Pairing in Fermionic Systems: Basics Concepts and Modern Applications (Series on Advances in Quantum Many-Body Theory)

Series on Advances in Quantum Many-Body Theory - Vol. 8

PAIRING „

FERMIONIC SYSTEMS Basic Concepts and Modern Applications

If World Scientific

PAIRING FERMIONIC SYSTEMS Basic Concepts and Modern Applications

Series on Advances in Quantum Many-Body Theory Edited by R. F. Bishop, C. E. Campell, J. W. Clark and S. Fantoni (International Advisory Committee for the Series of International Conferences on Recent Progress in Many-Body Theories)

Published Vol. 1:

Proceedings of the Ninth International Conference on Recent Progress in Many-Body Theories Edited by D. Neilson and R. F. Bishop

Vol. 3:

Proceedings of the Tenth International Conference on Recent Progress in Many-Body Theories Edited by R. F. Bishop, K. A. Gernoth, N. R. Walet and Y. Xian

Vol. 4:

Microscopic Approaches to Quantum Liquids in Confined Geometries £ Krotscheck and J. Navarro

Vol. 5:

150 Years of Quantum Many-Body Theory A Festschrift in Honour of the 65th Birthdays of John W Clark, Alpo J Kallio, Manfred L Ristig and Sergio Rosati Raymond F. Bishop, Klaus A. Gernoth and Niels R. Walet

Vol. 6:

Proceedings of the Eleventh International Conference on Recent Progress in Many-Body Theories Edited by Raymond F. Bishop, Tobias Brandes, Klaus A. Gernoth, Niels R. Walet and Yang Xian

Vol. 7:

Introduction to Modern Methods of Quantum Many-Body Theory and Their Applications Adelchi Fabrocini, Stefano Fantoni and Eckhard Krotscheck

Vol. 9:

Proceedings of the Twelfth International Conference on Recent Progress in Many-Body Theories Edited by J. A. Carlson and G. Ortiz

Vol. 10: Proceedings of the Thirteenth International Conference on Recent Progress in Many-Body Theories Ed/fed by S. Hernandez and H. Cataldo

Forthcoming Vol. 2:

Microscopic Approaches to the Structure of Light Nuclei Ed/fed by R. F. Bishop and N. R. Walet

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Basic Concepts and Modern Applications

World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING « SHANGHAI • HONGKONG • TAIPEI • CHENNAI

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Series on Advances in Quantum Many-Body Theory — Vol. 8 PAIRING IN FERMIONIC SYSTEMS Basic Concepts and Modern Applications Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-256-907-3

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To the Memory of Adelchi Fabrocini

Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry. Richard P. Feynman

Preface

As we approach the Golden Anniversary of the publication of the Bardeen-CooperSchrieffer (BCS) theory of superconductivity, appreciation for its conceptual fertility and ubiquity as a paradigm for the behavior of quantum matter has never been greater. The last few years have seen remarkable progress in our understanding of fermionic pairing across an impressive range of physical systems. Beyond the familiar examples of electronic systems in condensed-matter physics, these include ultra-cold atomic vapors, superfluid 3 He, finite nuclei, nucleonic matter, and quark matter. One common feature that has emerged is a tension between factors that favor standard Cooper pairing (simple attractive forces between the fermions) and factors that disfavor it (typically, conservation laws that separate the Fermi momenta of different species). Other complications arise from the state dependence and momentum-space structure of the realistic fermion-fermion interactions, medium-modification of these interactions and of consistent fermion self-energies, and retardation effects due to finite propagation times of exchange bosons. To understand the implications of such conflicts and complications, we need to extend the standard Bardeen-Cooper-Schrieffer (BCS) theory to cover a much wider class of possible fermionic pairing patterns. The topic of non-conventional fermion pairing is particularly rich in opportunities for interdisciplinary cross-fertilization, because it draws together theoretical ideas from atomic physics, condensed-matter physics, nuclear physics, particle physics, and astrophysics. These ideas have direct relevance to experiment. Pairing in cold, high-density quark matter will drastically affect the transport properties underlying the astrophysical signatures that are sought by astronomers studying neutron stars. Pairing in isospin-asymmetric nuclear systems is an important facet of the physics of nuclei far from stability being created at radioactive ion-beam facilities and in natural stellar nucleosynthesis, as well as the physics of neutron-star interiors. Experimental studies of ultra-cold, trapped atomic gases continue to energize one of the most successful and fast-growing areas of mainstream physics, at the crossroads of new thrusts in condensed-matter physics, atomic physics, quantum optics, and quantum-information science. This book collects a coherent set of expositions on fermionic pairing that grew out of lectures given in the Institute for Nuclear Theory at the University of

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Preface

Washington. The chapters cover theoretical aspects of fermion pairing in various areas of intense current research. In assembling the volume, it has been our intent to provide an accessible and useful introduction to basic principles of the theory of superfluid systems, and to their specialized application in a number of physical contexts. This aim is reflected in the arrangement of the chapters, which divide naturally into three blocks. The first block, comprising Chapters 1-5, examines fermionic pairing at the subnuclear level through explorations of color-superconducting phases of deconfined quarks at high baryon densities. The second block (Chapters 5-8) is devoted to pairing at the nuclear level, focusing on the formation of pair-condensed nucleonic superfluids below the deconfinment density in neutron-star matter and nuclear matter; some attention is given also to the higher-order clustering phenomenon of quartetting in both infinite nuclear matter and finite nuclei. The third block (Chapters 9 & 10) is concerned with pairing at the atomic level, as exemplified in dilute, ultracold systems of trapped fermionic atoms; at the same time, it addresses important theoretical issues of wider interest that arise when the pairing is between two fermionic species with unequal densities. Each block starts with an opening chapter which gives a broader perspective of the corresponding topical area. A. Sedrakian, J. W. Clark, M. Alford Tubingen, St. Louis June 2006

Contents

Preface 1.

Color Superconductivity in Dense, but not Asymptotically Dense, Quark Matter

vii

1

Mark Alford and Krishna Rajagopal 2.

Larkin-Ovchinnikov-Fulde-Ferrell Phases in QCD

37

Giuseppe Nardulli 3.

Phase Diagram of Neutral Quark Matter at Moderate Densities

63

Stefan B. Riister, Verena Werth, Michael Buballa, Igor A. Shovkovy and Dirk H. Rischke 4.

Spontaneous Nambu-Goldstone Current Generation Driven by Mismatch

91

Mei Huang 5.

The CFL Phase and ms: An Effective Field Theory Approach

109

Thomas Schdfer 6.

Nuclear Superconductivity in Compact Stars: BCS Theory and Beyond

135

Armen Sedrakian and John W. Clark 7.

Pairing Properties of Dressed Nucleons in Infinite Matter Willem H. Dickhoff and Herbert Miiiher

175

x

Contents

8.

Pairing in Higher Angular Momentum States: Spectrum of Solutions of the 3P2-3F2 Pairing Model

201

Mikhail V. Zverev, John W. Clark and Victor A. Khodel 9.

Four-Particle Condensates in Nuclear Systems

221

Gerd Ropke and Peter Schuck 10.

Realization, Characterization, and Detection of Novel Superfluid Phases with Pairing between Unbalanced Fermion Species

253

Kun Yang 11.

Phase Transition in Unbalanced Fermion Superfiuids

269

Heron Caldas Subject Index

283

Chapter 1 C o l o r S u p e r c o n d u c t i v i t y in D e n s e , b u t n o t A s y m p t o t i c a l l y D e n s e , Quark M a t t e r

Mark Alford Physics Department, Washington University Saint Louis, MO 63130, USA Krishna Rajagopal Center {or Theoretical Physics, Massachusetts Institute of Technology Cambridge, MA 02139 USA Nuclear Science Division, Lawrence Berkeley National Laboratory Berkeley, CA 94720, USA At ultra-high density, matter is expected to form a degenerate Fermi gas of quarks in which there is a condensate of Cooper pairs of quarks near the Fermi surface: color superconductivity. In this chapter we review some of the underlying physics, and discuss outstanding questions about the phase structure of ultradense quark matter. We then focus on describing recent results on the crystalline color superconducting phase that may be the preferred form of cold, dense but not asymptotically dense, three-flavor quark matter. The gap parameter and free energy for this phase have recently been evaluated within a Ginzburg-Landau approximation for many candidate crystal structures. We describe the two that are most favorable. The robustness of these phases results in their being favored over wide ranges of density. However, it also implies that the Ginzburg-Landau approximation is not quantitatively reliable. We describe qualitative insights into what makes a crystal structure favorable which can be used to winnow the possibilities. We close with a look ahead at the calculations that remain to be done in order to make quantitative contact with observations of compact stars.

Contents 1.1. Introduction 1.2. Review of color superconductivity 1.2.1. Color superconductivity 1.2.2. Highest density: Color-flavor locking (CFL) 1.2.3. Less dense quark matter: stresses on the CFL phase 1.2.4. Kaon condensation: the CFL-K 0 phase 1.2.5. The gapless CFL phase 1.2.6. Beyond gapless CFL 1.2.7. Crystalline pairing 1.2.8. Single-flavor pairing 1

2 3 3 3 4 5 5 6 10 10

2

Color Superconductivity

in Dense Quark

1.2.9. Mixed phases 1.3. The crystallography of three-flavor quark matter 1.3.1. Introduction and context 1.3.2. Model, simplifications and ansatz 1.3.3. Ginzburg-Landau approximation 1.3.4. General results 1.3.5. Two plane wave structure 1.3.6. Multiple plane waves 1.3.7. Free energy comparisons and conclusions 1.3.8. Implications and future work 1.4. Coda Bibliography

Matter

11 11 11 14 19 22 24 25 28 30 32 33

1.1. Introduction The exploration of the phase diagram of matter at ultra-high temperature or density is an area of great interest and activity, both on the experimental and theoretical fronts. Heavy-ion colliders such as the SPS at CERN and RHIC at Brookhaven have probed the high-temperature region, creating and studying the properties of quark matter with very high energy density and very low baryon number density similar to the fluid which filled the universe for the first microseconds after the big bang. In this paper we discuss a different part of the phase diagram, the low-temperature high-density region. Here there are as yet no experimental constraints, and our goal is to understand the properties of matter predicted by QCD well enough to be able to use astronomical observations of neutron stars to learn whether these densest objects in the current universe contain quark matter in their core. We expect cold, dense, matter to exist in phases characterized by Cooper pairing of quarks, i.e. color superconductivity, driven by the Bardeen-Cooper-Schrieffer (BCS) 1 mechanism. The BCS mechanism operates when there is an attractive interaction between fermions at a Fermi surface. The QCD quark-quark interaction is strong, and is attractive in many channels, so we expect cold dense quark matter to generically exhibit color superconductivity. Moreover, quarks, unlike electrons, have color and flavor as well as spin degrees of freedom, so many different patterns of pairing are possible. This leads us to expect a rich phase structure in matter beyond nuclear density. Calculations using a variety of methods agree that at sufficiently high density, the favored phase is color-flavor-locked (CFL) color-superconducting quark matter 2 (for reviews, see Ref. 3). However, there is still uncertainty over the nature of the next phase down in density. Previous work 4,5 had suggested that when the density drops low enough so that the mass of the strange quark can no longer be neglected, there is a continuous phase transition from the CFL phase to a new gapless CFL (gCFL) phase, which could lead to observable consequences if it occurred in the cores of neutron stars. 6 However, it now appears that some of the gluons in the gCFL phase have imaginary Meissner masses, indicating an instability towards a lower-energy phase. 7-18 Analysis in the vicinity of the unstable gCFL phase cannot determine the nature of the lower-energy phase that resolves the instability. However, the instability is telling us that the system can lower its energy by turning

M. Alford and K. Rajagopal

3

on currents, suggesting that the crystalline color superconducting phase, in which the condensate is modulated in space in a way that can be thought of as a sum of counterpropagating currents, is a strong candidate.

1.2. Review of color superconductivity 1.2.1. Color

superconductivity

The fact that QCD is asymptotically free implies that at sufficiently high density and low temperature, there is a Fermi surface of weakly-interacting quarks. The interaction between these quarks is certainly attractive in some channels (quarks bind together to form baryons), so we expect the formation of a condensate of Cooper pairs. We can see this by considering the grand canonical potential Q = E — fiN, where E is the total energy of the system, fi is the chemical potential, and N is the number of quarks. The Fermi surface is defined by a Fermi energy Ep = n, at which the free energy is minimized, so adding or subtracting a single particle costs zero free energy. Now switch on a weak attractive interaction. It costs no free energy to add a pair of particles (or holes), and if they have the right quantum numbers then the attractive interaction between them will lower the free energy of the system. Many such pairs will therefore be created in the modes near the Fermi surface, and these pairs, being bosonic, will form a condensate. The ground state will be a superposition of states with all numbers of pairs, breaking the fermion number symmetry. A pair of quarks cannot be a color singlet, so the resulting condensate will break the local color symmetry SU(3)co\or. The formation of a condensate of Cooper pairs of quarks is therefore called "color superconductivity". The condensate plays the same role here as the Higgs condensate does in the standard model: the colorsuperconducting phase can be thought of as the Higgs phase of QCD.

1.2.2. Highest

density:

Color-flavor

locking

(CFL)

It is by now well-established that at sufficiently high densities, where the up, down and strange quarks can be treated on an equal footing and the disruptive effects of the strange quark mass can be neglected, quark matter is in the color-flavor locked (CFL) phase, in which quarks of all three colors and all three flavors form conventional Cooper pairs with zero total momentum, and all fermionic excitations are gapped, with the gap parameter 5$ ~ 10 — 100 MeV. 2,3 This has been confirmed by both Nambu-Jona-Lasinio (NJL) 2 ' 3 ' 20 and gluon-mediated interaction calculations. 3 ' 21 " 23 The CFL pairing pattern is 2 (q?C^)

= 50(K + l)5?5f + 60(K - l)SfSf

[Stf(3)coior] x SU(3)L x

SU(3)RXU(1)B

= *(• • •) -*

SU(3)C+L+R

XZ2

(1.1)

4

Color Superconductivity

in Dense Quark

Matter

Color indices a, ft and flavor indices i,j run from 1 to 3, Dirac indices are suppressed, and C is the Dirac charge-conjugation matrix. The term multiplied by K corresponds to pairing in the (65,65), which although not energetically favored breaks no additional symmetries and so K is in general small but nonzero. 2 ' 21 ' 22,24 The Kronecker 2*

gCFL

\* \

%

u

I

15

S3 °
5U3 > Sds. The calculation of the gap parameters in the crystalline phases has been done in a Ginzburg-Landau approximation. Recall that the splitting between Fermi surfaces is proportional to M^/n, and that small (large) M^/fi corresponds to high (low) density.

of two phases in one of which standard BCS pairing occurs and in the other of which no pairing occurs. 53-55 Phase separation into electrically charged but color neutral phases is also a possibility in two-flavor quark matter. 56 In three-flavor quark matter, where the instability of the gCFL phase has been established in Refs. 8, 12, phase coexistence would require coexisting components with opposite color charges, in addition to opposite electric charges, making it very unlikely that a phase separated solution can have lower energy than the gCFL phase. 5 ' 42 Furthermore, color superconducting phases which are less symmetric than the CFL phase but still involve only conventional BCS pairing, for example the much-studied 2SC phase in which only two colors of up and down quarks pair 57-59 but including also many other possibilities,60 cannot be the resolution of the gCFL instability. 33 ' 60 It seems likely, therefore, that a ground state with counter-propagating currents is required. This could take the form of a crystalline color superconductor 19 ' 52 ' 61-71 — the QCD analogue of a form of non-BCS pairing first considered by Larkin, Ovchinnikov, Fulde and Ferrell. 72 Or, given that the CFL phase itself is likely augmented by kaon condensation, 37 ' 38 ' 40 ' 41 it could take the form of a phase in which a CFL kaon condensate carries a current in one direction balanced by a counter-propagating current in the opposite direction carried by gapless quark quasiparticles. 14 ' 18 The instability of the gCFL phase appears to be related to one of its most interesting features, namely the presence of gapless fermionic excitations around the ground state. These are illustrated in Fig. 1.3, which shows that there is one mode (the bu-rs quasiparticle) with an unusual quadratic dispersion relation, which is expected to give rise to a parametrically enhanced heat capacity and neutrino emissivity and anomalous transport properties. 6 The instability manifests itself in

8

Color Superconductivity

.

0

.

,

,

in Dense Quark

i

50

,

100

150

,

,

,

[

Matter

,

200

.

.

.

i

,

250

M^/n [MeV] Fig. 1.2. Free energy U relative to that of neutral unpaired quark matter versus M%/n for the CFL, gCFL and crystalline phases whose gap parameters are plotted in Fig. 1.1. Recall that the gCFL phase is known to be unstable, meaning that in the regime where the gCFL phase free energy is plotted, the true ground state of three-flavor quark matter must be some phase whose free energy lies below the dashed line. We see that the three-flavor crystalline color superconducting quark matter phases with the most favorable crystal structures that we have found, namely 2Cube45z and CubeX described in (1.31) and (1.33), have sufficiently robust condensation energy (sufficiently negative fi) that they are candidates to be the ground state of matter over a wide swath of M%//Li, meaning over a wide range of densities.

imaginary Meissner masses MM for some of the gluons. MM is the low-momentum current-current two-point function, and MM/(g2S2), with g the gauge coupling, is the coefficient of the gradient term in the effective theory of small fluctuations around the ground-state condensate. The fact that we find a negative value when the quasiparticles are gapless indicates an instability towards spontaneous breaking of translational invariance. Calculations in a simple two-species model 10 show that imaginary MM is generically associated with the presence of gapless charged fermionic modes. The earliest calculations for the three-flavor case show that even a very simple ansatz for the crystal structure yields a crystalline color superconducting state that has lower free energy than gCFL in the region where the gCFL—•unpaired transition occurs. 69 ' 71 It is reasonable, based on what was found in the two-flavor case in Ref. 65, to expect that when the full space of crystal structures is explored, the crystalline color superconducting state will be preferred to gCFL over a much wider range of the stress parameter Mf/(<Sn/z), and this expectation has recently been confirmed by explicit calculation. 52 It is conceivable that the whole gCFL region is actually a crystalline region, but the results shown in Fig. 1.2 that we describe in Section 1.3 suggest that there is still room for other possibilities (like the currentcarrying meson condensate) at the highest densities in the gCFL regime. An alternative explanation of the consequences of the gCFL instability was advanced by Hong 73 (see also Ref. 49): since the instability is generically associated with the presence of gapless fermionic modes, and the BCS mechanism implies that

M. Alford and K. Rajagopal

9

Gapless CFL phase

Fig. 1.3. Dispersion relations of the lightest quasiquark excitations in the gCFL phase, at p. = 500 MeV, with Ms = 200 MeV and interaction strength such that the CFL gap parameter at Ms = 0 would be