Sokolov, A. I. 128,5E9 Solomon, A. I. 118,5t9 Soni, V. 294,589 Sonin, E. B. 244, 254, 262, 337 , 381 , 399, 416, 491 , 5t9 Spencer, G. F. 285, 56ó,5E9: see also Carless. D. C.

K.241,

243,390 Simon, F. 57E: see also Kurti,

N. 94 sköld, K. 45, 46,5EE: see ¿/so Scherm, R. 45 Skrbek, L. 588: see also

Spohr, R. N. R.

also

198,262: Sauls, J. A.22,

Volovic,Y.Z.233,

243,262,371, 389, 390 Steinback. M. 573: see abo Guernsey, R. W. 106 Stern, M. R. 508,5t0,5E9 Stewart, G. R. 22,5E9 Stirling, W. G. 45, 46,588, 5E9: see ¿/so Scherm. R.

45i.

Schopohl, N. 546 Struik, D. J. 349,589 Sun, Y. 493,5E9 Swift, G. W. 98, 182,56Í, 569,5t9: see also Buchanan, D. S.489; Eisenstein, J. P. 98, 326, 490

Swithenby, S. J. 587: see also Saunders, J.374

Sydoriak, S. G. 1,5t9 Sykes, J. 38, 40, 565,5E9

Taconis, K. W. 5(ß: see also Das, P.93 Tagirov, M. S. 5E0,5E1:see ø/so Lounasmaa, O. V.

494; Manninen, M. T. 326

Takagi, H. 434, 507, 5t3, 589: see a/so Miyake, K. 416

Takagi, S. 71, 87-8, 177,

205,286,301,340,344, 346, 363, 365 , 395-7

,

399, 401,403, 478, 483, 495, 499,511, 563, 579-

E0,5t!) Takano, Y. 561, 56i2,577: see a/so Ahonen, A. [. 101, 103, 363, 495,497,498;

Tankersley, L. L. 182,590 Te5anovió,

SES: see

Pekola, J.P.326 Sprenger, W. O. 584: see alsoPaalanen, M. A. 182 Stamp, P. C. E. 53,5E9 Stare, C. l14, 117,129,165, 169,171, 5E2-3 Steglich, F.22, SEg Stegun, l. A. 278, 372, 561 stein, D. L. 294, 495, 570, 575,5t7, 5E9,592: see ¿/.ço Fetter, A. L. 197, 253;

Stone, M. 205, 572,5E9: see also Garg. ^.251 see also Streckwall, H. 588:

Kokko, J.380,497

491

Spohr, D. A. 57t: see also

Kurti, N. 94

Shivaram, B. S. 526, 542,

Simola, J.

Simola, J. T. 110, 263, 390,498 Smith, E. N. 403, 572: see ¿Iso Berthold, J. E. 106, 107; Giannetta, R. W.

Z.

494, 495, 590

Tewordt, L. 127 , 180, 311, 344,346, 494, 521, 523, 526-7,533,542,5E3-5, SEE.590: see øLço

Passvogel,'1. 237, 245, 246, 247, 262; Schopohl,

N. 526, 533,542,546 Theodorakis, S. 246, 248, 256,385,388, 570, 590 Thompson, K.574: see ølso

Hall, H. E.93 Thomson, A. L. 565: see also Bozler, A. L.495 Thouless, D. J. 5,495,578, 590

Thoulouze. D.572:

see also

Godfrin, H.495 Thuneberg, 8. V.248-9, 251, 336,391,394, 498, 571,590, 594: see also Freeman, M. R.364, 495;'Zhang, W.487

Author Index

60s

Tilley, D. R. 24,590 Tilley, J.24,Sm Tinkham, M.,92,590 Tkachenko, V. K. 258, 259, 590

Topsoe, F. 489,590 Torizuka, K. 578,587: see also Kyynàräinen, J. M. 98; Salmelin, R. H. 542 Torng, C. J. 593: see abo Wu, M. K.22 Tough, J. T. 251, 257,329, 590

Toulouse, G. 213, 216, 228, 250,266,337, 563, 59{l Träuble. H. 109,570 Truscott, W. S. 374,56E, 575,5E7, 59O:. see also Dahm, A. J. 330,336; Hutchins, J. D. 330; Sachrajda, A. S. 490; Saunders, l-374 Tsakadze, J. S. 574: see also Hakonen, P. J. 109

Tsubota, M. 576: see also Ishikawa, O.401 Tsuneto, T. 238, 261, 272,

278,355-6,376, 505, 521.571,5t3, 590: see ø/.so Fujita,'1. 251, 259, 261,495: Nakahara, M. 261; Ohmi, T.237,245, 254,292

Tvalashvili, G. K. 5t7: see ¿/so Salmelin, R. H. 542

A.1.74 Varoquaux E. 108, 494,526, 542,546,5ú1, 565, 5t23, 585, 5E7, 590: see also Avenel, O. 105,380, 399, 526, 542, 546¡' Bloyet, D. 398; Meisel,

M. W.542; Movshovich, R. 526; Piché, L.544; Salmelin, R. H. 542 Vdovin. Y. A. 6,80, 524, 590

Veuro, M. C. 561, 562: see ¿/so Ahonen, A. I. 495; Alvesalo, T. A. 106,491 Vibet, C. 5úi, 565:. see also Avenel, O. 380,399; Bloyet, D. 398 Vinen, W. F. 257, 328,590 Vollhardt, D. 24, 52-4, 59, l2l, 150, 16l, 163,167, 171, 309-10;, 314-15, 317- 18, 324-6, 349, 365, 367,374-5,381, 382, 394.5ffi,5t0, 582, stE, 590-91: see also Lin-Liu, Y. R. 304. 309, 3l1,376:' Seiler. K. 37,55; Tewordt, L.533 Volovik, G.8.21, 23, 150, 155,204-6,208,213, 217.220,223,230-t, 233-7 . 239-48, 250-4, 256,259,261,263-8, 270,278,280,285,292, 322, 333, 335 ,

Ueda, K. 150,5tt, ø/so Seiler, K. 37,55 Ugulava, A. l. 402, 576, 590 590: s¿e

Uhlig, K. 570: see a/so Eska, G. 102, 105,479,503 Ullah, S. 326, 369,372,570, 590

Unal, N. 311,569 Usmani, Q. N. 581: see also Manousakis , E. 47 Usui, T. 205, 420, 434, 576, 5t3: .çee ø/.ço Ishikawa. M.416; Miyake, K.416 Valatin, J. G. 68,590 Valls, O. T.37,53, 56, 59, tzl. l3t,135, 136,494, 5?9,590 van Roosbroeck, W. 5t4: s¿e ø/so Osheroff, D. D. 32, 283, 387, 510; Paalanen,

M. A. 182 Varma, C. M. 117,339,341, 5lxi, 590: see alço Millis,

337

, 37

389-91, 393-4, 403,

I,

4 16,

433, 434, 489,495,496, 511,5t2,515, 518, 542, 550, 5Ái-4, 574, 576, 5E2, 583, 5t5, 587, stt,

591-3:

see

abo Balatskii,

A. V. 205,208; Balinskii, A. A.234:' Hakonen, P. J. 109, ll0, 129,254,388, 391, 392, 394,399; Ikkala, O. T. 109,391; Mermin, N. D. 220;Pekola, J. P. 330, 391, 393; Seppälä, H. K.

241,243,390 Vulovic, V . Z. 233, 243, 263, 369,371, 3t9-90, 592 Vuorio, M. 186-7, 283, 399, 498,571,5Eó, 592 Wagner, H.562:

see also

Amit, D. J.36 Walecka, J. D.92,570 Wang, Y. Q. 593: see ølso

Wu. M. K.22 Ward, P. L. 568,588: see also Daniels, M. E. 542: Saunders, J.542

Warkentin, P. A. 5E7: see ølro Sager, R. E. 102, 399

Warnke, M. 584,Stt:

see

also Passvog el , '1 . 262;

Schopohl, N. 526,542 rilartak, M. S. 533, 534,592 Webb, R. A. 98, l0l-3,283, 345, 356, 360,399,402, 479, 576-7, 592: see also

Johnson, R.

T.

10ó;

Kleinberg, R. L. 99 Wellard, N. V. 473,592 Wennerström. P. 562: .ree a/so Alvesalo. T. A. 106 Werthamer, N. R. 6, 7, I l, 80, 117, 165,167,341, 5ó4, 590

Wheatley, J. C. 4, 14,23-4,

30-1, 36, 54, 59, 92,93, 95,98-9, 104, il1, 136, 182,336, 402, 403, 484, 534, 544, 546, 547 , 555, 561, 566,573, 576-8,

5t5,587, 592-3: see also Abel, W. R. 28,45; Buchanan, D. S. 489; Greytak, T. J. 96; Johnson, R. T. 106; Kleinberg, R. L. 99; Kojima, H. 105,504; Krusius, M. 336; Paulson, D. N.96, l0l, 103, 104, 336,484,540, 541,544,545; Sager, R.

8.102,399; Webb, R. A. 98, 101, 102, 103, 283,345,356,360 Whitham, G. B. 376,593 Wiedemann, W. 570: see also Eska, G. 102, 105,403, 479,503 Wilczek, F. 496, 564, 575, 593: see a/.ço

559; Ho,

Bialek, W.

T. L. 208

Wilkins, J. W. 575, 579: see also Højgaard Jensen, H.38,40; Lee, P. A.22 Wilks, J. 24, 59, 328, 576, 593: see ¿lço Keen, B.

8.45 Willers, H. G. 570: see also Eska, G.403 Williams. G. A. 109,563, 590,593: see abo Avenel, O.542;

Author Index

606 Williams, G. A.-{ont'd. Varoquaux, E. 526,542 Williams, M. R. 259,303,

311,319,570,593 Wiringa, R. B. 47,584 Witten, E. 173,593 Wojtanowski, W .73, 534, 58{1, 593: see also Ling, R.-2.544 Wölfle, P. 20, 23, 43, 45, 49, 5l-2, 56,58,59,73, 103, 194, 203,394, 438, 454, 457 , 462, 464, 470, 497 , 499,503, 51 1, 520, 524, 521, 529,534, 536, 540-7,569, 570-1, 575, 577,5t3, 5t5,5E9, 591, 593: see also E,inzel. D. 403, 486, 488, 492; Eska,

G. 105,503; Fomin, I. A. 403; Højgaard, Jensen, H. 489,491, 492,505; Sun, Y. 493; Vollhardt, D.52,54 Wolfke,

M.2,576

Woo, C. W. 47,53, 593 Woods,

A. D. B.

SEE: see

a/.so Scherm, R. 45 Wu, M. K.22,593

Yakovenko, V. M. 495,496, s92

Yamaguchi, Y. 359,361, 576,593: see also Katayama, T. 359, 361, 402

Yamasaki, H. 575: see also Ichikawa, K. 488 Yan, S. S. 568, 5t7: see also de Vegvar, P. G. N. 546; Sagan, D. C. 98 Yang, C. N. 140,593

Yanof, A. W. 105,504,593 Yanof, J. M. 5t4: see also Parpia, J. M. 106 Yarmchuk, E. J. 109,593 Yip, S. 129, 205, 247, 489, 499,579,5E9, 593, 594: see aho Sun, Y. 493 Yosida, K. 89,593

,t¡

i' i. !;..

Ytterboe, S. N. 564: see also Bates, D. M. 311 Zakrzewski, S. 217,5E0 Zaringhalam, A. 58, 565,

5t5: see ¿/so Brown. G. E. 59 Zee, A.496, 593 Zeise, E. K. 572,5tó: se¿ abo Giannetta, R. W. 105; Polturak, E. 546 Zhang, W.487,490,594

Zhao,Z.56l:

see also

Adenwalla, S. 542

Zhu, Y.-Q. StS:

see ølso

J.P.326 Ziercher, E. L. 587: see also Sagan, D. C. 98 Pekola,

Zimmermann,W. 492,575 Zinov'eva, K. N. 495, 594 Zotos, X. 230, 233, 243, 263, 2&,390,5E1,594 Zwicknagl, G. 185, 487,566, 571: see ¿/so Fulde, P. 22

Subject lndex

A

phase 7, 95 see also ABM state Bogoliubov-quasiparticle relaxation rate 46L collective modes 526-33, 515-19,

543-6 definition 7,95,8L dipole interaction 178 dissipation of superflow 331-6 finite normal density at T :0 206-8 free energy 120 gap parameter 82,'/-.L6 gradient free energy 195-8 hydrodynamic equations for 4t6-29 invariance properties of L64 Leggett equations for 348-9 linear defects 223-33, 240-5 longitudinal NMR linewidth 399 in a cylinder 288-9 in a magnetic field 430-2 in a slab 286-8 in a sphere 292-4 magnetic susceptibility 87, 101, 133 NMR 348, 355, 363, 365-83 NMR in rotating superfluid 389-90 nonlinear spin dynamics 355-9 normal fluid density 84 orbital magnetic moment of 553-5 order parameter 7,81, L48 orientation induced by an electric or magnetic field, and by superflow 182

pair breaking critical currents 326 permanent electric diPole moment 5s9

permanent orbital magnetic moment 553-5

persistent currents L07, 203, 294-320, 331,-7

planar defects 267-70 planar soliton 272-8 point defects265-6 rotating 109-1 1, 259-63, 389-90 solitons in 376-83 sound propagation 102, 502, 542-5 spin dynamics 342-5, 348-59 spin susceptibility 87, L01, 133 spin-wave mode 506 supercurrents L6, 203-8, 294-321, 331,-7

superfluid density 87, 198 superfluid helium 3 26L-3 surface energies and boundarY conditions 186 symmetry classification of collective modes 515-18 symmetry of order parameter 148, 158 textures 223 -33, 240- 5, 286-32L texture-induced NMR shifts in362-

76,389-90 thermodynamic identities for 420-3 transport coefficients of 394, 4L6-34, 479-84 vortices in 223-33, 240-5 A, phase 96,98,99,130-2, L59, L60, 169, L1l definition 7,148 longitudinal NMR resPonse 346 hydrodynamics 430 nonlinear spin dYnamics in 359 order parameter 7, L32,'l'48 second sound in 508 specific heat t32 spin-wave modes.507-8

608

Subject Index

Ar phase 96, 98, 99, L30, 133, 158, 430 hydrodynamics 430-2 order parameter 7, L30 specific heat 133 spin-wave modes 507-8 A, state lzt,169 see also At phase At state L21 see also A2 phase A transition 95, 96 A-B interface 489 A-B phase transition 99,128,247 ABM state 6, 7-8,12,157 see also A phase anisotropic gap parameter 83 equal spin pairing state 82 specific heat discontinuity for 84 spin susceptibility of 89 Absolute temperature scale 95 Acoustic shear impedance 105 Adiabatic demagneti zation 93 - 4 Aharanov-Bohm interference effects 245 Alpha (ø) state 164, L68, L69,I72 Anderson-Bogoliubov mode 524, 540

Anderson-Brinkman-Morel

see

ABM

state

Anderson-Higgs mechanism 172 Anderson-Toulouse (AT) textu re 228, 230,233,26L lattice 262 nonaxisym metric 243 Andreev equations 488 Andreev scattering at boundaries 48690

Anisotropic superfl ui dity 7 Attenuation of sound L03-4,540 see also sound attenuation Average energy dissipation of spin motion 400 Axial state see ABM state, A phase Axiplanar state249 Axisymmetric vortice s 234, 235-8, 240-

1,245-9,252-6 B phase 7,98,99, 160, 169 see also BW state Bogoliubov-quasiparticle relaxation rate 457-6L collective modes 502-IL, 512-15, 524-7 , 534, 539, 546 definition 7 , 8I, 95, 147 dipole interaction L79-80

dissipation of superflow 329-31 electric dipole moment 555-9 free energy 120 gap parameter 82,116 gradient free energy L99-200 hydrodynamics equations for 41,1,-L6 hydrodynamics for finite magnetic freld 432-3 in a cylinder 283-5 in a slab 28L-3 Leggett equations for 348-9 linear defects 220-3, 233, 245-57 , 259-63 magnetic susceptibility 89, 102 NMR 348, 359, 363, 383-9,390-4, 40L NMR in rotating superfluid 390-4 nonlinear spin dynamics 359-61 nucleation of L29 order parameter 7, 80, 147 orientation induced by magnetic and electric field and by superflow L81, 183-5 pair breaking critical currents 323-6 permanent electric dipole moment s55-9 persistent currents I07, 208. 323, 329 planar defects 270-2 planar soliton 278-80 point defects 264-5 sound attenuation in 104, 502,541 sound propagation in 502, 538-42 specific heat 83-4 spin dynamics 353-5 spin susceptibility 89, L02 spin-wave mode 508-10 supercurre nt 203 , 208 , 323 , 329 superflow dissipation in 329-31 superfluid density 87, L94 surface energies and boundary conditions 186-8 symmetry classification of collective modes 512-15 textures 220-3, 233, 245-57, 259, 264,

279,290-6 texture-induced NMR shifts 362-75. 383-8, 390-4 thermodynamic identities for 412 transport coefficients 464-7 5 vortices in 245-5L topological propertie s 220-3 B, phase (planar phase) 98, L34, B, phase (B phase in magnetic field) 98, 99, 134, L35, 157, 170, I7g, 1gl, 182.347

1

Subject Index

order parameter 7,134 B-transition 95. 96 Backflow 48 Balian-Werthamer, see BW state, B phase

Ballistic quasiparticle regime 461, 484, 491.

Barnett effect 433 BCS pair amplitude, properties of 67, s50-3 BCS pairing theory 61 BCS superconductor 141,-2 BCS theory 4,5 BCS weak-coupling theory 64 Bend spatial change of order parameter 196

Bending 189 of the order-parameter field 15, 196 Bending free energies L95, 199,206, 21L,238 in A phase 196-8 in B phase 199-200 Berezinski-Kosterlitz-Thouless vortexbinding transition 495 Beta (B) state 159, 169 Binary collision processes 450-3 Bipolar state 164, 168 Bogoliubov quasiparticle (BQP) 76, 85, 206 collision integral lor 454-6 definition 69 distribution function 72 energy change in 443 energy spectrum 70 kinetic equation lor 442-5 mean free path 46L,484 relaxation rate 454, 457 -61, scattering of, off at boundary 487 velocity 443 Bogoliubov transformation 68, M2 Bohr-Sommerfeld quantization rule 209 Boltzmann equation 443 for Bogoliubov quasiparticles distribution functions 437 for normal phase 38 Boojum 250, 266, 293, 294, 331, 332, 334 Bose condensate of diatomic molecules 205 Bose-Einstein condensate 4, 66, L40, 549

Bose-Einstein statistics 4 Bose system 66 Bound quasiparticle states 488, 498

609

Boundary condition 211 on the distribution function 486 Broken symmetries 9-12, 137-74, L3g, 14L-9, 179, 1gg chiral symmetry 306 discrete symmetries 162-5, 243-4 gauge symmetry 65, L41,189,327 ,

4t0 in BCS theory 141 in ferromagnet 142 in high-energy physics 172-3 in superfluid 4He 141 relative gauge-orbit symmetry 295 relative symmetry 144-7, L54,173 rotational symmetry in orbital space l44,4Ll in spin space L42,144,4L0 spin-orbit symmetry 11, 19, L46,L54, 339

Bulk energy 238 Bulk viscosity 33, 4L5,467-7L Bulk-viscosity coefficients 426, 480 BW state 6, LL,80-2 see also B phase Catastrophic instability 320 Chaotic motion 359 Chemical potential 26, 406, 468 Chiral symmetry, breakdown of 302 Cholesteric liquid crystals in a magnetic field 319 Circular disgyration 289 Circulation 16, 208, 209, 214, 226 Clapping mode 245, 5ll, 544 Coherence length 64, 195 Collective modes L8, 42, 245, 501-46 in the A phase 502-L1,515-19, 527-35 in the B phase 502-LL,512-15,524-7 collision-induced broadening 534 frequencies, splitting 526 observability of 534-5 residual interactions, effect of 533-4 spectrum, multip^let structure of 5L2 Collision integral (l) 33,37 , 450-61. for Bogoliubov quasiparticles 454-6 conservation properties of 453 for normal quasiparticles 37 linearized 38-40,452 Collisionless sound 535-46 see also zero sound Complex order parameter 6L, I90, 549 Composite bosons 549 Composite soliton 273,275-7 ,377

Subject Index

610 Composite soliton-Cont'd. creation of273 global instability of 316 Compressibility 30-2,502 Condensation energy 79 Conservation laws 32,405 Conserved densities 27,440 Continuity equation 32 Continuous degeneracy of ground

Diffuse boundary scattering 1.85, 484, 487

Diffusion constant 497 Dilution refrigerator 93 Dimensionality of defects 2L6 Dipole coherence length 20L,217 Dipole coupling parameter 177 Dipole-dipole interaction 13-14, 102, state 169, L74-9, 18L, 201 65,272 in A phase 178 Continuous mappings, theory of 213 in B phase 179 Continuous symmetries 150-60, 407 Dipole length 259 Conventional superconductors 74 Dipole-locked regime 218,231 Cooper instability 61-4 Dipole torque L9,342,358, 505 Cooper pair 4, 63,I37 Direct interaction 49,51 as diatomic molecules 551 DisclinationZZL internal degrees of freedom 9 Discontinuity of specific heat C" 83, internal structure 8-9 L32-3 Cooper-pair susceptibility 521 Discrete degeneracy 267 ,306 Cooper-pair wave function 64 of the ground state 270 Coreless vortices 333 Discrete symmetries 148, 150, 158, Correlated basis function (CBF) 47 162-5,223,267 Correlation length 64 DisgyrationZ42 Couette flow 485 circular 289 Critical angular velocity 258 Dissipation of superflow 326-36 Critical current 110, 309, 370,3L2, 3t7 , Domain wall 208, 2L2,267 ,268 definition 15 327 edge222 Critical field 135 Critical supercurrent 315 effect of boundaries on277 effect of superflow on 313-19 Critical spin supercurrents 331 global instability of 318 Critical superfluid velocities in the A and B phases 108,32L-6,329-36, lattice of 270 491 local stability of, for small currents Critical temperature 76 3L6 planar 269 Current density 27,193,44I propagating3T6 Cutoff energy 74,76 regular array of320 spontaneous formation of 318 d vector in A phase 82, 148 Double-core vortex 248,249 d(k) vector 70 Double-half disgyration 289,29I,373 dsoliton 273,275,377,378 d-wave pairing L14,170 Double sine-Gordon equation 384 Defects of order-parameter field Doubly quantized nonsingular vortices dimensionality of 216 390 linear 219-57 Duhem-Gibbs relation 422 planar 267-80 Dynamical pattern selection 321 point 263-7 Dynamical structure factor 497 topological investigation of 2LL-I9 Dynamical symmetry variables 418 topological equivalen ce 21ó Degeneracy space of order parameter Edge of a domain wall222 141., 212, 2L7 , 219, 220 Effective Hamiltonian 72-3, 341, 369 see also order parameter space Effective mass 27, 54 Density of states 26 Effective-mass relation 29,31 Density response function 48, 535-8 Effective pair interaction 73 Diatomic molecule 92,55I Effective-potential approximation 55

ffi

IË

IF

Subject Index

611

i,

i Ë,

Effective relaxation time 398 Effective-viscosity coefficient 489, 490 Effective viscous penetration depth 505 Einstein-de Haas effect 433 Einstein relation 497 Electric dipole moment (EDM) of a Cooper pair 558 induced 781-2 permanent 555-9 Electrohelicity 558 Electron-nucleon interaction 557 Elementary excitations 25 Energy-conservation equation 32 Energy current 34, M[,448 Energy density 440,448 Energy gap, drawing of 81 Entropy 28-30, 7 6, 407 -8, 420 Entropy current 408, 415 Entropy-production function 415 "( phase" 157 Equal spin pairing state 81 Equation of motion for hydrodynamic variables 32,341, 409 derivation of 340-2. 406-9 for intrinsic angular-momentum 423 for 1423.427 for spin density 341, 415, 505 for spin-orbit angles 414 for spin superfluid velocity 430-3, 505 for symmetry variables 409-11 for u. 4I3,427 e state 170 Euler angles 366 Euler characteristic 280 Euler-Lagrange equations 27 4, 367 Exchange symmetry 51 Extended Fermi-liquid theory 49-52 f-wave pairing 114 Fermi-Dirac statistics 4 Fermi energy 4,26 Fermi function 28 Fermi gas 4,25 Fermi liquid 25-60,84 see øbo quasiparticles collective modes 42-6 compressibility 30 extended theory 49-52 Landau parameters 27 , 29,31 Landau theory 25-46 specific heat28 spin diffusion coefficient 36, 42 spin susceptibility 30 thermal conductivitv 36, 42

thermodynamic properties 28-32 transport properties 32-42 viscosity 36,41 Fermi-liquid corrections 85, 89, Ig7,

rgg,200 Fermi-liquid interacti on 6, 27, 86, 89, 198,323 Fermi momentum 25 Fermi statistics 25 Fermi temperature 4 Fermi velocity 26 Ferromagnetic spin fluctuations 6-7 Field-turn-off experiment 399 Finite normal density at T :0 in A phase 206-8 Finite orbital magnetic moment 554 First sound 42,502 Flapping mode 529 Flare-out texture 284 Flow in restricted geometries 484-98 Flow-induced domain-wall lattice 318. 382 Flow-induced dynamical textures 3192L

Flow-induced precessing domain walls 327

Flow-induced soliton lattice 382-3 Flow through narrow channels 326 Fountain pressure 105 Fourth-order invariants 1 14 Fourth sound 105, 491, 504, 513 Free energy 76-9, I43 invariants of l22 symmetry and stationary points of 165-9 Free energy per unit length of a vortex, 239,252 local minimum of 301 Free-induction signal 358 Freedericksz transitio n 286 Galilean invariance 28, 421 Gap Á in the excitation spectrum 70, 83, 84 average magnitude 79 Gap equationT3-4,75 Gap parameter'10, 73, 7 4, 75, 82-4, 135 of the A phase, anisotropic 83, 120 for the B phase 83,120 near 7i 79,82-3, l\7 at zero temperaturc 79, 82-3 Gapless modes 501 Gauge invariance 65

Subject Index

612 Gauge-invariant densities and currents 447-50 Gauge-invariant distribution fu nction

45

Gauge-invariant energy change 445 Gauge-orbit symmetry 10, 4L8, 430 Gauge-orbit symmetry angles 418 Gauge symmetry L0, 15L, 407 in the theory of electroweak interactions 173 Gauge transformation of the BCS wave function 66 of the kinetic equation M5-7 operator for 151 of the order parameter 14L Gauge-wheel effect 420, 433 Generalized BCS wave function 65-7 Generalized Fermi-liquid interactions 439

Generalized pairing theory 64-71 Ginzburg-Landau free energy functional 79,lL3-18,149, L65, 195 parameters of t'1.6, L26-7 B including magnetic freld L29-36 including gradient terms 194-200 Global gauge symmetry, spontaneous breaking of 4 Global symmetries L72 Goldstone mode 139, I72, 5O5, 513, 524 Gradient free energy 194-202, 213, 238 Ground state, degeneracY of 65, L39 Ground-state energy 47 of normal phase 47 of superfluid phases 79 Group-theoretical approach to collective modes 513-19 Group-theoretical classification of defects 2ll-36,267-72 Gutzwiller-Hubbard lattice gas model 52-s Gyromagnetic ratio 18, 25, 30, 17 5 Gyromagnetism of rotating superfluid tHe Lo9, 252-7, 259-63, 392

Half-quantum vortice s 244-5 , 250 , 25L Hamiltonian approach to spin dynamics 365,369-71,383 Healing lengths 200-2, 27 0, 281 induced by dipolar interaction 20L induced by magnetic freld 202 induced by surfaces 202 Heat current 425-6,448 through boundary 493 Heavy-fermion superconductors 150

Heavy-fermion systems 22 Hedgehog order parameter configuration 2ó5 Heisenberg spins 213, 214

in aplane2lT Helical texture of I field 303 for qll^tÍ306-11

foru,

IH3ll-Lz

stability regions of 309, 310 time-dependent 320 unstable 319 Helium 3 (3He) 1-4 condensation 2 dilute solutions in helium 4 21 films, superfluidity in 494-5 layers at a solid surface 495 phase diagrams 2, 3 production of 1 superfluid 2 symmetry group 143 two-dimensional model lM-7 Helium 3-A (3He-A), see A Phase Helium 3-AL, see A, Phase Helium 3-n (3He-n), see B Phase Helium 3-helium 4 dilution refrigerator 93

Helium 4 (oHe) 1,-4 condensation 2 dilute solutions of helium 3 in2I phase diagrams 2, 3 Helium physics, early history of L, 5-7 Higgs-potential minima 165 High-energy physics 165 High-7i oxide superconductors 22, 550 Homogeneous spin precession, instability in 402 Homotopy theory 213-4, 216, 2L9-20, 263-4,280 Hubbard model52 Hydrodynamic equations 32-3, 405 for the A phase 41,6-29 for the B phase 411-16 for paramagnetic fluid 32 Hydrodynamic modes 502-t2 Hydrodynamic regime 18, 435 Hydrodynamic sound propagation 42, LOs,502 Hydrodynamic theory, microscopic derivation of 435 Hydrodynamic variables 405, 406 Hydrodynamics for finite magnetic freld 429-33 slip correction to 484-6 Hyperbolic texture 230

tç

ìI Subject Index

Induced electric dipole moments L8I-2 Induced interaction model 131 Induced intrinsic angular momentum

4t8 Inert state Il7,154,156, 160, L64, 168, 169,172,234 in plane texture 283,284 Interpenetrating square lattices of vortices 261 Intrinsic angular momentum 205, 4L618,422,429 induced 418 Intrinsic spin relaxation 394-403, 475-9 energy change by 399 Invariants of free energy 122 Ion mobility 495-8 effect of interaction with vortices 498 experiments 110 lsing spin 214,2L7 line-shaped defects for 218 pointlike defects 218 Isotropic superfluid 1,40, 257 -9, 327 Jastrow factors 47 Josephson effect 20, I08,494 Josephson equation 426, 433 transport coefficients in 481 Josephson flow oscillations 494 Josephson oscillator equation 355

Kelvin waves244 Kinetic equations 33-4, 437 -50 for Bogoliubov quasiparticles 422-5 gauge transformation of 445-7 matrix 438-40 Knudsen regime 460-1, 484, 489-91

i texture in a cylinder 288-91 in a slab 286-8 in a sphere 292-4 relaxational motion of 336 ^ I disgyration 227, 230, 289, 290, 293 I monopole 293,294 I vector in A phase 82, 148 Lagrangian formulation of spin dynamics 365-9 Landau-Boltzmann equation for Bogoliubov quasipar ticles 437 Landau critical velocity 321,327 Landau damping 43 Landau molecular fre\d,27,85, 88, 103, 440,462 Landau parameters 27 , 3L,54, 85-6,

6ti lgg, 440, 462, 520, 534, 542 Landau's theory 6 see also Fermi liquid Large-angle helix 311 Larmor frequency 18 Lattice-gas model of normal liquid 52_5 Lattice of planar domain walls 3tZ Lattice of vortices 258 Leggett angle l7g, lBB,2Zl Leggett configuration 199 Leggett equations 342, 34g Linear defects 2lg-57 see also vortices Local equilibrium 37 -8, 456, 462 approach to 462 of the BQP system 456 Local oscillator model 362 Local identity 406, 4IZ, -thermodynamic 421.424 Localization transmission 54 London limit 195 Long-range order 137-49, 174 Longitudinal driven NMR 361 Longitudinal (first) sound 42, 102, 502 Longitudinal NMR frequencies 345, 375 Longitudinal NMR linewidth in the A phase 399 Longitudinal resonan ce lO2 Longitudinal spin resonance lg, I0Z, 343 Longitudin al zero sound 43, 535_46 see also collisionless sound Macroscopic orbital magnetic moment 55s Macroscopic quantum coherence 549-Sg of the pair-correlated state 549 in superfluid helium 4 549 Macroscopic quantum effects 2 Macroscopic quantum state 190 Magnetic energy density 129-35, 1g0 Magnetic healing length 201-2, Zl7, 254,259 Magnetic resonance frequencies 364 see also nuclear magnetic resonance Magnetic susceptibility 181 see also spin susceptibility in the A phase 87, I01.,I33 anisotropy 87 , 180,201 in the B phase 89, L34 in the normal phase 30 see also spin susceptibility Magnus force 266 Manifold of internal states 14L,272

Subject Index

614 Mapping of a contour 213 Mass current 32,84,202, 412, 440, 448 Mass density 27,406 Mass supercurrent, see supercurrent Massive modes 501 Matrix distribution function 437 Matrix kinetic equation 439,5L9 Matrix of transport coefficients 409 Mean free path 36,460,484 Mean-field Hamiltonian 67 -9 Mean-field solution, effect of fluctuations on 128 Meissner effect in a superconductor 257 Mermin-Ho-tv{T vortices, lattice of 263 Mermin-Ho relation 210, 250, 295, 296, 422

Mermin-Ho (MH) texture 229,230, 233, 243, 250, 261, 262, 289, 291, 3L3,373 Microscopic derivation of hydrodynamic theory 428,461-84 Microscopic theories for the normal liquid 46-8 Mixt-twist (MT) texture 230, 26I, 262 Model states 80 classification of 1 18-23, 149-72 relative stability of 122-3 Molar volume 31 Molecule picture of Cooper pairs 552 Momentum conservation equation 32, 4L4,425 Momentum current densitY 4L4,425, 44r, M8 see also stress tensor Momentum density 27, 441, 448 see also mass current Monopole order-Parameter configuration 15, 265, 266, 292 Monopole-antimonoPole Pair 294 Multiple spin echoes 403 Multiplet structure of collective mode spectrum 5L2

ñ vector in B phase 146,199 fi soliton 272,279,280 effect of superflow on 3L4 ñ texture induced by surfaces 280-6 Neutron scattering 45 Neutron stars2Z bulk matter of 171 NMR, se¿ nuclear magnetic resonance Noether's theorem 172 Nonaxisymmetric AT texture 243

Nonaxisymmetric vortices 240, 241-3,

248-50.256-7 Noninert state 155, 159, 160, 168 Nonsingular point defect 265,266 Nonsingular structure 227 ,228 Nonunitary state 89 Normal-flapping mode 5lL, 53L, 544 dispersion of 533 Normal-fluid density 80, 84-6, 194,198, 207

Normal-fluid slip 504 Normal phase 25-46 Nuclear demagnetization 94, 96 Nuclear dipole interactions lO2, L74 see aßo dipole-dipole interaction Nuclear magnetic moment 30,175 Nuclear magnetic resonance (NMR) 18-20,95, L00, 339-403 frequency shifts 342-8, 362-90 effect of magnetic field on 345-7 effect of superflow on 364 effect of tilted uniform texture 363-4 limit of zero magnetic field 344 longitudinal 103, 345-7, 363 transverse L03, 345-7, 358, 3ó1, 363,369,371 Hamiltonian approach 369 Lagrangian approach 365 linear in nonuniform texture A phase 365-83 B phase 383-9 longitudinal driven 361 nonlinear in uniform texture 348-62 response of the helical texture 374-5 resonance linewidths 398 tHe 388-93 in rotating superfluid satellite peak 109, 110 signature of solitons 377-82 under linear spatially homogeneous conditions 342-8 Nuclear magnetic susceptibility, see magnetic suscePtibilitY Nucleation of B phase 128 o vortices 238, 240, 245-9, 251 Oblate state 157, 170

Off-diagonal distribution function, kinetic equation for 443 Off-diagonal energy 68 Off-diagonal long range order 140 Off-diagonal mean field ó8,70 Onsager relations 409

IE lç

Subject Index Onsager's symmetry principle 415 Orbital dissipation 333-6 Orbital magnetic moment of the A phase 553-5 Orbital relaxation 335, 482-4 Orbital supercurre nt 204 orbital viscosity 320, 427, 483, 5L1 Orbital wave 511, 533 Order parameter 9-12,71, 137 , 139, L40,236 complex-valued 6'1., 190, 549 matrix form 70 of A phase 7,8L, L48 of A, phase7,132,1.48 of Ar phase 7,l3O ofB phase 7,80,I47 of Br phase 7 , L34 orientation of. L92, 238 induced by dipole interaction 17480

induced by electric field 181-2 induced by magnetic field 180-1 induced by superflow 182 induced by surfaces 185-8 preferred diiection 1î¡ tt+ spatial variations of 185 see also textures symmetry of 144-7 transformation prope rties L44-7 vector form 70 Order-parameter manifold 212 see also order-parameter space Order-parameter modes, symmetry classification of 512-19 Order-parameter space 1,47, 212 see also order-parameter manifold, degeneracy space Orientational energies of orderparameter field 174-88 Orientational transiti on 286

P-T-H

phase diagram 99

Parity-violating interaction 557 p-wave pairing 80, 114 Pair amplitude 67, I40,557,552 Pair breaking20,326,329, 487 , 53L-2, 541

'e &

g Ê ìgi ,Ë

ffi

saturation of 546 Pair-breaking critical currents 32L-6 in the A phase 326 in the B phase 325 Pair-breaking critical velocity 207 ,322, 324 Pair condensation, energy due to 69,79

615 Pair-correlated state 6L, t40 Pair interaction 5, 57-9,74,76, \77 , 534 Pair vibration modes 20,5L3,524 Pair wave function 8, 62,92, 55L, 552 Pairing theory at finite temperature

7t-9

Pam-Am texture 289 Parallel ringing 359 Paramagnon theory 53 Parity-violating dipole moment in 3He-B 558

Partial spin susceptibility of the condensate 395 Particle-hole asymmetry 130, 469, 523 Particle-like soliton 266, 268 Particle number current 28 Particle number density 27,447 Particle physics 2L, 251 Persistent current 106-8, 190-4, 209, 294-321,326-37 see also superflow Phase diagram 95-7, 109, I22, 287 experimental investigation 95, 311 of 'He 3 of aHe 3 of helical texture 311 of rotating liquid'He 393 Phase

rigidity 4,6L,66

Phase-slip processes 250,327 , 494 Phase vortex 214-7, 220, 258 see also pure phase vortex Planar defects 267-80 topological classifi cation 267 -7 2 Planar soliton 267 in the A phase 272-8 in the B phase 278-80 Planar state 119, 150, 157, L&,168, '/..69,249, 25L, 323 Point defects 263-6, 2L6-8, 263-6, 33L see also monopole order-parameter

configuration in the d-vector field of '}]re-A266 in the l-vector field of tHe-A 265,331 in the ñ-vector field of t}le-B 264 Poiseuille flow 489-91, 505 Polar state 118-19, L55,160, L69, 242

Polarization potential theory 48-9 Polycritical point 96, 102 Pomeranchuk effect 94, 95 Preferred direction of order parameter, see long-range-order Pseudo-isotropic state 11, 81 see abo B phase and BW state

Subject Index

616 Pure phase vortex in the

A

phase 227

,

230-r,240,241,242 Q balls 129 Quantization of circulation 208-1 1 Quantum of circulation 209 Quasiparticle 6 see also Bogoliubov quasiparticle collision processes 450 compressibility 30-2 concept 25-8 definition 25,26 distribution function 33 dynamics in ballistic regime 491 effective mass 26, 29 energy 26,62 entropy 28-30 interaction2T lifetime L7,34-7 , 479 scattering amplitude 34, 49-52, 451 phenomenological model of 55-7 velocity 34,443 Radial disgyration 230, 242, 373 Real squashing mode 524,539,541 Relative gauge-orbit symmetry t57, L60 Relative gauge-spin symmetry L59, 160 Relative orbital angular momentum 8, 63

Relative rotation of spin and orbital spaces 147

Relative spin-orbit symmetry 154 Relative symmetry 146-8, 160 Relaxation time 396, 476,478 Relaxation-time approximation 456 Remaining symmetry 149 Remanent vorticity 331 Resonating-valence-bond 550 Rigidity phase 4,6L,66 Ringing of magnetization 102,355 tHe 109-1 I, 257 Rotating superfluid 63,389-94 s-p approximation 56 Scattering amplitude 34-5, 51, 55, 451 see aßo quasiparticle scattering amplitude Scattering of light 508 Schafroth condensation 550 Second sound 105, 502, 503, 505, 508, 513

Second-order (A) transition 78, 95, 96 Self-induced transparency 546 Shear flow 485

Shear modes 5L1. Shear viscosity 33,36,

4'/-.,

4L5, 426,

464-7,480 for 3He-B 468 for 3He-A 481 Sine-Gordon equation 26T, 37 6 Sine-Gordon solitons 272 ,Single-particle excitations 61, 69-70 Singlet pairing 70-1 Singular defect 2Ll-12, 214 Singular vortices 220, 259, 329, 390, 498 Slip correction to hydrodynamics 484-6, 489

Slip length 485, 486 Small-angle helix 311 Solid 3He 7 ,94-5 Soliton 268, 272, 273, 3L3 in the A phase 272-8,376-7 in the B phase 278-80 composite 273,275-7 ,3L6, 377 creation of 377-8 energetics of273-5 NMR signature of 377-82 planar 272-80 in the presence of superflow 382 Soliton lattice 277 -8, 380-3 Soliton propagation 382 Sound attenuation 20, 42, 44, 103, 536 in the A phase 104,545 in the B phase 103, 541 Sound dispersion relation 44,538 Sound modes 42-6, 102-5, 491-4,

502-5,535-46 first sound 42, 105, 49I, 502 fourth sound 105, 504 second sound 105, 503, 508 zero sound 44, 102, 49I, 535-46 Sound propagation L02-5, 491-4 in the A phase 542-5 in the B phase 538-42 nonlinear effects 546-6 in the normal state 31 ,44,I04 in restricted geometries 491-4 Sound velocity 3I, 44, 536 Specific heat 80,83-4,98-100, 117 in the A phase 84 in the At.r phases 132-3 in the B phase 83-4 discontinuity of 83, 100, ll7,132-3 in the normal state 28-30 Specific-heat curve, experimental 98 Specific-heat discontinuity 100, II7, 132, 133

for the ABM state 84

Subject Index Specular scattering of quasiparticles from a plane boundary 185, 484, 487

Spin conservation law 32,432 Spin current 34, 198, 431,44I,449 Spin density 27,448 continuity equation for 32, 505 equation of motion for 341, 415, 505 Spin diffusion 402, 415, 473-5 Spin diffusion coefficient 33, 36,42,415, 474 Spin dynamics 18-20, 102,339-402 in uniform texture of the A phase 342,348 ofthe B phase 343,348 nonlinear regime 348-62 in zero magnetic field 349-55

in nonuniform texture 362-89 of the A phase 365-83 of the B phase 383-9 including spin relaxation 394-403 in rotating superfluid tHe 389-94 Spin-echo method 403 Spin effective mass 447 Spin fluctuations 6,53, L24 Spin fluctuations model 123,I27 Spin-orbit angles 413 Spin-orbit forces L3, L74,341,394 Spin-orbit-gauge symmetry 430 Spin-orbit interaction 78, 174, 341, 394 Spin-orbit rotation symmetry 10 Spin-orbit waves 5L3, 524 Spi n - re ax ation 393 - 402 Spin-relaxation rate 394, 479 Spin supercurrent 194, 204, 33L, 403, 449-50 Spin superfluid velocity 194,446 Spin susceptibility 30, 80, 87-90, 180, 252,256 see also magnetic susceptibility Spin transport along capillaries 490 Spin-triplet pairing 70 p-wave 8, 80, 114 Spin wave 505-10, 546 damping of 508 standing 282-3,570 velocity 506, 510 Spin zero sound 43,45,546 Splay composite soliton 277 ,3L4-t6, I

r

i

N

I ? L

dt

tåt )L'

380 Splay spatial change of order parameter 196 Spontaneously broken continuous symmetry, see broken symmetries

617 Squashing modes 524, 547 Stability of superflow 190, 294,321,326 Standing spin waves 282-3,510 Static helical configuration of i-texture 302 Stress tensor 32, 33, 414, 417, 424-7, 464 Strong-coupling corrections 77 , IZZ-T, 198.300 SU(5) grand unification theory 165 Superconductivity 4, 61, 7 5 Superconductors, high- [ oxide 22 Supercooled A phase 247 Supercurrents 16-7, 193, 198, 202-11,

4t0 Super-flapping mode 531, 544 Superflow 16-17, 106-8, 193, 198, 202-11 collapse 333 in a cylindrical geometry 313 decay mechanisms 327 dissipation 326-36 effect of 182-3 on orientation of order parameter

t82 on domain walls or solitons 313-19 global stability 296 in a slab geometry 312 stability of in the bulk liquid 295-301 and textural transitions 294-32I Superfluid current 195 see also supercurrent, superflow Superfluid density 87,107,194, t98 Superfluid mass currents 16-17, 198, 204,410 see abo superflow Superfluid spin current 331 see also spin supercurrent Superfl uid-to-normal interconversion 492

Superfluid velocity 16-17, 174, 191, 193, 4r2, 4L3, 4L9, 431, Superfluidity 2, 4, 189, 190-4 in a Bose system 66 SuperheatedB phase247 Surface defect 266, 280, 281, Surface dipole energy 187 Surface-dominated orientation 283 Surface energies 217, 282 and boundary conditions 185-8 Surface healing length 202,283 Surface impedance 492, 503 Surface-induced textu r es 280-94

Subiect Index

618 Surface singularitie s 292-3 , 331 , 332, 335 see also surface defects 88 Susceptibility susceptibility magnetic see also

SymmetrY and order-parameter structure 141-73 discrete symmetry transformations 1,64

reduction due to a magnetic field or spin-orbit couPling t69-70 of a state. connection with energy as stationarY Point 165 Symmetry breakings, sequence of fi2 see also broken sYmmetries Symmetry classification 137 -49 of equilibrium states 149-72 of order-parameter defects 2Ll-38, 263-72 of collective modes 511-19 3He 143, Symmetry group of superfluid

t69-70 Symmetry variable 407

T. sptitting 131 Temperature-dePendent coherence length 185 Temperature-dePendent mean-field Hamiltonian 72 Tensor order Parameter 70 Textural critical velocities 322 Textural instabilities due to superflow in A phase 294-319 Texture-induced NMR shifts 362-89 in the A Phase 3ó5-83 in the B phase 383-9 see also nuclear magnetic resonance Textures 15, 189-200, 208-319 see also defects of order-parameter field induced by surface in 'He-A 286-94 ''He-B 280-6 3He-A 294-321 induced by flow in Thermal boundarY resistance 494 Thermal conductivity 33, 36, 471-3, 482 of 3He-A 482 of 3He-B 473 Thermal expansion coefficient 98 Thermodynamic identitY 406 for the A Phase 420-3 for the B Phase 412 Three-dimensional rotation of order parametet L42

Time-dependent mean-field theory 52035

Time-dependent texture 319, 320 Topological charge 15, 214, 220, 226, 232,267,268,273 Topological classification of defects

2ll-19,233 Topological ProPerties óf n-þnase vortices at large distances 231-3 of A-phase vortices at small distances 223-3r of B-phase vortices 220-3 Topological quantum number, s¿e topological charge Topological space of order parameter 142,1,43

Topological stability 216 of linear defects 2L9-20 of planar defects26T of point defects 263 of a singularitY or defect2l3 Topologiðally equivalent states 216 Topology 141 ôf nod". on the Fermi surface 250-L Torsional oscillator 106, 490 Transition temPeratures 7 5, 97 dependence on magnetic field- L29-36 vaiiation of, with channel width 494 Transport coefficient 426, 436, 461-84 ofthe A Phase 479-84 of the B phase 464-75 in the JoiePhson equation 481 matrix of 409 Transverse NMR resonance 344,345, 347 , 36L, 363, 375, 398 see also nuclear magnetic resonance Transverse sPin-wave modes 390 Transverse zero sound 43,45,105, 546

Triplet pairing 6,70-I Tritium 1 Twist comPosite soliton 277,314, 316-18, 378 insHe-A276 Twist spatial change of order parameter

196

ì

Two-dimensional Fermi liquid 495 Two-dimensional suPerfluid 216 Two-fluid model 16

U-tube oscillation experimen t 326, 330' 492

Ultralow temperatures, experimental techniques for attainin g 93-5

619

Subject Index

Ultrasound attenuation, see sound attenuation Ultrasound excitations 20-L Ultraweak interaction 555 Uniform I texture global stability of 295-8 helical instability of 302-t2 hydrodynamic stabilitY of 298 local stabilitY of, in Presence of superflow 298-30I tilted NMR oT363-4 Unitary states 71,118 Unwinding of the Phase 327 uvw-vortex 238, 241, 247 v-vortex 238, 24L, 243, 246, 247,

249-5r.256 van der Waals Potential 2, 49 Vibrating-wire experiments L06, 491 viscosity 33, 4'1.,105, 415, 426, 464-80 in A phase 105, 481 in B phase I05,468,47I in normal phase 33,36, 4L second (or bulk) viscositY 33,4I5, 426,467-7 1, 480 shear viscosity 33, 36, 41, 415, 426,

t I

I

É

464-7,480 Viscous penetration dePth 491 Vortex-antivortex Pair 217 Vortex core hard233 soft232 spontaneous magnetization density in 237,254 structure 250-L Vortex-core transition 25t, 391, 392 Vortex lattice 257,259 Vortex line2l7,226 see abo linear defect in superfluid helium 4,212 Vortex nucleation 327 ,329 Vortex ringsZ1l,329 Vortex tangle 328,330 Vortices L5, 109, 2L9-57 in the A Phase 223-33,240-5

associated with the phase @ (pure phase vortex) 219, 222, 226 axisymmetric 235-8, 240-1, 245-8,

252-6 in the B phase 220-3,245-51 magnetic proPerties of 252-7 energetics of238-57 half-quantum244-5 nonaxisymmetric 241-3, 248-50

vorticity 221,

23'1.,

232, 233

remanent 331

w-vortex 238, 24'l', 243, 244, 247 Wall-pinned configuration 360 rilall-pinned ringing mode, damping of 40r Walls influence on superfluid 174 and orientation of orbital angular momentum 14 Weak neutral currents 556, 557 Weak-coupling collective frequencies 516

Weak-coupling free-energy functional 77

Weak-coupling limit 63, 196 Weak-coupfing theory'17, 1L3, 127, L94, L97, r99 of pair correlations 61 Weak-coupling values of B parameters in free energy LL6 Weinberg-Salam model L73, 557 Wide-angle helix 320 Winding number 2I4, 215, 2L6, 221, 226,303

XY spin model t40,2I3 Yosida function 86, 89

z-vortex lattice262 Zeeman energy 30 Zero-point motion 2-3 Zero sound 43,45, L02,538-46

The Superfluid Phases of Helium 3

The Superfluid Phases of Helium 3

Superfluids, Vol. II: Macroscopic Theory of Superfluid Helium

The Electronic States of the Helium Molecule

Phases of Gravity

Phases of Gravity

The Phases of Quantum Chromodynamics

Interpretation of the Hydrogen and Helium Spectra

Phases of War

Phases of Gravity

Phases of Interpretation

Phases of Gravity

Phases of Gravity

The Structure of the Helium Atom I

Dan Simmons - Phases of Gravity

Dan Simmons - Phases of Gravity

On Phases

Dan Simmons - Phases of Gravity

The Major Phases of Philip Roth

The universe in a helium droplet

Radial Velocities of 368 Helium Stars

Perspectives on Phases

Dan Simmons - Phases of Gravity 4 0

The Fine Structure of the Helium Arc Spectrum

The universe in a helium droplet

Systematic Motion among Stars of the Helium Type

The Superfluid Phases of Helium 3