FRACTIONAL STATISTICS /
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QUANTUM THEORY 2ND EDITION
FRACTIONAL j ~
STATTSTTCS AND
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FRACTIONAL STATISTICS /
r^
AND
QUANTUM THEORY 2ND EDITION
FRACTIONAL j ~
STATTSTTCS AND
QUANTUM THEORY 2ND EDITION
• Avinash Khare Institute of Physics, India
\fc World Scientific N E W JERSEY • L O N D O N • S I N G A P O R E • BEIJING • S H A N G H A I • H O N G K O N G • TAIPEI • C H E N N A I
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
FRACTIONAL STATISTICS AND QUANTUM THEORY (2nd Edition) Copyright © 2005 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.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-256-160-9
Printed in Singapore.
Dedicated to My Parents Tai and Late Bhaiyasaheb
Preface to the Second Edition
The need for a second edition of the text "Fractional Statistics and Quantum Theory" has given me an opportunity to correct several misprints, some errors and unclear formulation and a few inconsistencies of notation. I am specially grateful to Mukunda Das for drawing my attention to some of them as well as for making several useful suggestions regarding topics which could be included in the second edition. I also would like to thank Mr. Rajeev Kapri and Mr. Venkatesh Shenoi for helping me in putting everything together properly. It is clearly impossible to keep up with the expanding literature on all the topics covered in the book. One major development after the first edition of my book has been the possibility of quantum computations using anyons. This field is in its infancy and it is not at all clear how this field will develop in the coming years. I have added a section (3.14) on quantum computing with anyons where I have very briefly discussed some of the key ideas involved. I have included a new section (3.7) on scattering of anyons with hard disk potential and have also rewritten section (3.13) on pseudointegrability of N anyon system. Similarly, I have included a new section (5.4) on negative probabilities, explaining why they occur in connection with fractional exclusion statistics. Further, I have also rewritten section (5.2) on the distribution function for fractional exclusion statistics. In Chapter 11,1 have added a brief description about several more omitted topics and have also provided a few references for each of these topics. In particular, a brief discussion is included about (i) Pseudo-Integrability of N-Anyon System (ii) Random Matrix Theory and Virial Coefficients (iii) Bose-Einstein Condensation in Two Dimensions (iv) Equivalence of Ideal Bose and Ideal Fermi Gases in Two Dimensions (v) Chern-Simons Term at Finite Temperature. vii
viii
Fractional Statistics and Quantum Theory
I shall be grateful to readers if they can inform me of further misprints and errors in the second edition. I hope to make these corrections available on my homepage. Avinash Khare Bhubaneswar, October, 2004
Preface
Sometime in early 1994,1 received a letter from World Scientific, Singapore, inviting me to write a book on anyons. That made me think about the whole thing. Initially I was not sure if I should spend about two years of my research career in such a venture. At that time I went through the literature on this field and I had a careful look at the only monograph on this subject by Lerda as well as the book on anyon superconductivity edited by Wilczek. While I liked parts of these books (and their influence is apparent in parts of the book), I also felt that our perceptions are some what different. Having worked on both the non-relativistic as well as the relativistic anyon models, I was finally convinced that perhaps the time had come to accept this opportunity and put forward my point of view and hence this book. As I started the planning of the book, I realized that a lot of work has been done in the last two decades in this field and it was not possible to cover everything in this monograph. I then decided that instead of pretending to be objective, it was better to include those topics which I consider important. However, I felt that I should at least give a brief description of the omitted topics and also give a couple of references for each of these topics so that the interested reader can follow the developments in these fields. I am somewhat lucky in that many of these topics have been adequately covered in the literature. Even though so much work has been done in this field in the last two decades, I feel that this area is still in its early developmental stage and it is not completely clear as to what direction it will take in the future. This is because even the basic problem of the statistical mechanics of a noninteracting anyon gas is still an unsolved problem. I strongly believe that unless one can solve this basic problem, no qualitative progress is possible ix
x
Preface
in this field, since in the absence of this bench-mark study, any calculation including interactions will always be unreliable. I am grateful to R.K. Patra who typed the first draft of the book, and to Umesh Salian, Sandip Joshi, and Koushik Ray for help with drawing figures and putting everything together properly. I also thank my daughter Anupama for a careful proof reading. Some of the chapters were written while I was visiting Los Alamos National Laboratory, Univ. of Oslo and Univ. Of Trondehim. It is my pleasure to thank Fred Cooper, Jon Magne Leinaas and Jan Myrheim for the kind invitations and for a warm hospitality. It is a great pleasure to thank all my colleagues and collaborators who, over the years, have enhanced my knowledge of theoretical physics in general and anyons in particular. I would specially like to thank Surjyo Behera, Rajat Bhaduri, Alan Comtet, Fred Cooper, Trilochan Pradhan and Uday Sukhatme. Farther, I am extremely grateful to all the members of Institute of Physics, Bhubaneswar, and specially the three Directors, Trilochan Pradhan, V.S. Ramamurthy and Surjyo Behera who in the last twenty two years have given all the help and encouragement and most importantly provided a very stimulating atmosphere to carry out research. I am grateful and indebted to my parents, Tai and late Bhayasaheb who in spite of difficult economic condition always encouraged us to look for intellectual pursuits and emphasized the need for hard work with single minded devotion. It is only because of their sacrifices that I am today what I am. Finally, I am grateful to my wife Pushpa who always set very high standards and encouraged and inspired me to try to achieve these standards. Avinash Khare Bhubaneswar, July, 1997
Contents
Preface to the Second Edition
vii
Preface
ix
1.
2.
3.
Introduction
1
1.1 Historical Review 1.2 Plan of the Book References
3 12 15
Fractional Statistics in Two Dimensions
19
2.1 Introduction 2.2 Why Fractional Spin? 2.3 Why Fractional Statistics? 2.4 Basic Properties of Anyons 2.5 Flux Tube Model of Anyons 2.6 The Braid Group References
19 20 21 29 32 38 42
Quantum Mechanics of Anyons
43
3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8
43 44 47 51 57 58 62 65
Introduction Two Noninteracting Anyons Two Anyons in an Oscillator Potential Two Anyons in a Uniform Magnetic Field Magnetic Field with Coulomb Repulsion Scattering of Two Anyons in Coulomb Field Scattering of Two Anyons with Hard-Disk Repulsion . . . N Anyons in an Oscillator Potential xi
xii
4.
5.
6.
Contents
3.9 Three-Anyon Ground State 3.10 General Results for N Anyons 3.11 N Anyons Experiencing a N-body Potential 3.12 N Anyons in a Uniform Magnetic Field 3.13 Pseudo-Integrability of N Anyon System 3.14 Quantum Computing and Anyons References
69 77 83 87 95 96 100
Statistical Mechanics of an Ideal Anyon Gas
103
4.1 Introduction 4.2 Ideal Fermi Gas in Two Dimensions 4.3 Ideal Bose Gas in Two Dimensions 4.4 Ideal Anyon Gas: Second Virial Coefficient 4.5 Ideal Anyon Gas: Third Virial Coefficient 4.6 Ideal Anyon Gas: Higher Virial Coefficients References
103 104 110 115 130 140 142
Fractional Exclusion Statistics
145
5.1 Introduction 5.2 Distribution Function 5.3 Thermodynamics of an Ideal Gas 5.4 Negative Probabilities 5.5 Gentile Statistics 5.6 New Fractional Exclusion Statistics References
145 146 150 157 159 162 164
Introduction to the Chern-Simons Term
167
6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11
167 168 170 172 173 175 175 176 177 181 182
Introduction What is the Chern-Simons Term? Gauge Invariant Mass Term Behavior Under C, P, and T Coleman-Hill Theorem Magneto-Electric Effect Chern-Simons Term by SSB Lorentz Invariance from Gauge Invariance Quantization of Chern-Simons Mass Parity Anomaly Topological Field Theory
Contents
References 7. Soliton as Anyon in Field Theories 7.1 Introduction 7.2 Charged Vortex Solutions 7.3 Relativistic Chern-Simons Vortices 7.4 Non-relativistic Chern-Simons Vortices 7.5 CP1 Solitons with Hopf Term References
xiii
183 185 185 186 197 208 213 217
8. Anyons as Elementary Field Quanta
221
8.1 Introduction 8.2 Non-relativistic Field Theories 8.3 Relativistic Field Theories References
221 222 231 237
9. Anyon Superconductivity 9.1 Introduction 9.2 Mean Field Theory 9.3 The Random Phase Approximation 9.4 The Effective Lagrangian 9.5 Anyons and High-Tc Superconductivity 9.6 Anyon Metal References 10. Quantum Hall Effect and Anyons 10.1 Introduction 10.2 The Landau Levels 10.3 Basics of Quantum Hall Effect 10.4 Trial Wave Function and Anyons 10.5 Mean Field Theory References
239 239 242 245 249 251 253 254 257 257 259 261 264 278 285
11. Omitted Topics
287
References
293
Index
297
Chapter 1
Introduction
O brave new worlds, that have such people in them!
— E.A. Abbott in Flatland Many of us have wondered some time or the other if one can have nontrivial science and technology in two space dimensions; but the usual feeling is that two space dimensions do not offer enough scope for it. This question, to the best of my knowledge, was first addressed in 1884 by E.A. Abbot in his satirical novel Flatland [1]. The first serious book on this topic appeared in 1907 entitled An episode of Flatland [2]. In this book C.H. Hinton offered glimpses of the possible science and technology in the flatland. A nice summary of these two books appeared as a chapter entitled Flatland in a book in 1969 edited by Martin Gardner [3]. Inspired by this summary, in 1979 A.K. Dewdney [4] published a book which contains several laws of physics, chemistry, astronomy and biology in the flatland. However, all these people missed one important case where physical laws are much more complex, nontrivial and hence interesting in the flatland than in our three dimensional world. I am referring here to the case of quantum statistics. In last two decades it has been realized that whereas in three and higher space dimensions all particles must either be bosons or fermions (i.e. they must have spin of nH or (2n + l)h/2 with n = 0,1,2,... and must obey BoseEinstein or Fermi-Dirac statistics respectively), in two space dimensions the particles can have any fractional spin and can satisfy any fractional statistics which is interpolating between the two. The particles obeying such statistics are generically called as anyons. In other words, if one takes one anyon slowly around the other then in general the phase acquired is exp(±i9). If # =0 or TT (modulo 2?r) then the particles are bosons or fermions respectively while if 0 < 6 < n then the particles are termed as anyons. 1
2
Fractional Statistics and Quantum Theory
In this book I plan to explore in detail the various facets of anyons. Before I go into the details, one might wonder if our discussion is merely of academic interest? The answer to the question is no. In fact it is a surprising fact that two, one and even zero dimensional experimental physics is possible in our three-dimensional world. A few lines of digression are called for here to explain how this is possible in our three dimensional world. The point is that because of the third law of thermodynamics, which states that all the degrees of freedom freeze out in the limit of zero temperature, it is possible to strictly confine the electrons to surfaces, or even to lines or points. Thus it may happen that in a strongly confining potential, or at sufficiently low temperatures, the excitation energy in one or more directions may be much higher than the average thermal energy of the particles, so that those dimensions are effectively frozen out. An illustration might be worth while here. Consider a two dimensional electron gas on which the first experiment was in fact done in 1966 [5]. The electrons are confined to the surface of a semiconductor by a strong electric field, and they move more or less freely along the surface. On the other hand, the energy E required to excite motion in the direction perpendicular to the surface is of the order of several milli-electron-Volt (meV). Now at a temperature of say T = 1-ftT, the thermal energy is kT, where k is the Boltzmann constant. Thus if the transverse excitation energy is say 10 meV, then the fraction of electrons in the lowest excited transverse energy level is e-w
= e - 1 0 0 « 10" 44 .
(1.1)
which is zero for all practical purposes. Thus the electron gas is truly a two-dimensional gas. Few examples where planar experimental physics is possible are electron gas, surface layer studies and copper-oxide materials. Of course, even there, at the most basic level, the fundamental particles are certainly fermions or bosons. However, the most direct and appropriate discussion of the low energy behavior of a material is usually in terms of the quasi-particles. The hope is that at least in some of these cases the quasi-particles could be anyons. This hope has in fact been realized in the case of the fractionally quantized Hall effect where the quasi-particles are believed to be charged vortices i.e. charged anyons [6,7,8]. Three rather different experiments [9] seem to confirm the existence of fractionally charged excitations and hence indirectly of anyons. There is another place where, at one stage many believed that anyons could play a major role. I have in mind here the high-Tc superconductors. To date the mechanism of superconductivity in these high-Tc materials is
Ch 1.
Introduction
3
not known. Few years ago, several people were excited by the suggestion that anyons could provide the mechanism for superconductivity in these materials [10]. It soon turned out that these models provide a unique test of these ideas. In particular, they predicted the violation of the discrete symmetries of parity and time reversal invariance in these materials. Unfortunately the experiments performed in several laboratories [11,12] failed to observe the parity and the time reversal violation in the high-Tc superconductors. While these experiments have certainly dampened the interest of the physics community in anyons, they also showed that the anyon ideas are not merely esoteric and have experimental consequences which could be tested in the laboratories. Another reason why I believe that the anyons would have relevance to the real world is because of the unwritten first law of physics which states that 'anything that is not forbidden is compulsory!'. Finally, anyons represent a challenge to all those people who think that they know quantum mechanics and statistical mechanics and that they could have contributed significantly to the development of these fields if only they had been born 60-70 years earlier! At this stage it might be worthwhile to give a short historical review of the field. By its very nature, such a review is bound to be subjective and I apologize in advance to those authors who may feel that their contribution has not been given its due credit.
1.1
Historical Review
The concept of the indistinguishability of the identical particles has a deep meaning in quantum mechanics. Actually, this concept was introduced by John Willard Gibbs in classical statistical mechanics, much before the advent of quantum mechanics in order to resolve the famous Gibbs paradox. However, its ramifications are far deeper in quantum mechanics vis a vis the classical mechanics. For example, it was realized quite early that in quantum mechanics, the identical particles always interact simply because they are identical. As a result, the physical behavior of a collection of identical particles is influenced not only by the conventional interactions but also by the statistics they obey. In particular, it was realized that there are two kinds of quantum statistics in nature. It was shown that all particles have either half-integral or integral spin (in units of the Planck constant K) and accordingly they satisfy Bose-Einstein [13,14] or Fermi-
4
Fractional Statistics and Quantum Theory
Dirac [15,16] statistics respectively. It was also soon realized that there is an effective attraction between the bosons and an effective repulsion between the fermions [17,18], both of which are purely quantum mechanical in nature, and are refereed to as statistical interaction. It may be noted that it is this statistical attraction which gives rise to the accumulation of the Bose particles in the ground state which is at the heart of the phenomenon of Bose condensation. Similarly, it is the statistical repulsion between the fermions which gives rise to the famous Pauli exclusion principle. In fact, the stability of matter very much depends on the fermionic nature of the matter. Recall that according to our current picture, all matter in nature consists of quarks and leptons which are fermions. The question that one wants to ask is whether Bose-Einstein and FermiDirac are the only possible forms of quantum statistics in nature? For almost fifty years, it was believed that the answer to this question is yes. To understand why, let us go back in history a little bit. Immediately after quantum mechanics was formulated by Schrodinger and Heisenberg as it is known today, Heisenberg and Dirac extended the theory to systems of identical particles [16,19,20]. Their key observation was that all operators representing observables in such systems are necessarily symmetric under the interchange of the particle labels, if the particles are really indistinguishable. This statement has profound consequences since the symmetry operators preserve the symmetry of the wave functions. Clearly, if the operator O and the wave function ip are both totally symmetric, then Otp is also totally symmetric while if O is symmetric but %p is anti-symmetric, then Otp is totally anti-symmetric. This immediately explained as to why one has quantum theories of identical particles using only symmetric or only anti-symmetric wave functions. Since several consequences of both the Bose-Einstein and the Fermi-Dirac statistics were soon experimentally verified, hence no one really bothered to construct a more satisfactory theory. The Heisenberg-Dirac theory, even though experimentally so successful, could however be questioned on philosophical ground. Consider for example the case when two particles are so far apart that they cannot be physically interchanged. Then, clearly, it does not matter if we symmetrize or antisymmetrize the wave functions or do neither! So the question arises whether there is some postulate which is more fundamental than the symmetrization or anti-symmetrization postulate. Strictly speaking, the interchange of particle labels is a slightly misleading concept. If the particles are strictly identical, then an interchange of the identical particles is obviously an iden-
Ch 1.
Introduction
5
tity transformation. Now in quantum mechanics it is not uncommon that a physical identity transformation may be represented mathematically by a phase factor. As is well known, any permutation of bosons gives the trivial phase factor of +1 while even and odd permutation of fermions gives the phase factor of +1 or —1 respectively. An obvious question is, can we also have a more general complex phase factor instead of just +1 or —1? This question was partly answered by Laidlaw and DeWitt [21] in 1971. They applied the Feynman path integral formalism to systems of identical particles. Note that in the path integral formalism, the interchange of identical particles has a clear physical meaning as a continuous process in which each particle moves along a continuous path. The path dependence of the interchange is all important since it relates the quantum mechanical concept of particle statistics to the topology of the configuration space. The phase factors associated with different interchange paths must define a representation of the first homotopy group of the configuration space. Unfortunately, Laidlaw and DeWitt confined their attention to only three dimensions and hence concluded that only bosons and fermions can exist thereby missing the more exotic possibilities in two space dimensions. In 1977, using a more traditional approach to quantization, Leinaas and Myrheim [22] derived the same relationship between the particle statistics and topology but were bold enough to enquire about the possible quantum statistics in two dimensions. Their approach was based on the geometrical interpretation of wave functions which is the basis of the modern day gauge theories. They showed that in two dimensions, the space is multiply connected which results in the possibility of what they termed as the intermediate statistics. In particular, they showed that there exists a continuously variable parameter, which one can choose to be the phase angle 6 (or a = 0/n), that characterizes different statistics: a equal to 0 or 1 corresponds to bosons or fermions respectively while 0 < a < 1 corresponds to more exotic possibilities. It may be noted that in principle, a can be a rational or an irrational number. As an illustration of their ideas, they explicitly solved the spectrum of two such exotic particles in a twodimensional harmonic oscillator potential, and showed that as a goes from zero to 1, there is a continuous interpolation between the bosonic and the fermionic spectra. This simple calculation also showed that even the twoanyon spectrum is not related to the corresponding single particle energy levels thereby suggesting that even the simplest problem of a noninteracting anyon gas may not be easy to solve. A clarification is in order at this stage. It should be noted here that the
6
Fractional Statistics and Quantum Theory
intermediate (or anyonic) statistics that we are talking about has nothing to do with para statistics. As we shall see in detail, while para statistics can exist in any dimension and correspond to higher dimensional representation of the permutation group, anyons can exist in only two space dimensions and the underlying group is the Braid group. Few years later, Goldin, Menikoff and Sharp obtained the same results by an entirely different approach [23]. They studied the representations of the commutator algebra of of particle density and current operators. This algebra has commutation relations that are independent of the particle statistics, but the inequivalent representations correspond to different statistics. It is fair to say that the idea of the intermediate statistics (in two space dimensions) did not receive enough attention in the physics community till the papers of Wilczek [24]. It is he who coined the name anyons for the twodimensional identical particles obeying the intermediate statistics and since then it is being called as the anyonic or more generally as the fractional statistics. Apart from proposing the name, wilczek's main contribution was the flux-tube model for anyons in which anyons turn out to be point particles having both the electric charge and the magnetic flux i.e. they are point charged vortices. Wilczek also clearly spelled out the concept of statistical transmutation i.e. the fact that one can treat the noninteracting anyons as interacting bosons or interacting fermions. This idea has proved extremely useful in trying to work out the statistical mechanics of an ideal gas of anyons by treating it as an interacting Bose (or Fermi) gas. Now it is well known that the interacting Bose or Fermi gas problems are notoriously difficult and so this analogy is extremely useful in appreciating the difficulties involved in understanding even the seemingly simple problem of the noninteracting anyon gas. The flux-tube model also showed that the statistical interaction between the anyons is a vector and not a scalar long ranged interaction. This is a very important point because if it would have been a scalar long ranged interaction, going like 1/r at long distance, then in fact the virial expansion would not have existed for the noninteracting anyon gas! Around the same time, Wilczek and Zee [25] constructed a model for neutral, extended anyons within the relativistic field theory formalism. In particular, they considered the O(3) cr-model in 2+1 dimensions and showed that the solitons of this model acquire fractional spin and statistics in the presence of the Hopf-term which is an incarnation of the Chern-Simons term. This was an important development as the questions like the spin-
Ch 1.
Introduction
7
statistics theorem can be handled rigorously, only within the formalism of the relativistic quantum field theory. Soon afterwords, Wu and Zee [26] showed that the same (extended) anyon solutions can also be constructed within the CPl model. The advantage over the a model is that, in this case, the Hopf term can be written in a local form in terms of the CP1 fields. It must be added here that the relativistic anyons are invariably extended objects and hence it is extremely difficult to do any calculations with them. Thus so far as the application of anyons to the real world is concerned, we shall mostly be using the non-relativistic flux tube model of anyons in which anyons are treated as point objects. In 1983, the anyon ideas received a tremendous boost when it was realized that these ideas are not merely esoteric but can have applications in the real world. I have in mind the application in the context of the Fractional quantum Hall effect which was experimentally discovered in 1982 [27], soon after the discovery of the Integer quantum Hall effect [28]. It was Laughlin who proposed an explanation of the fractional quantum Hall effect [6]. According to him, the fractional quantization of the Hall resistance is the manifestation of a new state of matter, the incompressible quantum fluid, with elementary excitations which have fractional charge. Whereas Laughlin [6] and Haldane [7] suggested that these elementary excitations are fermions and bosons respectively, it was Halperin who correctly suggested that they are in fact anyons [8]. By using Berry phase calculations, soon it was proved by Arovas, Schrieffer and Wilczek [29] that these excitations indeed carry fractional charge and obey fractional statistics. Over the years, a number of improved theories of the fractional quantum Hall effect have been constructed [30], but all of them agree to the basic fact that the elementary excitations in the quantum Hall effect are indeed anyons. It might be added here that three rather different experiments seem to confirm the existence of the fractionally charged excitations [9] and hence indirectly of anyons. In 1984, Wu [31] emphasized the fact that whereas the first homotopy group of the configuration space of identical particles in three and higher dimensions is the permutation group SN , in two dimensions the corresponding group is the braid group B^. It is amusing to note that the mathematicians arrived at exactly the same configuration space concept from the opposite direction, namely as a useful tool for studying the braid group [32]. Wu also made another important contribution [33]. He wrote down a class of exact solutions for three anyons in an external harmonic oscillator potential. In particular, he showed that an exact anyon state starting from the
8
Fractional Statistics and Quantum Theory
three-boson ground state does not interpolate to the three-fermion ground state but interpolates to an excited state so that there must be a crossing in the ground state of the three anyons. This was a remarkable result because it immediately showed that unlike the two-anyon case (where the ground state energy smoothly interpolated from the bosonic to the fermionic end), in the three-anyon case the three-body potential between the anyons must be playing a nontrivial role. This gave the first hint that the multi-anyon problems are going to be highly nontrivial. The next obvious question to investigate was the behavior of an ideal gas of anyons. This is a kind of bench-mark study which is an absolute must before one can take into account the effect of interactions. Let us recall here that a similar study for an ideal Bose and Fermi gas was done right in the early days of quantum statistical mechanics. Of course that was easily done since the wave function for TV-bosons or ./V-fermions is merely the product of the single particle wave functions, but with appropriate symmetry or anti-symmetry factors. Unfortunately, as was clear from the seminal paper of Leinaas and Myrheim [22], even the two anyon spectrum in an oscillator potential had no correlation with the single particle spectrum. The first step towards determining the equation of state of an anyon gas was taken by Arovas, Schrieffer, Wilczek and Zee [34,35] who calculated the second virial coefficient of a noninteracting anyon gas by treating it as an interacting Bose gas and showed that it has cusps at the bosonic values of 8 = 2mr. In particular, they considered two anyons in a circular box with hard walls. Few years later, Comtet, Georgelin and Ouvry [36] simplified the calculation by confining the particles in an external harmonic potential in the same way as Fermi had done for fermions [15], and essentially using the spectrum derived earlier by Leinaas and Myrheim. This calculation clearly showed the regularization independence of the second virial coefficient. In fact Arovas et al. [34] have also calculated the second virial coefficient by path integral methods without using any regulator (since there is no need to impose any finite area restriction) and of course all the methods give the same answer. The fact that the second virial coefficient of the noninteracting anyon gas turned out to be finite is a nontrivial statement since the noninteracting anyons experience a long ranged but vector interaction and it is not at all obvious that a cluster expansion should exist for the noninteracting anyon gas.. In this paper Arovas et al. also showed that one way to impart fractional statistics to the particles is to couple them with the Chern-Simmons term. In 1986, Samir Paul and myself [37] considered an abelian Higgs model
Ch 1.
Introduction
9
with the Chern-Simons term [38] in 2 + 1 dimensions and showed that this model admits charged vortices of finite energy (in 2 + 1 dimensions). As an extra bonus, we found that these vortices had non-zero angular momentum which in general could take any arbitrary value. This strongly suggested that these could in fact be extended charged anyons. This was rigorously proved by Frohlich and Marchetti [39]. Thus this is the first field theory model for an extended charged anyon thereby generalizing the point charged vortex model of Wilczek. Sometimes later, Jatkar and I [40] showed that the abelian Higgs model with pure Chern-Simons term also admits charged vortex (i.e. extended anyon) solutions. This was important from condensed matter point of view since at long distance, the Chern-Simons term having only one derivative is expected to dominate over the Maxwell term which has two derivatives. Soon afterward, two groups simultaneously obtained the self-dual pure Chern-Simons vortex solutions [41]. The interesting point is that these are the noninteracting, though extended vortices. The other interesting point was that for the first time, one had found simultaneously the topological as well as the non-topological [42,43] self-dual solutions in a theory. Jackiw and Pi [44] considered the non-relativistic limit of this abelian Higgs model with pure Chern-Simons term and showed that this non-relativistic theory which can also be looked upon as an N-body problem with an attractive delta function interaction, admits self-dual charged vortex (i.e. anyon) solutions. The important point is that at the self dual point, the vortices are non-interacting though extended. Around the same time, Jackiw and Nair [45] obtained a wave equation for anyons. Around 1988, Laughlin suggested that perhaps anyons could provide a mechanism for the high-Tc superconductors [10]. This really attracted the attention of the physics community to the ideas of fractional statistics. Two issues are involved here. Firstly, whether anyons indeed provide mechanism for superfluidity and superconductivity? Secondly, are the high-Tc superconductors anyonic in nature? It is the second question which attracted a lot of attention since we do not know the mechanism of high-Tc superconductivity even till today. Now as we have seen, anyons must necessarily violate the discrete symmetries of parity (P) and the time reversal invariance (T). It soon became clear that if anyons have anything to do with the high-Tc superconductors, then one must be able to see such a P and T violation in these materials. Unfortunately, precision measurements showed that these materials do not really show [11,12] such P and T violation. This dramatically reduced the enthusiasm of many people towards the fractional statistics. I must however add that in a way even this neg-
10
Fractional Statistics and Quantum Theory
ative result is interesting because it shows that the anyon ideas are not esoteric but they could be tested in laboratories. After all physics is an experimental science and so a theory is useless unless it has some experimental consequences which can be tested in the near future. As far as the first issue is concerned, various calculations strongly suggest that anyons can certainly provide mechanism for superfluidity and hence superconductivity. What is not clear is if nature has made use of this mechanism in some superconductors. In 1989, Comtet and Ouvry wrote an interesting paper [46] where they showed that the second virial coefficient of a noninteracting anyon gas is related to the axial anomaly in a 1+1 dimensional field theory. As soon as we saw this paper, it occured to us that perhaps the semi-classical approximation is also exact for the second virial coefficient. The point is, it is well known that the chiral anomaly gets contribution from only one loop (i.e. lowest order term in the semi-classical approximation) and since it is related to the second virial coefficient, hence that must also be one loop exact. This was subsequently proved by us [47]. The attractive point about the semi-classical approximation is that it does not require a knowledge of the quantum spectrum, which, in most cases including anyons, is very difficult to find. Besides, since there is no other scale in the problem apart from the thermal wave length, and further, since the semi-classical approximation for the virial expansion must be exact at very large temperatures, hence I believe that the semi-classical approximation must be exact for all the virial coefficients of the non-interacting anyon gas. Of course, it is not at all clear if one can really compute the higher virial coefficients within the semi-classical approximation. Around 1991, a lot of work was done regarding the multi-anyon spectrum in a harmonic oscillator potential. Several people generalized the exact solutions of Wu [33] for three anyons and obtained a class of exact solutions for iV-anyons in an oscillator potential [48] as well as in a uniform magnetic field [49] for all of which the energy varies linearly with the anyon parameter a. It soon became clear that these linear solutions form only a small subset among the full class of solutions, and that for the missing solutions, the energy must vary nonlinearly with the anyon parameter a. It is unfortunate that till today not even one of these nonlinear state is known analytically. It might be added here that recently a class of exact solutions for 7V-anyons have also been obtained in an Af-body potential for which energy does not vary linearly with the anyon parameter a. It turns out that for all these levels, it is not the energy but E~1/2 which is linear in
Ch 1.
Introduction
11
the anyon parameter [50]. The perturbative [51], numerical [52] and semiclassical [53] studies have revealed several unusual and interesting features of the ./V-anyon spectrum in an oscillator potential as well as in a uniform magnetic field. For example, it is found that unlike the two-anyon case, the effective interaction around the three-fermion ground state is repulsive and not attractive as one had thought. As a result the ground state energy of three-anyons is not maximum at the fermionic end but at around a = 9/-K = 0.71. This has defied explanation so far. By now it is known that the same thing happens in the case of 6,10,15,... anyons i.e. one finds that the effective interaction near the ground state of ./V-fermions very much depends on the value of N. In particular, so long as N is such that the fermions form closed shells (as happens for N = 3,6,10...), the effective interaction is repulsive while in all other cases it is attractive. Further, it has been shown that for large N, there must be at least v/JV number of crossings in the ground state of ./V-anyons in a harmonic oscillator potential [54]. The fact that the iV-anyon interaction around the fermionic ground state depends so sensitively on whether the shells are closed or not raises doubts about the validity of the approximate schemes like mean field theory which do not take into account this important fact. At around the same time, there was also lot of activity regarding the third and higher virial coefficients of a noninteracting anyon gas. It was proved that unlike the second virial case, none of the higher virial coefficients contain a term which is linear in the anyon parameter a [55]. The semi-classical calculation of the third virial coefficient was also performed [47] which indicated that it is symmetric around a = 1/2, i.e. 6 = TT/2. The same symmetry was also apparent in the spectra of three anyons in an oscillator potential. Inspired by these observations, Sen [56] rigorously proved the symmetry of the third virial coefficient around a = 1/2. Soon afterward, two groups [57] independently computed the third to sixth virial coefficients to second order in the anyon parameter. From the fourth virial coefficient onward, these expressions are quite complicated, having logarithms of algebraic numbers. It thus appears that the virial expansion for a noninteracting anyon gas may have a very complicated expansion in the power of the anyon parameter. Even though so much work has been done in this field in the last two decades, I still feel that this area is still in its early developmental stage and it is not completely clear as to what direction this area will take in the future. This is because even the most basic problem of the statistical mechanics of a noninteracting anyon gas is still an unsolved problem. I
12
Fractional Statistics and Quantum Theory
strongly believe that unless one can solve this basic problem, no qualitative progress is possible in this field since in the absence of this bench-mark study, any calculation including interactions will always be unreliable. As is clear from this (subjective) review of the field, several aspects of anyons have been explored in great detail in the last two decades and it is almost impossible to cover all these topics in this monograph. I have therefore, decided that instead of pretending to be objective, it is better to cover only those topics which I believe to be important. I have, however, included in Chapter l l a brief description of the omitted topics and given a few references for each of these topics so that the interested reader can go back and trace other references and study these topics further. I have been fortunate in the sense that many of these topics have been adequately covered in the literature. I must also apologize to several authors whose work may not have been adequately quoted in the monograph in spite of my best attempts.
1.2
Plan of the Book
This book can be broadly divided into two parts. The first part has five chapters of which the first four are devoted to the various aspects of the non-relativistic model of anyons, while in the fifth I discuss the fractional exclusion statistics. The first part of the book can be understood by any one having a reasonable background in quantum and statistical mechanics. In particular no knowledge of many-body theory or quantum field theory is required in understanding this part. The second part also has five chapters and is at a slightly advanced level. A first course in quantum field theory and many-body theory is a prerequisite for really appreciating this part. In this part, I discuss the field theory models for anyons. In particular the role of the Chern-Simons term, which naturally provides a model for anyons, is discussed at length. Fractional quantum Hall effect and anyon superconductivity are also discussed in some detail. Finally, in Chapter 11, 1 give a brief description of the topics which have not been included in the book and give few references for each of these topics. When one starts talking about anyons, the first obvious question is what are anyons and why are they possible only in two space dimensions and not in higher ones? This question is discussed at length in Chapter 2 where we shall see that whereas in two dimensions, the configuration space is multi-valued, it is only double-valued in three and higher space
Ch 1.
Introduction
13
dimensions. A flux-tube model of anyon is also introduced here according to which an anyon can be looked upon as a point particle with a flux tube attached at its sight. This has an immediate consequence that one can freely transmute anyonic statistics with interaction. In particular, an ideal anyon gas is equivalent to an interacting Bose (or Fermi) gas. This also explains as to why even the problem of a non-interacting anyon gas is so difficult. It turns out that the group structure underlying the anyons is in fact the braid group. An elementary introduction to the braid group is also given in this chapter. While talking about the new quantum statistics, the first obvious question is about the distribution function of an ideal gas of anyons. One would also like to know things like partition function and equation of state of the ideal gas. As a first step in that direction, the quantum mechanics of multi-anyons is discussed at length in Chapter 3. Exact solutions for two anyons in an oscillator as well as in several other potentials are discussed at length. The problem of three and multi-anyons in an oscillator potential and/or in a uniform magnetic field, as well as in an iV-body potential, are also discussed and the difficulties in finding the full spectrum are pointed out. Using the second order perturbation theory, the ground state energy of three anyons in an oscillator potential is computed and using these results, a detailed discussion is made about the nature of the ground state of three and multi-anyons in an oscillator potential and/or in a uniform magnetic field. Finally, a brief discussion is also presented about the role anyons could play in quantum computations. Chapters 3 and 4, in a sense are the heart of the book. After giving a brief introduction to the ideal Fermi and Bose gas in two dimensions, I discuss the various known results for an ideal gas of anyons. In particular, I discuss two different derivations for the computation of the second virial coefficient of an ideal anyon gas. Further, i show that the semi-classical techniques also reproduce the exact second virial coefficient of not only an ideal anyon gas but even of an interacting anyon gas so long as the interaction is scale invariant and hence does not introduce any extra dimensional parameter in the problem. It is worth emphasizing that to date, the full expressions for the third or higher virial coefficients are not known. However, some results for the higher virial coefficients are known which are discussed briefly and the difficulties in obtaining the exact results for the higher virial coefficients are pointed out. Recently a new quantum statistics called Fractional Exclusion Statistics has been introduced which is valid in any dimension. This is discussed in
14
Fractional Statistics and Quantum Theory
some detail in Chapter 5. Various quantities like distribution function etc. are computed for an ideal gas and it is pointed out that the CalogeroSutherland model represents an example of an ideal one dimensional gas with fractional statistics. The phenomenon of fractional statistics is most naturally discussed within the framework of Chern-Simons theories and that will be done in the second part of the book. As a prelude to these discussions, I discuss the various properties of the Chern-Simons term in Chapter 6. In particular its role as a gauge field mass term and its behavior under the discrete transformations of parity (P) and time-reversal (T) is emphasized. It is worth noting that in the long wave length limit, this term having only one derivative (in two space and one time dimensions) is expected to dominate over the usual Maxwell kinetic energy term which has two derivatives. In the presence of the Chern-Simons term, anyons can appear in two different ways and both approaches are elaborated in some detail. The charged vortex solutions in abelian Higgs model with Chern-Simons term are obtained in Chapter 7 and it is pointed out that these charged vortices represent the first relativistic model for (extended) charged anyons. I also construct the charged vortex solutions in pure Chern-Simons theory in both the relativistic and the non-relativistic settings. Finally, I also discuss an example of neutral relativistic anyons by considering the soliton solutions in the CP1 model with the Hopf term which is one of the avtars of the Chern-Simons term. In Chapter 8,1 elaborate upon the other approach in which fundamental fields of theories with Chern-Simons term themselves carry fractional spin and obey fractional statistics. A la Dirac equation for the spin-1/2 fields, I also discuss here a relativistic wave equation for anyons. Chapter 9 is devoted to a discussion about the many-anyon system within the mean field approach. Approximate schemes like RPA which go beyond the mean field approximation are also discussed. Qualitative arguments are also given here regarding the possibility of anyon superconductivity. It may be emphasized here that these arguments are quite robust and even though the high-Tc superconductors have perhaps nothing to do with the anyons, it is not unlikely that there may be other (yet to be discovered) superconductors in nature in which anyons could play an important role. No discussion of anyons will be complete without an account of its most outstanding success-its application to the fractional quantum Hall effect where anyons appear as a quasi-particle excitations. After a brief
Ch 1. Introduction
15
discussion to the Landau level problem and to the quantum Hall effect, the trial wave-function approach of Laughlin [6] is discussed in some detail in Chapter 10. The Landau-Ginzberg-Chern-Simons approach is also briefly introduced. Finally, in Chapter 11, a brief description of the omitted topics is given along with a few references for each of these topics so that the interested reader can trace back and study these topics further.
References [1] E.A. Abbot, Flatland (Princeton Univ. Press, New Edition, 1991). [2] CH. Hinton, An Episode Of Flatland (1907). [3] M. Gardner, in The Unexpected Hanging And Other Mathematical Diversions, ed. M. Gardner (Simon and Schuster 1969). [4] A.K. Dewdney, Two Dimensional Science and Technology, J. Recreate. Math. 12 (1979) 16 ; For a short summary of the book see M. Gardner, Sci. Ame., July Issue (1980) 18. [5] A.B. Fowler, F.F. Fang, W.E. Howard and P.J. Stiles, Phys. Rev. Lett. 16 (1966) 901. [6] R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. [7] F.D.M. Haldane, Phys. Rev. Lett. 51 (1983) 605. [8] B.I. Halperin, Phys. Rev. Lett. 52 (1984) 1583; 52 (1984) E2390. [9] R.G. Clark et al., Phys. Rev. Lett. 60 (1988) 1747 ; J.A. Simmons et al., 63 (1989) 1731 ; A.M. Chang and J.E. Cummingham, Solid State Comm. 72 (1989) 651 ; For a popular readable account see, A. Khurana, Phys. Today 43(1) (1990) 19. [10] R.B. Laughlin, Phys. Rev. Lett. 60 (1988) 2677 ; Science 242 (1988) 525. [11] S. Spielman et al., Phys. Rev. Lett. 65 (1990) 123. [12] R.F. Kieff et al., Phys. Rev. Lett. 64 (1990) 2082. [13] S.N. Bose, Zeits. f. Phys. 26 (1924) 178. [14] A. Einstein, Sitzungsber. Preuss. Akad. Wiss. (1924) 26 ; (1925) 3,18. [15] E. Fermi, Zeits. f. Phys. 36 (1926) 902. [16] P.A.M. Dirac, Proc. Roy. Soc. London A112 (1926) 661. [17] R. K. Pathria, Statistical Mechanics (pergamon Press, Oxford, 1972). [18] K. Huang, Statistical mechanics (John Wiley and Sons, Inc., 1963). [19] W. Heisenberg, Zeits. f. Phys. 38 (1926) 411. [20] P.A.M. Dirac, The Principles of Quantum Mechanics (Oxford Univ. Press, Oxford, 1935). [21] M.G.G. Laidlaw and CM. DeWitt, Phys. Rev. D3 (1971) 1375. [22] J.M. Leinaas and J. Myrheim, Nuovo Cim. B37 (1977) 1. [23] G.A. Goldin, R. Menikoff and D.H. Sharp, J. Math. Phys. 21 (1980) 650 ; 22 (1981) 1664. [24] F. Wilczek, Phys. Rev. Lett. 48 (1982) 1144 ; 49 (1982) 957.
16
[25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55]
Fractional Statistics and Quantum Theory
F. Wilczek and A. Zee, Phys. Rev. Lett. 51 (1983) 2250. Y.-S. Wu and A. Zee, Phys. Lett. B147 (1984) 325. D.C. Tsui, H.L. Stormer and A.C. Gossard, Phys. Rev. Lett. 48 (1982) 1559. K. von Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45 (1980) 494. D. Arovas, J.R. Schrieffer and F. Wilczek, Phys. Rev. Lett. 53 (1984) 722. J.K. Jain, Phys. Rev. B40 (1989) 8079 ; B41 (1991) 7653. Y.-S. Wu, Phys. Rev. Lett. 52 (1984) 2103. E. Fadell and L. Neuwirth, Math. Scand. 10 (1962) 111 ; R. Fox and L. Neuwirth, 10 (1962) 119. Y.-S. Wu, Phys. Rev. Lett. 53 (1984) 111; 53 (1984) E1028. D.P. Arovas, J.R. Schrieffer, F. Wilczek and A. Zee, Nucl. Phys. B251 (1985) 117. J.S. Dowker, J. Phys. A18 (1985) 3521. A. Comtet, Y. Georgelin and S. Ouvry, J. Phys. A22 (1989) 3917. S.K. Paul and A. Khare, Phys. Lett. B174 (1986) 420 ; Phys. Lett. B177 (1986) E453. S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48 (1982) 975; Ann. Phys. 140 (1982) 372. J. Frohlich and P.A. Marchetti, Comm. Math. Phys. 121 (1989) 177. D. Jatkar and A. Khare, Phys. Lett. B236 (1990) 283. J. Hong, Y. Kim and P.Y. Pac, Phys. Rev. Lett. 64 (1990) 2230 ; R. Jackiw and E.J. Weinberg, 64 (1990) 2234. R. Jackiw, K. Lee and E.J. Weinberg, Phys. Rev. D42 (1990) 3488. A. Khare, Phys. Lett. B255 (1991) 393. R. Jackiw and S.-Y. Pi, Phys. Rev. Lett. 64 (1990) 2969; Phys. Rev. D42 (1990) 3500. R. Jackiw and V.P. Nair, Phys. Rev. D43 (1991) 1933. A. Comtet and S. Ouvry, Phys. Lett. B225 (1989) 272. R.K. Bhaduri, R.S. Bhalerao, A. Khare, J. Law and M.V.N. Murthy, Phys. Rev. Lett. 66 (1991) 523. C. Chou, Phys. Rev. D44 (1991) 2533 ; D45 (1992) E1433 ; R. Basu, G. Date and M.V.N. Murthy, B46 (1992) 3139 ; A.P. Polychronakos, Phys. Lett. B264 (1991) 362. G.V. Dunne, A. Lerda, S. Sciuto and C.A. Trugenberger, Nucl. Phys. B370 (1992) 601 ; K. Cho and C. Rim, Ann. Phys. 213 (1992) 295. A. Khare, Phys. Lett. A221 (1996) 365. A. Khare and J. McCabe, Phys. Lett. B269 (1991) 330 ; A. Khare, J. McCabe and S. Ouvry, Phys. Rev. D46 (1992) 2714. M. Sporre, J.J.M. Verbaarschot, and I. Zahed, Phys. Rev. Lett. 67 (1991) 1813 ; M.V.N. Murthy, J. Law, M. Brack and R.K. Bhaduri, Phys. Rev. Lett. 67 (1991) 1817. F. Illuminati, F. Ravndal and J.A. Rudd, Phys. Lett. A161 (1992) 323; M. Sporre, J.J.M. Verbaarschot, and I. Zahed, Nucl. Phys. B389 (1993) 645. R. Chitra and D. Sen, Phys. Rev. B 46 (1992) 10 923. J. McCabe and S. Ouvry, Phys. Lett. B260 (1991) 113 ; A. Comtet, J. McCabe and S. Ouvry, Phys. Lett. B260 (1991) 372 ; D. Sen, Nucl. Phys.
Ch 1.
Introduction
17
B360 (1991) 397. [56] D. Sen, Phys. Rev. Lett. 68 (1992) 2977. [57] A. Dasnieres de Veigy and S. Ouvry, Phys. Lett. B291 (1992) 130 ; Nucl. Phys. B388 (1992) 715 ; R. Emparan and M.A. Valle Basagoiti, Mod. Phys. Lett. A8 (1993) 3291.
Chapter 2
Fractional Statistics in Two Dimensions Ah, the joy, ah, the joy of Thought! What can It not achieve by thinking! — E.A. Abbott in Flatland
2.1
Introduction
In this chapter we shall enumerate in some detail the arguments which lead to the conclusion that unlike in three and higher space dimensions, in two dimensions one can have fractional statistics [1,2,3,4,5,6,7]. To begin with, we first give arguments as to why unlike in three and higher dimensions, in two dimensions the eigenvalue of the angular momentum operator can be fractional in units of H. We then follow Leinaas and Myrheim [8] and argue that the key reason for the fractional statistics in two dimensions is the fact that the points which correspond to a coincidence of the position of two or more particles are singular points of the configuration space and hence must be excluded from it. It will turn out that the relevant group here is not the permutation but the braid group. Basic properties of the braid group will be discussed. Some of the important properties of anyonsthe particles obeying fractional statistics will be enumerated. Particular mention may be made of the fact that the anyons must necessarily violate the discrete symmetries of parity (P) and time reversal invariance (T). A dynamical model for anyons will be discussed in which the anyons are point particles with a delta-function flux tube attached to each particle so that the anyon is a point charged vortex. One of the remarkable consequences of this model is that it offers us the phenomenon of statistical transmutation i.e. one can also regard an ideal anyon gas as an interacting Bose or Fermi 19
20
Fractional Statistics and Quantum Theory
gas. This is very important in appreciating as to why the problem of an ideal gas of anyons is an unsolved problem till today. We shall be using this picture and study the quantum and statistical mechanics of an ideal anyon gas in this book. 2.2
Why Fractional Spin?
The possibility of fractional spin in two dimensions is easily understood. The point is that the spin in two dimensions differs fundamentally from the spin in higher dimensions. This is because whereas in three and higher space dimensions, the spin angular momentum algebra is non-commutative i.e. (ft = 1) [Si,Sj]
= ieijkSk;
i,j,k = 1 , 2 , 3
(2.1)
in two space dimensions, it is a trivial commutative algebra since only one generator (say 53) is available which obviously commutes with itself. As a result, there is no analogue of the quantization of the angular momentum, which arises in three and higher space dimensions from the nonlinear commutation relation (2.1). Here e,jfc is the completely antisymmetric tensor. Let us first show how the commutation relation (2.1) leads to the quantization of the eigenvalues of the spin angular momentum operator. Let \s,m) be the state satisfying S2\s,m) — s(s + l)\s,m);
S3\s,m) = m\s,m).
(2-2)
On applying the raising operator S+ on the state \s,m) we have S+\s,m) = [s(s + l ) - m ( m + l)] 1 / 2 |s,m+l) = \s,m').
(2.3)
Requiring this state to have positive norm leads to m<s.
(2.4)
One can now construct states (S+)2\s,m), (S+)3\s,m),.... It is clear that for some value of ml = m + integer, m' > s, one will have a state of negative norm unless s = m' i.e. s — m = integer.
(2-5)
Similarly by insisting that the state S~\s,m) has positive norm, we get s(s + 1)- m(m - 1) > 0
(2.6)
Ch 2.
Fractional Statistics in Two Dimensions
21
and again by constructing states (S~)2\s,m), (S~)3\s,m),..., it is easily seen that one can avoid the negative norm states by choosing s + m = integer.
(2.7)
On adding Eqs. (2.5) and (2.7) we then conclude that 2s = integer
(2.8)
thereby proving that the eigenvalues of the spin angular momentum operator must necessarily be either half-integer or integer (in unit of h). On the other hand, in two space dimensions, there exists only one axis of rotation (the axis perpendicular to the plane) and hence spin here only refers to S3 which obviously commutes with itself. As a result, there is no restriction on the eigenvalues of the spin angular momentum operator S3 in two dimensions and it can in fact take any arbitrary value. For completeness, it is worth pointing out that in one (space) dimension, there is no axis for rotation and hence there is no notion of spin in one dimension. Now, in relativistic quantum field theory, there is a deep and profound connection between the spin and the statistics i.e. particles with half integer spin are fermions, satisfying Fermi-Dirac statistics, while those with integer spin are bosons, satisfying Bose-Einstein statistics. This immediately suggests that in two dimensions the particles may exhibit fractional (i.e. any) statistics. We now provide rigorous arguments to show that this expectation is indeed justified. 2.3
Why Fractional Statistics?
Before we come to a proper discussion about the statistics, it is worth clarifying as to what exactly one means by quantum statistics. In most text books on statistical mechanics, the term "quantum statistics" refers to the phase picked up by a wave function when two identical particles are interchanged, i.e. under the permutation of the particles. But this is slightly misleading and has been correctly criticized in the literature [9]. If the particles are strictly identical, the word permutation has no physical meaning since a given configuration and the one obtained by the permutation of the particle coordinates are merely two different ways of describing the same particle configuration. The origin of the trouble lies in the introduction of the particle coordinates which brings elements of non-observable character into the theory and makes the discussion more
22
Fractional Statistics and Quantum Theory
obscure. The term quantum statistics actually refers to the phase that arises when two particles are adiabatically transported giving rise to the exchange. In this book, we shall be concentrating on this definition of quantum statistics. It is a coincidence that in three and higher dimensions, the two definitions, based on the permutation and the adiabatic exchange of two particles, coincide, but in two dimensions the two definitions give very different answers. The key reason for the fractional statistics in two dimensions is the principle of indistinguishability of identical particles. It is one of the most important characteristics of quantum mechanics (vis a vis classical mechanics) and it has profound physical consequences. The principle is in fact older than quantum mechanics. It was introduced by John Willard Gibbs even in classical statistical mechanics to resolve the famous Gibbs paradox. Even though this principle has been with us for a very long time, unfortunately, its full significance was not appreciated till 1977 and that is how one missed the possibility of fractional statistics in two dimensions for all these years. It may therefore be worthwhile to restate the Gibbs paradox and its resolution by Gibbs. Stated simply, the paradox was "if one mixes two gases of the same kind in classical statistical mechanics then the entropy seems to increase." Gibbs resolved the paradox by pointing out that one has made an error in calculating the phase space volume in the case of the identical particles. The correct volume for N identical particles is in fact N\ times smaller than what one thought it to be. It turns out that it is this principle which is at the heart of the whole problem. Following Leinaas and Myrheim [8], let us enquire about the configuration space of a system of identical particles? Normally one considers the full phase space in statistical mechanics but it turns out that configuration space is enough for this discussion. Suppose one particle space is X. Then what is the configuration space of N identical particles? The Naive answer is XN, which, even though true locally, is not correct globally. Why? The reason is, since the particles are strictly identical, hence there is no distinction between the points in XN that differ only in the ordering of the particle coordinates. For example, consider the point x = (x1)x2,...,Xiv)
(2.9)
in XN where x, € X for i = 1,2, ...,7V. Now consider another point x' in XN which is obtained from x by the permutation p of the particle indices
Ch 2.
23
Fractional Statistics in Two Dimensions
i.e. x' = P(x) = (xp-i ( 1 ) ) ...,Xp-i W ).
(2.10)
Clearly, both describe the same physical configuration of the system. Thus the true configuration of the TV-particle system is not X but it is the space XN/SN which is obtained by identifying points in XN that represent the same physical configuration, i.e. it is obtained from XN by dividing out by the action of the symmetry group SN- Note that SN is a discrete, finite group obtained by permutation of N identical particles. As a result, the space XN /SN is locally isomorphic to XN except at its singular points. However, the global properties of the two spaces are very different. Whereas XN has only regular points when X is regular, those points in XN/SN which correspond to a coincidence of the positions of two or more particles are in fact singular points of XN/SNThus to calculate the configuration space of identical particles, such singular points must be excluded by say so that we can determine if two particles have been exchanged or not. This of course does not make much difference classically. However, in the quantum case the global properties of the configuration space are of deep significance and this results in the possibility of fractional statistics. It is worth emphasizing that this is the crux of the whole matter and it is this fact which was missed for about fifty years! We shall see later that this hardcore constraint is actually unnecessary since for all particles except bosons, there is an automatic angular momentum barrier preventing the crossing of the trajectories. On the other hand, for bosons there is no potential barrier and the trajectories may indeed cross but that is fine since, in any case, the relative phase factor for the exchange of two bosons is one! in case the Let us now study in detail the configuration space XN/SN one-particle coordinate space is d-dimensional Euclidean space Rd. Let us first introduce the center-of-mass coordinate
R=
^i>
&Rd
2 n
(- )
where xi, ...,xjv are the coordinates of the TV-particles. Now, the center of mass coordinate R is invariant under SN and hence the TV-particle space can be written as a Cartesian product ^-=Rdxr(d,N)
(2.12)
24
Fractional Statistics and Quantum Theory
of the center of mass space Rd and relative space r(d, N) representing d(N — 1) degrees of freedom of the relative motion of particles. Note that the relative space r(d, N) is obtained from the Euclidean space RNd~d by identifying points connected through an element from Sjy. For simplicity, let us now consider the case of two particles i.e. N = 2. Generalization to arbitrary N is straightforward. In this case, the relative space r(d, 2) is obtained from Rd by the identification of points x = xi — X2 and —x = X2 — xi. The identification space has one singular point x = 0, corresponding to a coincidence of the positions of two particles. Thus the configuration space of two identical particles in (^-dimensions can be written as
*!
=
&
x
(fl'-W)
= {0; 00> x Pd_x
(2J3)
Here {0} means that the relative coordinate x = 0 is to be excised and Zi corresponds to the fact that x is to be identified with —x. Further, (0, oo) means the positive real line with r = 0 omitted while Pd-i is a (d— 1)dimensional real projective space for the direction ±T^T of X. It is at this stage that the difference between d = 2 and d > 3 comes out, so let us discuss the two cases separately.
Fig. 2.1 All possible close paths in the configuration space in d > 3.
(a) d > 3: In this case the relative configuration space r(d, 2) is the surface of a (d — l)-dimensional sphere Sd~l with diametrically opposite points identified. Now, to know the phase picked up by the wave function of a particle as it moves around the other particle, one has to classify all possible close paths in the configuration space, i.e. in the relative space r(d, 2). In the case of d > 3, there are three possibilities as shown in Figs.
Ch 2.
Fractional Statistics in Two Dimensions
25
2.1a, 2.1b and 2.1c [6] corresponding to (a) no exchange, (b) single exchange and (c) two exchanges respectively. Consider first the case (a). It defines a motion of the particles which does not involve any exchange and hence it is a close path which can be continuously shrunk to a point. Thus this path cannot impart any phase to the wave function. Case (b), on the other hand, involves the exchange of two particles and goes from a point on the sphere to its diametrically opposite point - again a close path. This path can cause a non-trivial phase in the wave function. This is because, having fixed the two end points, no continuous process can shrink this path to a point. Finally, the case (c) which involves two exchanges, forms a close path on the sphere which can be continuously shrunk to a point. One can easily convince oneself of this by looping a ball with a closed string! We thus conclude that in three and higher space dimensions, only two classes of closed paths corresponding to one or no exchange are possible. Now, since two exchanges correspond to no exchange, hence it follows that if rj is the phase picked up during one exchange then r\ can have only two possible values i.e. ±1. Hence, the only possible statistics in d > 3 are the Bose-Einstein and the Fermi-Dirac statistics.
Fig. 2.2 Relative configuration space r(2, 2) having the topology of a cone without the tip.
(b) d = 2: The above discussion breaks down in the case of two space dimensions. This is because, unlike d > 3, the relative configuration space r(2,2) is a punctured plane with the opposite points identified, i.e. it is a circular cone of half-angle 30°. Now a cone is globally curved, although it is locally flat everywhere except at the singular vertex. All this is best visualized in Fig. 2.2 [8]. The identification of r(2,2) may be effected by
26
Fractional Statistics and Quantum Theory
cutting the plane along a line I from the origin O and then folding it into a circular cone of half-angle 30°. Notice that a circle C in the plane centered at O then revolves twice around the cone.
Fig. 2.3 All possible closed paths in the configuration space in two dimensions.
In this case, several closed paths are possible, as shown in Figs. 2.3a, 2.3b and 2.3c [6] corresponding to (a) no exchange, (b) one exchange and (c) two exchanges respectively. The case (a), involves no exchange and can obviously be shrunk to a point. However, the case (b), that exchanges the two particles, is clearly non-contractible (as in d > 3) since the end-points are fixed. However, the real surprise is that even the case (c) which involves two exchanges (either in the clockwise or in the anti-clockwise directions) cannot be contracted to a point. Clearly the same is also true for 3,4,..., n exchanges. Further, since the particles cannot go through each other, hence one can distinguish between the clockwise and the anti-clockwise winding. Thus unlike in d > 3, in two dimensions it is not enough to specify the initial and final configurations to completely characterize a system ; it is also necessary to specify how the different trajectories wind or braid around each other. Thus the time evolution of particles is important and cannot be neglected in two dimensions. Thus the main distinction between the configuration spaces in two and three (or higher) dimensions is that the removal of the origin (x = 0) in two dimensions makes the space multiply connected while for d > 3 it is only doubly connected and that is why in two dimensions it is possible to define paths that wind around the origin an arbitrary number of times counted with orientation. Mathematically, this distinction is expressed in terms of the first homotopy group TTI which is the group formed by the inequivalent paths (paths that are not deformable to one another) passing
Ch 2.
Fractional Statistics in Two Dimensions
27
through a given point in configuration space with the group multiplication being defined as traversing paths in succession and the group inverse as traversing a path in the opposite direction [10,11,12]. Thus for the two dimensional case, one has
(2.14)
4-,^^i^i_m^MRPl)=z while for d > 3 one has ^3
= vi
f(R^-{0})\
=
^Rp^j
= Z2.
(2.15)
It is worth repeating once again that if the vertex of the cone had been included-that is if we had allowed the particles to occupy the same position then the configuration space, even in two dimensions, would be simply connected. Thus if we do not impose any hard-core constraint on the particles, then we can describe only bosonic statistics! On the other hand, in three (and higher) dimensions we find that the projective space is only double connected. For example P2 can be described as the norther hemisphere with opposite points on the equator being identified. Generalization to N particles is now straightforward. Clearly, the configuration space in d dimensions is ON
JN
(2.16)
where A is the generalized diagonal given by A = {ri, ...,rjv, : Tj = rj for some I ^ J } .
(2-17)
We now have to find the first homotopy group of such a space. This is a problem in algebraic topology whose solution has been given by mathematicians [13,14]. It is remarkable that the mathematicians were interested in this problem at around the same time as the physicists! In particular, it turns out that the first homotopy group is given by
(
XN\ — = BN , for d - 2 SN ) = SN, for d> 3
(2.18)
where BN is the braid group of N objects while SN denotes the permutation group which incidentally is a finite subgroup of the braid group [15]. We thus see that whereas the permutation group SV is at the heart of the
28
Fractional Statistics and Quantum Theory
Fermi-Dirac and the Bose-Einstein statistics, it is the braid group BN which is at the heart of the anyonic statistics. It turns out that whereas there are only two one dimensional representations of the permutation group (the identical one and the alternating one, corresponding respectively to the bosonic and the fermionic statistics), the braid group BN admits a continuous parameter family of one dimensional representations. We shall identify the labeling parameter of these representations with the parameter 6 which characterizes fractional statistics. A brief introduction to the braid group is given at the end of the chapter. Summarizing, we have seen that whereas the configuration space of N particles in three and higher space dimensions is doubly connected, it is multiply connected in two dimensions. This fact has profound consequences when we quantize a system of identical particles. Let us discuss the quantization within the path integral prescription [16,17]. According to Feynman [18,19], whereas in classical mechanics the particle trajectory follows the path of least action, in the quantum case, all paths contribute but with appropriate weight factor. In particular, the transition amplitude from (xi,ii) to (XJV,£;V) is given by %
-
I
Ldt) . (2.19)
Without any loss of generality, let us consider only the closed paths (i.e. xi = XJV) which begin and end with the same configuration of particles. Each such trajectory is a closed loop in the configuration space. What is the proportionality constant? Normally that is unimportant since it is independent of the paths which can be continuously deformed into one another and hence unobservable. But why should the proportionality constant be the same for those trajectories which cannot be deformed into one another? Remember, this is the case in two dimensions where the configuration space is multiply connected. In such a case we must put appropriate weight x(a) i-e- w e c a n organize the sum over all loops into a sum over the homotopic classes a so that
<xJV,tAr|x1)t1)=
J2 XW) E all paths
q(t)e/3
eX
p Q [NL(q,q)dt\ ^
fl
(2.20) '
How much is x(/3)? Is it completely arbitrary or is there some principle which fixes it, at least partly? According to the general rules of quantum mechanics, we must demand that the time development of a trajectory over
Ch 2.
Fractional Statistics in Two Dimensions
29
the time interval to to t\ followed by its development over the interval t\ to £2 is assigned the same amplitude as we would assign to the trajectoryover the interval to to t^. This implies that X(01)X(P2) = x(Pifo)
(2.21)
whose solution is ±i9
x(f3)=e
,
0 y or x —• x, y —> — y and not x —> — x,y —> — y since the latter one is equivalent to a rotation). Similarly, if the counterclockwise winding is reversed in time, it will look like the clockwise winding. Thus the violation of the parity and the time reversal invariance offers a unique test of the theories of anyons. It may be noted here that in three (and higher) space dimensions, the single exchange path (Fig. 2.1b) can be smoothly deformed to its time reversed path by simple rotation about the north pole. (2) Anyons are sort of in between the bosons and the fermions i.e. the repulsion between two-anyons in the ground state monotonically increases as a goes from 0 to 1 (we shall demonstrate this in the next chapter), with there being no repulsion between two bosons. Thus, in a sense, anyons are closer to the fermions than to the bosons since all of them will satisfy a generalized form of Pauli exclusion principle. Put another way, the trajectories of two anyons cannot cross and one can, in principle, distinguish crossing in front from crossing behind i.e. counterclockwise from clockwise winding. (3) The technical name for the group structure of the paths for anyons is braid group BN which is both apt and picturesque. The braid in braid group, refers to the interpretation of the disconnected piece of the trajectory space as topologically distinct methods of styling coils of hair. The one dimensional representations of the braid group BM are labeled by the continuous parameter 9 as given in Eq. (2.22), periodic modulo 2?r. Thus by anyons we mean particles which belong to the one dimensional representations of the braid group. For comparison, let us note that there are two one dimensional representations of the permutation group SN assigning 1 to each permutation or assigning ±1 to each permutation depending on whether the permutation is even or odd and they correspond to bosons and fermions respectively. A brief introduction to the braid group is given at the end of the chapter. (4) How does one braid a third particle trajectory into two? For example it can be done in two different ways as shown in Figs. 2.4a and 2.4b [1]. In the first case, the third fellow is a sort of passive bystander
Ch 2.
Fractional Statistics in Two Dimensions
31
Fig. 2.4 Two possible forms of three particle trajectories.
while in the second case there is a nontrivial braiding which cannot be undone. As a result, we find that, whereas in the first case the phase factor is el6, in the second case the appropriate phase factor is e3lS. This is bad news because it means that the phase due to the exchange of two identical particles in two dimensions depends in principle on the position of all other particles. It is this fact which makes even the simplest many-body problem of an ideal gas of anyons to be highly nontrivial. In fact till now full solution is not known to any N-anyon problem (N > 2). (5) Is there a relation between the anyonic statistics and the parastatistics? There is a misconception among many physicists that the two are related. However, it is not true, and the two have nothing to do with each other. The two are totally different concepts since they are built on two different structures i.e. the braid group and the permutation group respectively. Parastatistics merely corresponds to the higher dimensional representation of the permutation group while anyons as discussed above correspond to the one dimensional representation of the braid group. Thus, whereas one can have particles in any dimension (d > 2) obeying parastatistics, anyons can exist only in two dimensions.
32
2.5
Fractional Statistics and Quantum Theory
Flux Tube Model of Anyons
This discussion suggests dynamical implementation of the fractional statistics. Following Wilczek [21] let us consider the interaction term Ls = ha^, at
(2.23)
a=-, TT
in the quantum mechanics of two identical bosons (or fermions). Here a is the numerical parameter (0 < a < 1) while <j> is the relative angle between the identical particles. Note that even though <j> is ambiguous by multiple of 2TT, d/dt is perfectly well denned. Now, according to the path integral prescription, this term will induce a phase
exp
Vk I
Lsdt
)
= exp {ia[m
~ ^a)])
(2 24)
-
which is proportional to the winding angle. Since only trajectories with common end points can interfere hence in evaluating the physical effects due to Eq. (2.23), we shall always be comparing paths with a common value for the relative angle (<j>(b) — <j>(a)) swept by the particles, modulo n. This also makes it clear that the numerical parameter 9 is indeed an angular variable with period 2-ir. These arguments can be easily generalized for an arbitrary number of particles. In this way we see that the statistical interaction term (2.23) concretely implements the ^-statistics, or to be more precise, implements the transmutation of statistics. In particular, if we consider a Lagrangian for two non-interacting identical bosons (or fermions) i.e.
L = y(*? + *2)
(2-25)
and add the statistical interaction term Ls to it, then the total Lagrangian can either be interpreted as the Lagrangian for two noninteracting anyons or as the Lagrangian for two interacting bosons (or fermions). This interpretation is very illuminating and makes it clear why the problem of even noninteracting anyon gas is rather nontrivial. The point is, a noninteracting anyon gas problem is equivalent to an interacting Bose (or Fermi) gas problem which is in general quite difficult to solve. We shall be using this equivalence throughout this book to study the quantum and statistical mechanics of a noninteracting anyon gas and we shall see that till today only a few two-anyon problems have been solved exactly and there exists
Ch 2.
Fractional Statistics in Two Dimensions
33
no multi-anyon (N > 2) problem for which the complete solution is known as yet. This concrete implementation of the anyonic statistics nicely illustrates the following two points. (1) Anyonic statistics is a pure quantum mechanical effect with no classical analogue. Note that Ls is a total time derivative and hence cannot contribute to the (classical) equations of motion. In quantum mechanics, its effect arises because it contributes to the canonical momentum, which in turn appears in the commutation relations that defines the quantum dynamics. (2) Since d(j>/dt changes sign under the parity as well as the time reversal invariance, hence we see that unlike the Lagrangian as given by Eq. (2.25), the statistical interaction term as given by Eq. (2.23) is odd under parity as well as time reversal invariance (6 ^ TTITT) thereby confirming that anyons must necessarily violate these two discrete symmetries. Let us start from the Lagrangian for two noninteracting anyons as given by Eqs. (2.23) and (2.25) i.e.
L=^(rl + rl) + Ha^
(2.26)
where a = ^ is the anyon parameter with 0 < a < 1. Let us separate the center of mass and the relative motions i.e. define R = ( r 1 + r 2 ) / 2 ^ P = p1+p2
(2.27)
r = n - r 2 =>• p = (pi - p 2 )/2.
(2.28)
In terms of these coordinates, L can be rewritten as L = LR + Lr where LR = mR 2 Lr = jr2
+ Mj> = j (r2 + r2j>2) + haj>.
(2.29) (2.30)
We thus see that the center of mass motion is free (HR = P^/im) and is independent of the statistical parameter a. This feature is also valid in the case of TV anyons. Thus to understand the anyon dynamics, it is enough to concentrate on the Hamiltonian for the relative motion of N anyons and to
34
Fractional Statistics and Quantum Theory
forget about the trivial center of mass motion. The corresponding relative Hamiltonian turns out to be =
pj + { p ^ l
(2-32)
mrz
m
The generalization to the case of N particles is straight forward. In particular the Lagrangian for N noninteracting anyons is given by
* = yX>? +
fcȣ^
(2-32)
where faj = arctan[(j/j — yj)/(xi — Xj)\. As mentioned above, this can also be regarded as the Lagrangian for N interacting bosons (or fermions).
Dynamical Realization of the Statistical Interaction What could be the physical origin of the statistical interaction term (2.23)? Can one realize it dynamically? The answer is yes provided (fictitious) electric charge and magnetic flux are attached to the particles. We want to make it absolutely clear that this fictitious charge and flux have nothing to do with the ordinary electro-magnetism; rather a new gauge field is introduced purely as a mathematical device. In order to avoid any possible confusion, we shall therefore use a for fictitious fields and A for electromagnetic fields. Let us imagine that on each point particle a delta-function flux tube is attached. Such a flux tube can be realized by the (fictitious) vector potential ^ )
=|
^
-
(2-33)
Thus ax — ^ j s f ^ r , ay - i ^ ^ q ^ r , or in polar coordinates ar = 0 , a^ = $/2TT. Hence, by minimal prescription, the Hamiltonian for any one of the particles is given by H=^{pl-eai)2.
(2.34)
Now, the field strength fa = didj — djdi vanishes except at the origin where it is a (singular) delta-function so that the corresponding flux is given by
<j)akdxk = $
(2.35)
Ch 2.
35
Fractional Statistics in Two Dimensions
for any path that winds around the origin once in anti-clockwise direction. It may be noted that there is no classical force since the magnetic field vanishes everywhere away from the origin. However, as is well known from the Aharonov-Bohm effect [22] (actually, essentially the same effect was discussed ten years earlier by Ehrenberg and Siday [23]), there are important long range effects associated with a point flux tube. For example, in the path integral prescription, to get the total amplitude we add the contributions of all possible paths with appropriate weights. For our case of the charged particle (of fictitious charge q), in the presence of a gauge field the weight includes a factor exp I / Lint d t ) = exp l i q v - a . d t ) = \J / \ J /
exp l i q a - d x j \ J /
(2.36)
representing the interaction of the particle with the gauge field. In our case q J a.dx = q/2Tr. Thus this integral simply measures the angle swept out by the particle as viewed from the origin. But as is clear from Eq. (2.24), this sort of factor is precisely what is needed to implement the fractional statistics. In this way we see that the dynamical effect of attaching the fictitious charge and delta-function flux tube to particles is indistinguishable from the general kinematic possibility of fractional statistics and further q$/2 (actually q$/2hc if we do not choose H = 1 = c) can be identified with the anyon parameter 8 (an apparent factor of two arises because each particle moves in the potential of the other). Thus with the flux tube model, we see that if a particle has charge q and flux $ then it can be identified with anyon and it will satisfy fractional statistics with anyon parameter 8 — q - ^ / - V ^ " 2 ^ |r._r.2
a/r A
(2 38)
'
where 6ij = arctan [(j/j — yj)/{xi — Xj)\ is the relative angle between the particles i and j . Notice that it is a highly nonlocal gauge potential. Further, the charge in each anyon sees the vector potential due to the flux
36
Fractional Statistics and Quantum Theory
tubes in all the other anyons. Using the standard minimal prescription (see (2.34)), the HamiltonianforN noninteracting anyons is then given by 1
N
#=^E(P'- ea (r,R)=^(R)V>2(r)
(3.9)
Ch 3.
Quantum Mechanics of Anyons
47
where V'i(R-) = e l K R is the free particle center of mass wave function while the relative wave function ip2(r) is a s given by Eqs. (3.5) and (3.7) i.e. V>2(r) = e"*J|,_ a |(fcr).
(3.10)
Here the energy corresponding to the relative motion is related to k by E = h2k2/m, and / = 0, ±2, ±4.... If we now consider free anyons in a circular box (of radius R) with hard walls, then the relative wave function must vanish at r = R i.e. J\i_a\(kR) = 0, so that the allowed bound state energy eigenvalues are TP
f^\
K,i(a
=
H2Jj2
\l-a\,n
frz— mR2
,„ 1 r ^
(o.ll)
where j v > n represents the values at which the Bessel function Jv is zero i.e. Jv{jv,n) = 0. The zeros of the Bessel function have been well tabulated [2] from where we find that for anyons (0 < a < 1) there is no degeneracy in the spectrum, while for bosons and fermions the spectrum is two-fold degenerate. Further, there are no crossing of energy levels. I might add here that historically, the second virial coefficient of the noninteracting anyon gas was first calculated by using these eigenvalues [1].
3.3
Two Anyons in an Oscillator Potential
We now discuss the interesting example of two anyons in an oscillator potential [3]. As in the case of two free anyons, the problem is separable in the center of mass and relative coordinates i.e. \mu2{r\ + r|) = muj2(R2 + ^r 2 ).
(3.12)
Unlike the free particle case, the spectrum corresponding to both the center of mass and the relative motion is now discrete and we shall see that the energy levels of a two-anyon system cannot be obtained as a sum of single anyon energy levels (except of course when a = 0 or 1). Notice that the center of mass motion is again independent of the statistics of the particles and it just corresponds to the quantum mechanical problem of a single particle of mass 2m in a two dimensional harmonic oscillator. As for the relative part, the eigenfunctions in the <j> variable are
48
Fractional Statistics and Quantum, Theory
again given by Eqs. (3.3) to (3.5) while the radial equation takes the form [dr
r2
r dr
Ah2
h2 J
v
v(3.13) ;
'
On using the ansatz R(r) = r l ' - Q l e - T ^ x ( O ,
(3.14)
it is easily shown that xir) satisfies the differential equation „. . f(2|Z — o; +1)
X»+ [—
;
mw 1 , , . \mE ,,,
.
.mui~] , ,
- -^-rj X'(r)+ ^ - (\l -a\ + 1 ) — j x (r) = 0. (3.15)
On further substituting ^ ^
(3 16)
= ^
-
we find that x(p) satisfies
PX"(P) + [\l ~ a\ + 1 - p}X'(p) + \ [ ^ - (\l ~ a\ + 1)] X(P) = 0 (3.17) which is the differential equation satisfied by the associated Laguerre polynomials. Using any standard book on Quantum Mechanics [4], we can immediately write down the energy eigenvalues for the problem Enrti(a)
= (2nr + \l-a\
+ l)hu
(3.18)
where as before / = 0, ±2, ±4,... while the corresponding eigenfunctions are W,*(^,«) = e^r-K-le-^i^l (^r 2 )
(3.19)
with L\ being the associated Laguerre polynomial. Several comments are in order at this stage. (1) Not surprisingly, for a = 0,1 we get the usual energy levels for the bosons and the fermions respectively. (2) The energy eigenvalue spectrum and the corresponding degeneracy can be compactly written in the form En(a) = (2n + 1 + a)hw , deg. = n + 1 for I < 0 = (2n + 1 - a)hu , deg. =n
for I > 0
(3.20)
Ch 3.
Quantum Mechanics of Anyons
49
where n — nr + | = 0,1,2,... since I is an even integer. We shall be using these energy levels and the corresponding degeneracy in the next chapter to compute the second virial coefficient of an anyon gas. (3) What is the origin of the large degeneracy that one finds for the spectrum of two anyons in an oscillator potential? It turns out that there is a SU{2) symmetry in the problem which is responsible for this degeneracy [5]. It has also been shown that in this problem one has two independent supersymmetries [6].
Fig. 3.1 Spectrum of two anyons in an oscillator potential.
(4) A plot of the energy levels (in units of tkJ) as a function of a is given in Fig. 3.1 [3]. The degeneracy of the levels are also mentioned. Prom the figure it is clear that the energy eigenvalues of the two anyon system are a monotonic function of a, and change continuously as one goes from the bosonic (a = 0) end to the fermionic (a = 1) end. In particular, notice that the ground state energy of two-anyons monotonically increases as one goes from a = 0 to a = 1; it being least for bosons and maximum for fermions. This is understandable, since the repulsion between two anyons monotonically increases as a goes from 0 to 1, being maximum for fermions. We also note that the energy levels do
50
Fractional Statistics and Quantum Theory
not cross each other as a function of a. We shall however see that for some other two-anyon problems and for almost all known multi-anyon problems (N > 2), there are always level crossings. (5) The energy spectrum is not equi-spaced except when a = 0, 1/2 or 1, and even at a = 1/2 (semions), the spacing between the energy levels is half of that for the bosons and the fermions. There is, however, one curious fact about the spectrum which is worth noting. In particular, notice that for 0 < a < 1, the alternate energy levels are equi-spaced with spacing 2fkv while for a = 0,1, the whole spectrum is equi-spaced with spacing 2hu>. (6) The energy eigenfunctions corresponding to the center of mass motion are easily written down. They are simply obtained from those of the relative motion by putting a = 0, replacing l,<j> by L, $ respectively, and also changing the mass m by Am. Thus the energy eigenvalues of the center of mass Hamiltonian are given by Ecm = (ni + \L\ + l)hu .
(3.21)
On combining the energy eigenvalues corresponding to the center of mass and the relative motion it is easily seen that except for a = 0 and 1, the two-anyon energy levels bear no simple additive or combinatoric relation to the levels of the one anyon system. The same thing is also true of the eigenfunctions. For example, the ground state wave function for the two-anyon system is given by 1/>(R, $ , r, <j>) OC r»ei»e-^(R2+r2/4)
=
r,e-m,(fi
2
+r
2
/4) _
( 3 .2 2 )
Written in terms of the original two particle coordinates, the ground state wave function of the two-anyon system is thus
^(n,r 2 ) oc (n -T2)°e-n*Lrt+'$K
(3.23)
Except for a = 0,1, this two-anyon wave function does not bear any simple relationship to the single anyon wave function. Thus even the two-anyon wave function cannot be written as the product of the single particle wave functions. A similar thing can also be shown in the other cases, including that of two free anyons discussed above. This in essence is the root cause of the difficulty, and that is why, till today, the problem of the statistical mechanics of an ideal anyon gas has remained an unsolved problem.
Ch 3.
51
Quantum Mechanics of Anyons
(7) Prom Eq. (3.19) we also notice that as in the free anyon case, the two anyon-relative wave function vanishes as r —> 0 (a ^ 0), thereby again justifying the hard core assumption made in the last chapter. (8) The other two-anyon problem which can be analytically solved for all partial waves is when V(r) = —e2/r. It is easily shown [5,7] by using Eq. (3.6) that the energy eigenvalues are (n — nr +1 = 0,1, ±2, ±3,...)
^"W(|,,^|
+
i)»-
(3 M)
'
The corresponding eigenfunctions are also easily written down. Unlike the oscillator case, the energy here is a nonlinear function of a. However, E~1/2 is linear in a. Besides, as in the oscillator case, there is a large degeneracy in the problem. In particular, for positive n, one has \ (^^0 degenerate states if n is even (odd) while for negative n the degeneracy is -^ ( 2 ') depending on if \n\ is even (odd). What is the origin of this large degeneracy? It turns out that there is a 0(3) symmetry in the problem which is responsible for this degeneracy [5]. It has also been shown that there is a supersymmetry in this problem [5,6]. 3.4
Two Anyons in a Uniform Magnetic Field
Let us now consider the interesting problem of two anyons in a uniform magnetic field . This problem is not only of academic interest but is also of physical relevance since the only known place where the anyons do seem to play a role is the fractional quantum Hall effect, where the charged particles experience an external uniform magnetic field. Let us consider two spin-less, charged, identical anyons (charge —e, mass m) moving in the x-y plane under the influence of a static and uniform magnetic field B, which is perpendicular to the x-y plane (B = Boz). As a first step, we neglect the Coulomb repulsion between the anyons. The effect of Coulomb repulsion will be considered later in the chapter. The solution to this problem is very similar to the usual Landau level problem, with some interesting twists arising from the anyonic statistics. Hence we shall first solve the problem of two spinless, charged identical particles under the influence of an orthogonal uniform magnetic field and then come back to the anyon problem and point out the interesting twists. The Hamiltonian for two charged particles in a uniform magnetic field
52
Fractional Statistics and Quantum Theory
is given by [4]
^ = ^ [ ( P i - e A 1 ) 2 + (p 2 -eA 2 ) 2 ]
(3.25)
AJ = J B X r*
(3.26)
where
is the symmetric gauge vector potential of the particles. Without loss of generality, we assume BQ > 0 since the spectrum can be shown to be independent of the sign of Bo. In terms of the center of mass and the relative coordinates R and r denned in the last chapter, the Hamiltonian (3.25) is easily separated i.e. H — Hcm + Hrei, where #cm = ^ ( P - 2 e A c r o ) 2
fireZ = - ( p - ^ A r e J ) 2 . m
(3.27)
(3.28)
2
Here A c m = \BX
R; Arel
= ^B x r .
(3.29)
Thus the center of mass (relative) problem is effectively that of a single particle of mass and charge TTIR = 2m, qR = —2e (mr = m/2, qr = — e/2) i.e. both are the Landau level problems whose solutions are known from the early days of quantum mechanics [4]. For example, consider the Schrodinger equation for the relative motion in cylindrical coordinates i.e. HrenP{r,c}>)=ETJ){r,4>).
(3.30)
^(r,0)=ea*iJ(r)
(3.31)
On making the ansatz
where I = 0, ±2, ±4, ...(±1,±3,...) in the case of bosons (fermions), the radial Schrodinger equation can be shown to be (3.32)
Ch 3.
53
Quantum Mechanics of Anyons
which is very similar to Eq. (3.13). Hence the energy eigenvalues are immediately obtained (see Eq. (3.18)) as En,i = (2n + \l\ - I + l)huc
(3.33)
where u>c = ^ ^ is the cyclotron frequency, n = 0,1,2,..., while I — 0, ±2,... or ± 1 , ± 3 , . . . depending on if the charged particles are bosons or fermions respectively. The corresponding radial eigenfunction is (see Eq. (3.19)) ^.«(r)=r"'exp(-^r»)L"!"(^r»)
(3.34)
where L\ is the associated Laguerre polynomial. The solution of the center of mass problem can similarly be written down. It is worth noting that in both the cases, every state (including the ground state) is infinite-fold degenerate, corresponding to the all possible positive integer values of I. However, for samples of finite area, this degeneracy is finite since in this case the angular momentum is bounded from above. Let us now solve the two-anyon problem in a uniform magnetic field [8]. As seen in the last two sections, the center of mass problem is unaltered and hence the same as Eq. (3.27), while as far as the relative motion is concerned, one simply has to replace the angular momentum I by I—a in the relative radial Eq. (3.32) while the angular eigenfunction is still exp(il). On replacing I by I — a in Eq. (3.32), it follows that in this case the radial Schrodinger equation will be [dr2
r dr
r2
2h
16h2
h2 \
(3.35)
Hence a la Eqs. (3.18) and (3.33), it follows that the energy eigenvalues for the relative motion of two-anyons in a uniform magnetic field are given by Enj = (2n +\l-a\-(l-a)
+ l)huc
(3.36)
where 0 < a < 1. We thus find that the energy eigenvalues fall into two classes that we label / and 77 : Efo)
= (2n + \)hwc, I = 2,4,...
££ 7 (a) = (2n + 2\l\ + 2a + l)fnoc ,1 = 0, - 2 , - 4 , . . . .
(3.37) (3.38)
Notice that every energy level in the class / is infinite-fold degenerate, corresponding to the fact that for every positive even value of l(= 2,4,6...)
54
Fractional Statistics and Quantum Theory
the spectrum is the same. On the other hand, if we define n\ = n + \l\, (I = 0, —2, —4,...) then it easily follows that the degeneracy of the levels in the class / / is ^^i^1^) if n is even (odd). The corresponding energy eigenfunctions are also immediately obtained from Eqs. (3.31) and (3.34) by replacing I by I — a in the later equation 0
(3.39)
(3.40) A few comments are in order at this stage (1) One important difference between the class / and the class / / states is that, whereas the energy of the former is independent of a, the latter's energy is not. Further, whereas the class I states (which includes the ground state with energy tiujc) are infinite-fold degenerate, the class / / states have only finite degeneracy for 0 < a < 1. (2) The ground state energy is hioc which is independent of the anyon parameter a and further for every value of a, this state is infinitefold degenerate. This happens because of the complete cancellation between the statistical frustration (caused by the long range repulsive interaction between the anyons) and the external uniform magnetic field. (3) For each value of a (0 < a < 1), one has alternatively, Landau levels with infinite-fold degeneracy and intermediate states with finite degeneracy. Only in the case of a = 0 (bosons) and a = 1 (fermions), all states are infinite-fold degenerate and the spectrum is equi-spaced with the spacing of 2twjc. (4) Unlike a = 0,1, for 0 < a < 1 the spectrum is not equi-spaced (except for a = 1/2 when the spacing is however hujc). However, again as in the oscillator case, the alternate levels are indeed equi-spaced with the spacing of 2huic while for a = 0,1 the whole spectrum is equispaced with the spacing of 2tkoc. Thus, again for 0 < a < 1, it is not possible to build the two-anyon wave function by simple product or Slater determinant of the single particle basis functions. (5) One problem which has not been satisfactorily answered as yet is about the origin of the infinite-fold degeneracy for the alternate anyon levels.
Ch 3.
55
Quantum Mechanics of Anyons
Two Anyons in Magnetic Field with Harmonic Attraction Let us now consider the problem of two-anyons experiencing a uniform magnetic field as well as harmonic inter-particle attraction [9,8], i.e we add V(r) = \muj2r2 to the relative Schrodinger Eq. (3.35). Proceeding as above, the energy eigenvalues and eigenfunctions for this problem are (n = 0,l,2,...;Z = 0,±2,±4,...) E n ,z(a) = (2n + \l-a\ + \)hut T {I ~ ce)Juoc
*B,z(r,*,a) = W - l exp ( - ^ r
2
) ^ - ! (^r2)
(3.41)
(3.42)
where w2 = ui2, + LO2 with wc being e\Bo\/2m. Note that in this case the spectrum depends on the sign of Bo i.e. — (+) sign in Eq. (3.41) corresponds to BQ > 0 or Bo < 0 respectively. A few comments are in order at this stage. (1) From Eq. (3.41) we observe that the infinite-fold degeneracy of the pure magnetic field problem has been lifted because of the addition of the oscillator potential . However, so long as cjc/ut is a rational number, the spectrum always has some degeneracy for any a (0 < a < 1). (2) The ground state energy depends sensitively on whether Bo > ( 0 the levels starting from the bosonic and the fermionic ground states meet when (3 - a)fojjt =F (2 - a)hujc = (1 + a)fkot ± ahvc
(3.44)
i.e. when a = a* = (1 — ^ £ ) . Thus for Bo > 0, the ground state energy monotonically increases as one goes from a = 0 to a = a* and then it decreases as a increases to one from a*. In other words, unlike the other two-anyon problems, the ground state energy of two-anyons is not maximum at the fermionic end but is maximum at a = a* < 1. In Fig. 3.2 we have plotted the ground state energy (E/Tvjjt) of two anyons as a function of a when Bo > 0 and x = u>c/^t = 0.75. Note,
56
Fractional Statistics and Quantum Theory
Fig. 3.2 Ground state energy of two anyons in an oscillator potential plus a uniform magnetic field Bo (> 0).
however, that for Bo < 0 the ground state energy is E0(a) = hwt + ah(ut - wc)
(3.45)
which monotonically increases as one goes from a = 0 to a = 1. (3) Whereas in the special case of u>t = wc (i.e. u> = 0), there is a complete cancellation between the statistical frustration (caused by the repulsive long range interaction between the anyons) and the external electromagnetic field, there is only a partial cancellation between the two when an external harmonic interaction is added. That is why, whereas the ground state energy in the former case is same no matter what a is, in the latter case the ground state energy is a function of a and is always more in the case of fermions compared to bosons. Note also that whereas the ground state is infinite-fold degenerate in case to = 0, it is non-degenerate in case w > 0.
Ch 3.
3.5
57
Quantum Mechanics of Anyons
Magnetic Field with Coulomb Repulsion
In Sec. 3.4 we considered the problem of two like-charged anyons in a uniform magnetic field. However, we neglected the Coulomb repulsion between the two anyons. In this section we consider the full problem taking into account the Coulomb repulsion [10] i.e. we add a potential V(r) = +e2/r to the relative Hamiltonian as given in Eq. (3.35). An explanation is in order at this point. Since our world is three dimensional, while anyons could exist only as quasi-particle excitations in certain two-dimensional systems (like in fractional quantum effect), hence we take the Coulomb potential to be e 2 /r rather than lnr. In the case of a uniform magnetic field plus repulsive Coulomb potential, the relative radial Schrodinger equation has the form
[|i+i» < ^+ ^ [dr2
r dr
r2
2h
a )
_ ^ _ ^ l 16h2
h2r
+ ^ l2 s ( r ) = 0. h \
(3.46) Because of the confining harmonic potential coming from the uniform magnetic field, the spectrum will be purely discrete even in the presence of the repulsive Coulomb potential. On splitting off the asymptotic behavior as r —* oo and r —* 0, we look for solutions of the form R(r) = r^^e-T^xW •
(3.47)
On substituting this ansatz in Eq. (3.46) and after a little algebra we find that x{v) satisfies the differential equation yx"(y) + (a- 2y2)x'(y)
+ (gy - d)x(y) = 0
(3.48)
where lmujc ,, , , me2 I 2h y= \h^rr> a = 2l-a + 1 , d=—rJ , V 2h h2 V muic 9 = r— + 2(1 - a) - 2\l - a\ - 2. nu)c
(3.49)
In general, analytic power law solutions do not exist to this equation except when the magnetic field BQ has special values. To get these solutions, let us write x(y) m a power series form oo
x(y) = J2hykfc=0
(3-5°)
58
Fractional Statistics and Quantum Theory
We then find that Eq. (3.48) reduces to a three term difference equation (k + 2){k + 1 + a)bk+2 - dbk+1 + (g- 2k)bk = 0 .
(3.51)
Clearly, if xiv) n a s to terminate at some finite value of k, say k = n, so that the highest power of y in x(u) is yn(n > 1), then the following two conditions must be satisfied g = 2n,
bn+1 = Q.
(3.52)
The condition g = 2n gives the energy eigenvalues as En,i = [n+\l-a\
+ l-(l-a)]hioc.
(3.53)
On the other hand, the condition bn+i = 0 leads to a relation between LOC, a and I i.e. for a given magnetic field one is able to obtain eigenvalues for at most one value of 1. For example, for n = 1 one finds that an exact polynomial solution to Eq. (3.48) is X(y)
= 1 + -j
(3.54)
provided g = 2 and d2 = 2a i.e. if A
-^r = {2\l-a\ + l)hwc nr and the corresponding energy eigenvalue is E=[2+\l-a\-(l-a)}huc.
(3.55)
(3.56)
The above \{v) IS nodeless and so this is an exact ground state solution but only for a specific value of I for which Eq. (3.55) is satisfied. Similarly, exact solutions (again onlyfora specific value of I) can also be derived for n = 2,3,.... Thus even though the entire spectrum is discrete, for a given magnetic field at most a few levels can be obtained analytically while the rest of the spectrum has to be obtained numerically.
3.6
Scattering of Two Anyons in Coulomb Field
So far we have discussed several problems in most of which the two anyon spectrum is purely discrete. It may be worthwhile to discuss a few examples where the spectrum is continuous and to see how the scattering cross-section changes as a function of the anyon parameter a. To that effect we now discuss the problem of two like-charged anyons experiencing the Coulomb
Gh 3.
Quantum Mechanics of Anyons
59
repulsion [7] i.e. V(r) = +e 2 /r. As explained above, we have in mind a three dimensional system in which the motion of the charged particles is confined to a plane so that anyons exist as quasi-particle excitations in certain two-dimensional systems. In this example, we also hope to study the effect of the interference between the planar Coulomb and the statistical interaction on the differential cross-section. It is worth pointing out here that scattering in two dimensions is peculiar since at low energy there is no effective-range expansion due to the presence of logarithmic singularity. As shown in the previous sections, the two-anyon center of mass motion is again that of a free particle while the relative radial Schrodinger equation in the presence of the repulsive Coulomb potential is given by
f»L + I » _ ( L ^ ! _ ^ + ^ U ) = o 2
[or
r or
2
2
r
2
h r
(3.57)
h \
where the total relative wave function is as before ip(r,<j>) — ell(^R(r), with I = 0, ±2, ±4,... in the bosonic basis. Since the motion in this case is purely continuous hence we make the ansatz R(r) = r"- a 'e' f c r X (y)
(3.58)
where E = h2k2/m, and y = —2ikr. On substituting this ansatz in Eq. (3.57) we find that x{y) satisfies the confluent hypergeometric equation [2]
yX"(y) + [(2\l-a\ + l)-y]X'(y)-^\l-a\
+ ^j + l^y(y) = 0 . (3.59)
Hence the regular solution of the Coulomb problem can be expressed as a linear combination of the partial waves with appropriate coefficients C; and is given by
^ ( r ) = J2 Cleu*r»-a\eik\F1{\l-a\ + ±+l^2\l-a\ l= — oo
^
+ l-,-2ikr\. '
(3.60) Here i-Fi(a, b; z) is the confluent hypergeometric function and (3 = me2/h2 is the inverse Bohr radius. The procedure for extracting the scattering amplitude from here is lengthy and tedious but straightforward and similar to the three dimensional Coulomb problem discussed in books on quantum mechanics [4]. We therefore omit the details but simply mention that the coefficients C\ are chosen in such a way that when the planar Coulomb distorted asymptotic plane wave is subtracted from the solution (3.60),
60
Fractional Statistics and Quantum Theory
only an outgoing phase shifted spherical wave is left, modulated by the scattering amplitude fk{4>) with <j> being the scattering angle. We obtain fk(4>)= £
_1
e «*( e X p[2*7 7 | J _ a |+t7r(|i|-|/-a|)]-l)
(3.61)
where
e^-. =
££^±i±|).
(3.62)
As expected, this expression for the phase shift reduces to the correct well known limits for (i) anyonic scattering [11] when j3 = 0, and (ii) two dimensional Coulomb scattering (e 2 /r) when a = 0. Note that in order to match our expressions with those of [11], we have to replace (f> by TT — (f> in Eq. (3.61). We now calculate the differential cross-section by summing over I in Eq. (3.61). Actually, being in the bosonic basis, we should only sum over all even integer values of I. But instead we sum over all integral values of I but take care of the bosonic basis by taking the sum of fk () and fk () = > aA —— ^ \irkrj
e
^
d
cos kr cos0 = Sw+a-^n,
l):
M) = fiA)(4>) + f{kHD)(4>),
(3.74)
where .
r
f{kA\ + i sin 6 cos $)^
(3.100)
.
(3.101)
The ranges of (9, ,1>}->{0 + n, + n,4}, {0,(p,ip} -> {6» + 7r,0,^ + 7r},
{6>,^}-l-0+|,0+|,V>+|}.
(3.102)
Since the relative wave functions are really functions of ui and u2, we must demand that the wave function be invariant under these three transformations which leave u\ and 112 invariant. This essentially says that the whole wave function is denned by its values in the fundamental region of the angular coordinates. Now u\,u2 have quite complicated properties under the cyclic permutation (c.p.) of the three particles :{xi,x 2 ,x 3 } -^ {x 2 ,x 3 ,Xi}. For u\ and u2 this transformation becomes
{"1,1*2} ^
j ^ « 2 - l-uu -\u2 - ^Ul J .
(3.103)
The remaining generator of the permutation group xi x 2 on u\ and u2 is {«i,«2}-^{-ui,«2}-
(3.104)
Using the transformations (3.100) and (3.101), it is easily shown that the real advantage of the angular coordinates is that both the generators of the permutation group act simply on them i.e.
{#,,} ^ { M + ^ + TT}
(3.105)
{6,ci>,iP}^{-8,-4>,iP + n}.
(3.106)
Using P123 and P12, it is easy to construct projection operators which form fully (anti) symmetric wave functions for (fermionic) bosonic three-particle systems P± = ^[1 + ^123 + (Pl23) 2 ]i(l ±Pl2).
(3.107)
Radial Excitations It is worth pointing out that the variable ijj gives the overall orientation of the three anyon system. It is thus clear that its conjugate variable, the total (relative) angular momentum is conserved. Further, as expected, the total scale of the system, p, decouples from the statistical interaction. In
Ch 3.
71
Quantum Mechanics of Anyons
particular it is easily shown that the three-anyon relative Hamiltonian with harmonic interaction can be written in these coordinates as [24,23] H = H0 + -^(aH1+a2H2) mp2
(3.108)
where H
° = T~ ( " 1T2 ~ -pdp #" + T + 2m \ dp2 p2
m
W
)
= HurQad+^TO/0 2 . T
(3.109)
Here - A 2 is the Laplacian on the three-dimensional sphere i.e. A
2_
d2
~ dO2
2 sin 20 d
d2
1
cos 26» 86 + cos 2 20 d2
2*n26_d_d_ ,_J_d^ cos2 26> d di> cos2 20 0,
(3.120) (3.121)
and (k,v,l) is defined by (k, v, I) = C felV ,,e^(sin 29)eiv+eil* .
(3.122)
The numbers n',a,/3 are given in terms of the angular quantum numbers k, v, and I by 2n' = k-max{v,
\l\},a=
) - \ v + l\,(3=
\\v-l\.
(3.123)
The function O^f is defined in terms of the Jacobi polynomials P"P by G^Or) = (1 - x)a'2{l + xf'2P^{x)
(3.124)
74
Fractional Statistics and Quantum Theory
while Ck,u,i is the normalization factor for the angular eigenfunction Ck
_
1
/n'!(n' + a + /3)!(fc + l)
'"' ~ V 2 " + / V + 0) then the angular matrix element is given by [23] (^.olV^il^.o.o) _ 6(-l) V6(2p + l)(p - g)!(p + g)l + 1 , g _! P 2«(p+l)! P-9 (0) ~
(3 137)
-
where P™@(x) is the Jacobi polynomial. On combining Eqs. (3.135) to (3.137) we find that the O(/32) contribution to the three anyon ground state energy from H\ is given by
2
2
oo
oo 3r) is given by (see Eq. (3.99)) E(/3) = [2 + 3(1 - p)]w .
(3.142)
On comparing the three-anyon ground state energies starting from the bosonic and the fermionic ground states as given by Eqs. (3.141) and (3.142), we see that the two curves cross at (3* ~ 0.29 i.e. a* = 0.71. In other words, the three-anyon ground state energy is maximum at a* = 0.71 (and not at a = 1 as one would have naively expected) and at this point the ground state is two-fold degenerate. Thus the perturbative calculation from either the fermionic or the bosonic end will break down beyond a*. Summarizing, the three-anyon ground state energy monotonically increases from 2u> at a = 0 and reaches its maximum at a = a* = 0.71. As a further increases from a* to 1, the ground state energy monotonically decreases to 4w at a = 1. This is shown in Fig. (3.4).
3.10
General Results for N Anyons
The highly nontrivial result for the three-anyon ground state raises several questions.
78
Fractional Statistics and Quantum Theory
(1) Are there similar crossings in the ground state of the multi-anyon systems (N > 3)? (2) Are there similar crossings in the excited states? (3) Is the ./V-anyon effective interaction (N > 3) near the fermionic end always repulsive? (4) Are there avoided crossings in the ./V-anyon spectra? (5) Is the ./V-anyon system integrable?
Fig. 3.4
Spectrum of three-anyons in an oscillator potential.
Over the years, significant progress has been made in answering these questions at least qualitatively. For example, the three-anyon low lying spec-
Ch 3.
Quantum Mechanics of Anyons
79
trum in an oscillator potential has been numerically calculated [28,29]. In Fig. 3.4 [29] we have plotted the low lying spectrum of the relative Hamiltonian (in units of ui) as a function of the anyon parameter 6 (= it a). The angular momentum assignments of the bosonic states emerging from energy 2,4,5,6,7 (in units of to) are given by [0], [0,±2], [±1,±3], [02,±2,±4] and [±1 2 , ±3 2 , ±5] respectively with the upper power referring to the degeneracy of the angular momentum states. From the spectrum, one finds that there are two kinds of energy levels in the spectrum. In particular, apart from the exactly known linear states for which energy changes by ±3w as one goes from the bosonic to the fermionic end, there are several nonlinear states (i.e. where E is a nonlinear function of a) for which energy changes by ±u> as one goes from the bosonic to the fermionic end. Whereas all the linear states are analytically known, not even one exact solution is known as yet involving the nonlinear states. In fact this is the main difficulty in the three and multi-anyon problems. For example, the nonlinear state starting from the three-fermion ground state at 4ui, has an energy of 5UJ at the bosonic end. Note that the perturbative calculation if valid till /3 = 1 (note (3 = 1 — a) would predict the energy at the bosonic end to be 5.29u> rather than 5w, but of course the perturbation theory is expected to breakdown beyond/?* (=0.29). There are plenty of true crossings, but no avoided crossings in the threeanyon spectra. All energy levels are monotonic, i.e. dE/da does not change sign for 0 < a < 1. The numerical calculations also confirm the perturbative result about the cross-over between the ground states (and also the fact that there is only one cross-over). In fact both the calculations agree upto first two decimal places regarding the value of a*(= 0.71) at which the crossing occurs thereby showing the reliability of the perturbative calculation. Several pairs of states cross at the semionic point (a = i ) . Indeed, all the nonlinear states and those exactly known linear states which have slope —3u> have another state in the same family which is related to it by a mirror reflection around a = 1/2 [30] and hence most of the semion spectrum is at least two-fold degenerate. It would be interesting to explore the origin of this extra degeneracy in the semion spectrum. Let us now discuss the four-anyon spectrum which has been numerically calculated [31]. In Fig. 3.5 [31] we have plotted the low lying spectrum (in units of u>) of the relative Hamiltonian in an oscillator potential as a function of the anyon parameter a. The angular momentum assignments of the bosonic states emerging from energy 3,5,6,7,8 (in units of to) are
80
Fractional Statistics and Quantum Theory
Fig. 3.5 Spectrum of four-anyons in an oscillator potential.
[0], [0, ±2], [±1, ±3], [03, ±2 2 , ±42] and [±1 3 , ±3 2 , ±5] respectively. Here the upper power refers to the angular momentum degeneracy of the states. Several conclusions can be drawn from the figure. (i) There are both linear and nonlinear states in the spectrum out of which all the linear states are analytically known (see Eq. (3.99)) while not a single nonlinear state is analytically known as yet. (ii) Whereas the energy changes by ±6w for the linear states as one goes from the bosonic to the fermionic end, it changes by ±4u>, ±2w or 0 for the nonlinear states as one goes from the bosonic to the fermionic end.
Ch 3.
Quantum Mechanics of Anyons
81
(iii) Contrary to the three-fermion ground state (which is nondegenerate), the four-fermion ground state is three-fold degenerate with angular momenta I — 0, ±2. Whereas the I = 0 state has zero slope, the I = ±2 levels have nonzero slopes of equal magnitude but opposite sign (with I = — 2 being pushed down and I — +2 being pushed up). This fact persists for all energies. (iv) As in the three-anyon case, there is one cross-over in the four-anyon ground state. In particular, the linear state beginning from the bosonic ground state (at 3w) and the nonlinear state beginning from the fermionic ground state (at 7u> with I = —2) cross approximately at a* ~ 0.55. Thus at this point the four-anyon ground state is two-fold degenerate (just as the three-anyon state is at the cross-over point of a* = 0.71). (v) One very important difference between the three and the four-anyon spectra is that unlike the three-anyon case, in the four-anyon problem, the ground state energy monotonically decreases as one goes from the fermionic to the bosonic ground state. In particular, whereas the effective threeanyon interaction near the fermionic end is repulsive, the effective fouranyon interaction near the fermionic end is attractive and not repulsive. (vi) Another surprising result is the observation of a Landau-Zener type (or so called avoided) crossing at E = 12w between states with a nonlinear a dependence and the same angular momentum thereby suggesting that the many-anyon system may be non-integrable. We shall discuss the issue of integrability of the 7V-anyon system at the end of the chapter. By now the results about the three and four anyon spectrum (in an oscillator potential), have been generalized to the 7V-anyon case and the following conclusions can be drawn in the iV-anyon case: (1) In the 7V-anyon spectrum, there are both linear and nonlinear states and there are tower of states differing in energy by 2,4,... units from each basic state having zero radial node. All the linear states are analytically known with the spectrum being given by Eq. (3.99). For these states the energy changes by ±N(N2~1>)LJ as one goes from the bosonic to the fermionic end. On the other hand, for the nonlinear states, the energy (in units of w) changes by ^ ^ - 2, ^ [ = 1 1 - 4,..., - M ^ i ) + % While no general proof exists for this fact, it is supported by the WKB analysis [32,25] and is consistent with the numerical calculations for three and four anyons. (2) For any iV(> 3) there is always a cross-over in the ground state. In fact, it has been shown [33] that for large N, there are at least - 2 can be seen as follows. The Nanyon state starting from the bosonic ground state has energy (TV— l)u> at a = 0 with angular momentum I — 0 (see Eq. (3.99)). As a goes to one, this anyonic state has energy E = [(TV — 1) H— 2~ 1^ a n d angular momentum 2 • Clearly this is not the TV-fermion ground state but is an excited state. The TV-fermion ground state is obtained by filling the one particle oscillator levels from bottom to top. One can show that the total angular momentum of the /V-fermion ground state is always less than | ^ | — ^ | and its energy (excluding the center of mass motion) is
3). Here r denotes the number of electrons in the fc'th shell while (k — 1) shells are completely filled so that TV = r + fc(fc~1} (note that k = 2,3,... and hence TV - r - 1,3,6,10,...). Another way to see the crossing is to note from Eq. (3.143) that for large TV, the ground state energy of TV-fermions goes like TV3/2 while the state starting from the bosonic ground state has energy going like TV2 at the fermionic end. (3) The TV-anyon interaction around the fermionic ground state depends crucially on the value of TV. For example, when TV = 3,6, • • • (i.e. when V8TV + 1 is an integer) then at the fermionic point one has a closed shell so that the TV-fermion ground state is unique with zero angular momentum. In this case one can rigorously show that the effective TV-anyon interaction near the fermionic ground state is always repulsive. On the other hand, when TV does not correspond to these magic numbers, then the effective TV-anyon interaction near the fermionic ground state is always attractive. This can be proved in perturbation theory by noting that in these cases the TV-fermion ground state is degenerate and states with I — ±p(p being an integer) have non-zero and opposite slopes so that all states with I = — p will be pushed down because of the anyonic perturbation. However, we are not aware of any non-perturbative proof of this fact (It is of course clearly true for TV = 2 and 4). The fact that the effective TV-anyon interaction near the TV-fermion ground
Ch 3.
Quantum Mechanics of Anyons
83
state (in an oscillator potential) depends so sensitively on the value of TV (i.e. if shells are closed or not), raises doubts about the validity of the approximate schemes like mean field theory because they do not take into account this important fact. (4) There are several true crossings and a few avoided crossings in the excited state spectrum of the TV-anyon system. 3.11
TV Anyons Experiencing a TV-body Potential
As we have seen, so far only a class of exact solutions have been obtained in case TV-anyons (TV > 3) experience a harmonic oscillator potential. Further, all the known exact solutions are such that the energy eigenvalue spectrum is linear in the anyon parameter a. Besides, there is a cross-over in the ground state for any number of anyons (TV > 3). It is clearly of interest to enquire whether one can also obtain exact solutions in case N-anyons are experiencing some other potential and whether in those cases also the energy vary linearly with a. Further, is there a crossover in the ground state and if yes then at what value of a does it occur? The purpose of this section is to present one such example. In particular, we obtain a class of exact solutions in case TV-anyons are interacting via the JV-body potential [34] V(xi,x2)...,XAr) =
,
6
==• y jV Y^i<ji i ~ x j ) K
(3.144)
Note that even though this TV-body potential may appear to be very complicated, we shall see that, in terms of the hyper-spherical coordinates, it is just the Coulomb-like potential — e2/p and as is well known, the only two radial potentials for which the Schrodinger equation can be solved for all partial waves are the oscillator and the Coulomb potentials. In fact, it has recently been shown [35] that the -/V-body problem in one dimension with the inverse square interaction is analytically solvable when the TV particles are also interacting via the TV-body potential of the form as given by Eq. (3.144). The interesting point is that unlike the oscillator case, the exactly known energy spectrum here is not linear in a. However, (E)~1/2 is linear in a. Further, a la the oscillator case, these exact solutions include the ground state of TV-bosons but not the ground state of TV-fermions (TV > 3). We therefore perturbatively calculate the ground state energy of three-anyons near the fermionic end and show that for this potential
84
Fractional Statistics and Quantum Theory
also there is a cross-over between the ground states. We shall show that a similar cross-over must also occur in the case of N anyons (N > 4). The 7V-anyon relative problem is best discussed in terms of the hyperspherical coordinates in 2N — 2 dimensions i.e. in terms of the radial coordinate p and 27V — 3 angles and even in that case the decomposition as in Eqs. (3.108) and (3.109) is still valid except that now —A2 is the Laplacian on the (2N — 3) dimensional sphere and instead of | -§- one now has (2iV~3) A. in Ho. On using the fact that (i) only the angular part of the Hamiltonian is affected due to the anyons, (ii) the angular part is independent of the radial potential V(p) between the anyons and (iii) the radial equation for TV-bosons, TV-fermions and /V-anyons is the same but for the coefficient of the 1/p2 term, one can immediately write down a class of exact solutions for TV-anyons experiencing the TV-body potential (3.144). In particular, let us first note that the relative Hamiltonian for N particles can be formally written as H = Hrad +
mp2
(3.145)
where
V
(3.146) *<J
As remarked before, Hana is the same as in the oscillator case and only jjrad j s different. Now suppose that we have solved the angular eigenvalue equation Hangy =
^
+ 2]V
- 4)*
(3.147)
where S is obviously the same as in the oscillator case which in general is a complicated function of a. As a result, the radial Schrodinger equation in this case is (3.148)
This is a standard radial equation for the Coulomb problem whose solution is immediately written down [4]. In particular the energy eigenvalues are given by (3.149)
Ch 3.
Quantum Mechanics of Anyons
85
where E = me4e/2. It is worth repeating that as such this solution is not of much use unless one can determine 5 as a function of a. On using the fact that S is the same as for the exact solutions in the oscillator case, it is straightforward to figure out 5 and write down the exact solutions. In particular, on comparing with the oscillator exact solutions as given by Eq. (3.99) it is easily seen that 6 = I and hence the energy eigenvalues for the exact solutions are given by £nM) =
-in+\l-m^a\+N-=W
(3J50)
where I is the eigenvalue of the relative angular momentum operator. Several comments are in order at this stage. (1) Unlike the oscillator case, the exactly known spectrum here is not linear in a. However, (—e)~xl2 is indeed linear in a for all these solutions. (2) It is worth pointing out that for N = 2, the expression (3.150) gives the complete spectrum. For N > 3 however, it does not give the complete spectrum. For example, a la oscillator case, the ./V-fermion ground state is missing from these exact solutions and as in the oscillator case, one can show that a level-crossing must occur in the true ground state of the A^-anyon system (N > 2). (3) What is the nature of the missing states in the iV-anyon spectra? It turns out that whereas for the exact solutions (—E)~~xl2 is linear in a, for all the missing solutions (—E)~xl2 will have nonlinear dependence on a. Further, while all those states for which (—E)~1/2 varies linearly with a are known analytically, not even one of the states, for which (—E)~xl2 varies nonlinearly with a is known analytically. (4) Even though the nonlinear states are still not known, it follows from Eq. (3.149) that both the linear and the nonlinear states have a family structure i.e. once a basic solution with zero radial node is there, then one also has solutions with 1,2,... nodes and for them (—e)"1/2 is more by 1,2,... units respectively from that of the corresponding node-less solution. (5) Regarding the missing nonlinear states, it is clear that once we can obtain some result for them in one potential, we can always translate the results to the other case. In particular the entire discussion given about the N-anyon spectrum in the oscillator case applies here. For example, whereas for all the analytically known states (—e)"1/2 changes by ±N(N—l)/2, for the missing states it will change
86
fractional i
N(N-l)
o
Statistics and Quantum Theory N(N-l)
.
N(N-l)
,
n
-ii
2) for all partial waves. Of course in both cases one can always add the potential V(p) = g2/p2 and still the problem will be solvable analytically but now the degeneracy will be much less. All other potentials are at best quasi-exactly solvable and hence for them, eigenstates could be analytically obtained, if at all, for only specific values of the angular momentum I.
3.12
N Anyons in a Uniform Magnetic Field
The problem of ./V-anyons in a uniform magnetic field is more than just an academic problem. In the only known physical application of anyons (i.e. in the fractional quantum Hall effect) and in most of the other proposed realizations of the fractional statistics, the anyons feel an effective external magnetic field. Let us therefore discuss this problem [36,37,38]. Later on we will also consider the case of ./V-anyons experiencing a uniform magnetic field as well as an external harmonic interaction. The Hamiltonian for ./V-anyons experiencing a uniform magnetic field B is given by
H^^-jrL-a^^^-eAir,)}2
(3.153)
where A(TJ) = ^ z x r j is the external vector potential in the symmetric gauge. Without any loss of generality we choose B > 0 since the spectrum in this case is independent of the sign of B. After some algebra one can
88
Fractional Statistics and Quantum Theory
rewrite this H as 1
F
v^fr
v^zxiijl2
a
2
2
221
T
= ^ E { [ P « - E ^ ^ j +m u,yij-ucJ+
N(N-l)
2
Va
(3.154)
where J is the total angular momentum as given by Eq. (3.83) and u>c = e\B\/2m is the cyclotron frequency. We see that the Hamiltonian in the uniform magnetic field case is the same as in the oscillator case except for a constant term proportional to N(N — 1) and a term proportional to the total angular momentum J. Hence we can immediately write down the exact solutions for this case since the exact wave function is also an eigenfunction of the angular momentum operator J. Clearly the eigenvalues will be shifted from the oscillator eigenvalues by a constant as well as a term proportional to the eigenvalue of the total angular momentum. In particular, the exact energy eigenvalues (including the center of mass) for the problem of iV-anyons in uniform magnetic field are given by (n = 0,1,2,...) Enj(a) = \^n+N+\j-
N{N
~
l)
a[] u>c- (j-
^
^
1}
a ) LOC . (3.155)
The extra term (i.e. the last term on the right hand side) compared to the oscillator case (see Eq. (3.98)) changes the spectrum non-trivially. In particular, observe that the ground state energy is given by E0(a)=Nuc,
(3.156)
which is in fact independent of a and is infinite-fold degenerate a la the conventional Landau levels. Note that no matter what j is, so long as j > N(N — l)/2 the ground state energy is always Nuc. Further, because of the family structure, one has radially excited levels with n nodes having energy En(a) = (2n + N)wc, all of which are also infinite-fold degenerate. On the other hand, for j < 0 the energy eigenvalues take the form EnJ{a)
= [2n + N + 2\j\ + N(N - l)a] uc.
(3.157)
All these levels have only finite degeneracy and for all of them, the energy for a given n,N, and \j\, monotonically increases as one goes from the bosonic to the fermionic end. It may be noted that as in the oscillator case, one has only been able to obtain analytically, a class of exact energy eigenstates for all of which the energy eigenvalues vary linearly with a.
Ch 3.
Quantum Mechanics of Anyons
89
However among the huge class of nonlinear states not even one is known analytically as yet. Finally let us consider the case when the anyons experience an external harmonic interaction in addition to the uniform magnetic field. Following the oscillator problem as well as the above discussion, it is easy to see that H for the combined system is given by
(3.158) where
J^ = Jl + <J.
(3.159)
Hence the energy eigenvalues for the exact solutions (including center of mass) are now given by
EnJ(a)= L + A r + l i - ^ ' ^ a l l ^ T ^ - ^ " ^ ^ ^ (3.160) where the upper (lower) sign corresponds to B > 0(B < 0). We shall see below that the results now crucially depend on the sign of B. It is interesting to note that the infinite-fold degeneracy of the pure magnetic field case has been lifted by the addition of the external harmonic potential. Further, the ground state energy is now a function of a and its value crucially depends on whether B > ( = 0 (i.e. u>t = u>c), the infinite-fold degenerate bosonic ground states with energy 2uc are \n = 0, k = 2p, v = 0,1 = ±k) and |0, k, v ^ 0, ±fc) while the infinite-fold fermionic ground states with energy hojc are |0, k, v ^ 0, ±fc). Similarly all the excited states can be written down, all of which are infinite-fold degenerate. As the harmonic interaction is introduced, this infinite-fold degeneracy is lifted and each level has only finite degeneracy. When u / 0 , then clearly u>t > coc and Eq. (3.162) implies that the unique bosonic ground state is |0, 0,0,0) with energy 2u>t- On the other hand, the fermionic ground state depends crucially on whether B > 0 or B < 0 and the ratio u>c/u)t. In particular, the fermionic ground state is |0,3, 3,3)(|0, 3,3, -3)) with energy 5wt - 3wc if B > 0 (B < 0) and u>t < 3wc, while it is |0, 2,0,0) with energy 4u>t if u>t > 3wc (see the discussion in Sec. 3.9). In particular, at tut — 3uic the doubly degenerate fermionic ground states are |0, 2,0,0) and |0,3,3,3) (or |0, 3,3,-3)) with energy 4wt(= 12wc). Let us now work out the three-anyon ground state in the various cases by making use of the exact solutions with energy as given by Eq. (3.161) and the perturbative results around the fermionic end as discussed in Sec. 3.9. First of all, notice that the three-boson ground state is |0,0,0,0) with energy 2u>t and it interpolates to the state |0,3,3, —3) in the fermion spectrum at a = 1 with energy 5ut ± 3toc (if B > or < 0). We thus see that the results about the anyonic ground state will crucially depend on the sign of B. Let us therefore discuss the two cases separately. (i) B < 0 : The energy of the three-anyon exact solution interpolating to the three-boson ground state |0,0,0,0) is given by (see Eq. (3.161)) £n=o,;=o(oO = (2 + 3a) ut - 3a wc.
(3.163)
For a = 1, its energy is 5wt — 3wc which corresponds to the energy of the fermionic state |0,3,3, —3). This state will be the fermionic ground state for values of the magnetic field for which it has lower energy than the |0,2,0,0) state (whose energy is 4u>t). Thus we conclude that the exact solution as given by Eq. (3.163) is the exact three-anyon ground
Ch 3.
Quantum Mechanics of Anyons
91
Fig. 3.6 Ground state energy of three-anyons in oscillator potential plus uniform magnetic field in case B < 0 and (a) 0 < x = uic/^t < 1/3 (b) 1/3 < x < 1.
state for all values of a so long as 5u>t — 3wc < 4u>t i.e. u>t < 3wc. In this case, the ground state energy (3.163) monotonically increases as one goes from the bosonic to the fermionic end. On the other hand, when ujt > 2>u)c then the state |0,2,0,0) is the true three-fermion ground state. Hence, in this case, there must be at least one cross-over in the ground state of threeanyons. The value of a at which the cross-over occurs can be determined perturbatively using the results obtained in Sec. 3.9. These results were actually obtained for the case of an oscillator potential alone, but are trivially extended to the case at hand, since the fermionic ground state |0, 2,0,0} has zero angular momentum. The ground state energy of three-anyons near the fermionic end is given by (see Eqs. (3.127), (3.134) and (3.140) with UJ being replaced by u>t)
Eo(a)= [ 4 + ^ ( l - a ) 2 l n Q ) L t + O(l-a) 3 .
(3.164)
On comparing the two energies as given by Eqs. (3.163) and (3.164) we see that the two curves cross at , ,, - ( 1 - x) + [(1 - xf + 2(1 - 3x) ln(4/3)]V2 1- 1/3).
92
Fractional Statistics and Quantum Theory
(ii) B > 0 : The three-anyon state which interpolates to the three-boson ground state is again the state with n = I = 0 but the energy of that state is now given by En=0,i=o = (2 + 3a)cot + 3aojc .
(3.166)
At a = l,this state interpolates to the state |0, 3, 3, —3} and has energy 5u>t + 3wc. This energy is always greater than the energy of the two possible fermionic ground states |0, 2,0, 0} and |0,3,3,3) which have energy 4cot and but — 3LUC respectively. Hence, in this case, there is at least one cross-over in the ground state. The three-fermion ground state will be |0, 2,0, 2) or |0,3,3,3) depending on whether ut > 3wc or ut < 3wc respectively and the interpolation is different in the two cases. For wt > 3wc (i.e. when |0,2,0,2) is the three fermion ground state), the cross-over value of a is obtained by equating the two energies as given by Eqs. (3.164) and (3.166). We find that the two curves cross at (x ~ uic/u>t)
1 - a2(x) = -(l+*HV(l+f
' + 2(1 + 3*)m(4/3) _
(3.167)
o In 4y o
Prom this expression it is easy to see that the cross-over point c*2(x) increases from 0.71 to 0.8 as x increases from 0 to 1/3, the point beyond which |0,2,0,2) is no longer the ground state. What is the cross-over point in case W{ < 3wc i.e. when |0,3,3,3) is the true fermionic ground state? This can be obtained by making use of another exact three-anyon eigenstate which interpolates from an excited bosonic state |0,6, v, 6) with energy 8u>t — 6wc to the fermionic state |0,3,3,3) with energy 5wt — 3wc. In particular, the energy of the corresponding anyonic state is given by E(a) = (2 + |6 - 3a\)wt - (6 - 3a)wc .
(3.168)
The two exact solutions as given by Eqs. (3.163) and (3.168) cross at at3 = l - x .
(3.169)
We observe that for 0.33 < x < 0.48, a^ is greater than a2 as given by Eq. (3.167). This implies that whereas there is one cross-over (at a = a^) when 0.48 < x < 1, there are two cross-overs in the ground state in case 0.33 < x < 0.48. The second cross-over point is easily obtained by comparing the
Ch 3.
Quantum Mechanics of Anyons
93
two energies as given by Eqs. (3.164) and (3.168). We find
1
---
(1 - x) - J(l-l) 2 -2(3i-l)to(|)
- ^ w m
from which we see that the cross over point 0:4(0;) decreases from 1 to 0.55 as x increases from 0.33 to 0.48. To conclude, we find that for three-anyons in an oscillator potential and a uniform magnetic field with B > 0, there are at least two cross-overs in the ground state in case 1/3 < x < 0.48 while there is at least one cross-over otherwise. In Fig. 3.7 [39] we have plotted the three-anyon ground state energy as a function of a in case B > 0 and (a) x = uic/ujt = 0.25 (< 1/3), (b) x = 0.40 (1/3 < x < 0.48), and (c) x = 0.60 (> 0.48). What about the case of JV anyons (N > 3) experiencing both the oscillator potential and a uniform magnetic field? Unfortunately no general conclusions can be drawn because even though a class of exact energy eigenstate are again known, the ground state energy of ./V-fermions cannot be easily written down in this case. However in the special case when N = 3,6,10,15,..., there is a fermionic state containing only closed shells i.e. having I = 0. This state generalizes the |0, 2,0, 0) three-anyon state. Clearly, this I = 0 state has the same energy in an oscillator potential plus magnetic field as it would have with a purely harmonic oscillator of strength u)t. Its energy is given by
EfN= |"^vT+8JvU,
(3.171)
for all those values of N > 3 for which y/1 + 8N is a positive integer (i.e. N = 3,6,10,15,...). Following the arguments in the three-anyon case, it is then clear that when B > 0, the n = I = 0 exact 7V"-anyon energy eigenstate interpolates from the bosonic ground state with energy (N — l)uit to a fermionic state with energy greater than that of the 1 = 0 state as given by Eq. (3.171). Hence, there is at least one cross-over in the ground state when B > 0. Similarly, for B < 0 the same situation occurs when
(3.172) i.e. there is at least one cross-over in the ground state so long as uc
[3(Ar + l)-2y / rT8]V]
(3.173)
94
Fractional Statistics and Quantum Theory
Fig. 3.7 Ground state energy of three-anyons in an oscillator potential plus uniform magnetic field in case B > 0 and (a) 0 < x < 1/3 (b) 1/3 < x < 0.48 (c) 0.48 < x < 1.
Summarizing, we see that the multi-anyon (N > 3) problems are highly complicated due to the nontrivial braiding effects which give rise to the long ranged three-body potential.
Ch 3.
3.13
Quantum Mechanics of Anyons
95
Pseudo-Integrability of N Anyon System
In this chapter we have studied the TV-anyon problem in several different cases like in an oscillator potential, in an TV-body potential as well as in a uniform magnetic field and in all the cases we found a class of exact energy levels for all of which the energy (or (—E1)"1/2 in the iV-body potential case) varies linearly with a. Besides, numerically one has obtained several energy levels for which energy varies nonlinearly with a. The nonlinear spectrum displays many level crossings, some of which are of the LandauZener type (i.e. avoided crossings) [31]. This has led to the conjecture that the multi-anyon system (N > 3) may be non-integrable or even chaotic. On the other hand, if one looks at the classical Lagrangian for ./V-anyons in an oscillator potential, then one finds that the statistical interaction due to anyons is a total time derivative (see for example Eq. (2.32)). Hence the Euler-Lagrange equations of motion are the same as those of an Noscillator system which is known to be integrable. The puzzle then is, why do the numerical results suggest a non-integrable system? Secondly, what is the reason for the existence of a class of exact solutions in all these cases which is somewhat uncommon for a generic many-body problem? The point is that even though the system is naively integrable (the anyonic interaction being a total derivative), as shown in the last chapter, the classical relative configuration space on which the corresponding Lagrangian is defined is not simply connected but multiply connected and hence is topologically nontrivial [40]. In particular, the relative configuration space of N-particles is R^ — A, where A is as defined by Eq. (2.17) and denotes the generalized diagonal with set of all diagonal points removed. As a result, all the orbits which pass through these diagonal points are therefore modified. Hence, even though the number of constants of motion is the same as the number of degrees of freedom so that one has a potentially integrable system, it is not generically integrable via the action-angle variables. For example, even though one has 2N constants of motion in involution, there exist invariant surfaces which do not have the topology of a 2./V-dimensional torus. This bears a resemblance to the billiard system of Richens and Berry [42] although the reasons for the failure of integrability via the action-angle variables appear to be different. Following the arguments of [42], one can show that the system is only pseudo-integrable and this could be the possible origin of the level repulsion seen in the four-anyon spectrum. Further support for this picture comes from the study of the so called
96
Fractional Statistics and Quantum Theory
nearest-neighbour spacing distribution in the energy spectrum [43]. It has been shown that the spectrum exhibits features of both regular and irregular spectra as a function of the statistical parameter a. As far as we are aware off, this is probably the first example of a many body pseudo-integrable system. As far as the reason for the existence of a class of exact solutions is concerned (say in an oscillator potential), we have seen that in all the cases, the relative Lagrangian has a partial separability. In particular, we saw that (i) only the angular part of the Hamiltonian is affected due to the anyons, (ii) the angular part is independent of the radial potential V(p) between the anyons and (iii) the radial equation for iV-bosons, iV-fermions and ./V-anyons is the same but for the coefficient of the 1/p2 term. One can show that in this case the Hamiltonian has two collective degrees of freedom and the remaining are the relative degrees of freedom. Further, One can show [41] that the subset of exactly solvable solutions arise from the quantization of the collective coordinates (after the trivial center of mass is removed). It may be noted that these solutions do not carry any information about the internal dynamics of the system, which may be frozen as far as these solutions are concerned. Thus these solutions incorporate somewhat trivial aspects of anyon dynamics. The real anyon dynamics therefore resides in the solutions for which the energy eigenvalues depend nonlinearly on a. As we shall see in the next chapter, these trivial (linear) states conspire to cancel the divergent parts in the equation of state for an ideal anyon gas while the nonlinear states give the finite part which then defines the equation of state. Further, the spectrum shows evidence of many level crossings and level repulsions thereby indicating that the system is not integrable via action-angle variables but is only a pseudo-integrable system.
3.14
Quantum Computing and Anyons
In this section we shall briefly discuss about the possible role of anyons in quantum computation, a topic which is different from the rest of the material covered in this chapter. The idea of computational device based on quantum mechanics emerged when scientists were wondering about the question of fundamental limits of computations. It was Feynman [44] who produced the first abstract model in 1982 that showed how a quantum system could be used to perform
Ch 3.
Quantum Mechanics of Anyons
97
computations. He also explained how such a machine would be able to act as a simulator for quantum physics. Soon afterwords, Deutsch [45] wrote an important paper where he showed that any physical process, in principle, could be modeled perfectly by a quantum computer so that a quantum computer would have capabilities far exceeding those of any traditional classical computer. The next major development was a paper by Shor [46] in which he proposed a method for using quantum computer to solve the important problem of factorization in number theory which has potentially significant application in the area of encryption. Subsequently, remarkable algorithms have been found in three other areas: searching a data base [47], simulating physical systems [48] and approximate but rapid evaluation of invariants of three dimensional manifolds [49]. Let us recall that in a classical computer the basic task is to interpret and manipulate an encoding of binary digits (called bits) into a useful computational result. Thus bit is the building block of information, classically represented as 0 or 1 in conventional classical digital computers. On the other hand, in a quantum computer, the building block of information is called a quantum bit or qubit. It should be noted that qubit is not binary in nature. This is because of the profound difference between the laws of classical and quantum physics. In particular, a qubit can not only exist in logical state 0 or 1 as in a classical bit, but also in states corresponding to a superposition of these classical states with a numerical coefficient representing the probability for each state. Thus by performing a single operation on the qubit, one would have performed the operation on two different values. Using this parallelism and right kind of algorithm, it is possible to solve certain problems in a fraction of the time taken by a classical computer. For example, a system of 500 qubits, which is impossible to simulate classically, represents a quantum superposition of as many as 2500 states! Just as information is manipulated through Boolean logic gates arranged in succession, in the classical computers, a quantum computer manipulates qubits by executing a series of quantum gates, each a unitary transformation acting on a single or pair of qubits. By applying these gates in succession, a quantum computer can perform a complicated unitary transformation on a set of qubits. Even though quantum computing holds tremendous promise, it must be noted that the quantum computing technology is still in its infancy. At present, we cannot clearly envisage what the hardware of that machine will be like. But one thing is certain. Any practical quantum computer
98
Fractional Statistics and Quantum Theory
must incorporate some type of error correction into its operation. This is because, quantum computers are far more susceptible to making errors than conventional digital computers and clearly some method of controlling and correcting these errors will be needed to prevent a quantum computer from crashing. The most formidable enemy of quantum computer is decoherence. The point is, a quantum system inevitably interacts with the environment. The information stored in the computer decays, resulting in errors and the failure of computation. Note, however, that decoherence is not the only enemy of quantum computers. Even if we can somehow isolate the quantum computer from environment, we could not expect to execute quantum logic gates with perfect accuracy. Small errors in quantum gates can accumulate over the course of computation, eventually causing failure and it is not obvious how to correct these small errors. The prospects for quantum computing received a tremendous boost from the discovery that quantum error correction is really possible in principle [50,51]. But it is important to realize that this is not enough. To carry out a quantum error-correction protocol, we must encode the quantum information we want to protect and then repeatedly perform recovery operations that reverse the errors that accumulate. Secondly, to operate a quantum computer, we must not only store quantum information but must also process the information. Thus we must be able to perform quantum gates, in which two or more encoded qubits come together and interact with one another. And we must design gates to minimize the propagation of errors. Incorporating quantum error correction will surely complicate the operation of a quantum computer. Because of this necessary increase in the complexity of the device, it is not a priori obvious that error correction will really improve its performance. This brings us to the novel concept of fault tolerant quantum computation [52,53]. A device is said to be fault tolerant if it works effectively even even when its components are imperfect. One of the best example of this concept is human body itself! It is a very illuminating example of imperfect hardware as well as hierarchical architecture with error correction at all levels. It may be noted here that similar issues also arose in the theory of fault tolerant classical computation. In 1956, Von Neumann [54] suggested improving the reliability of circuit with noisy gates by executing each gate many times, and using the majority voting. One shortcoming of his analysis was that he assumed perfect transmission of bits through the wires connecting the gates. In 1986, Gacs [55] was able to go beyond this assumption. In the last few years the fault tolerant methods have been
Ch 3.
Quantum Mechanics of Anyons
99
developed sufficiently so that there is a feeling that it is now possible in principle for the operator of a quantum computer to actively intervene to stabilize the device against noisy (but not too noisy) environment. These techniques cope with sufficiently small errors. However, the error magnitude must be smaller than some constant (called an accuracy threshold) for these methods to work. According to rather optimistic estimates, this constant lies between 10~5 and 10~3, beyond the reach of current technologies. An obvious question is if one can design quantum gates that are intrinsically fault tolerant so that active intervention by the computer operator will not be required to protect the machine from noise? In 1997, Kitaev [56] suggested a novel way for achieving it-by anyon based computation. The central idea here to store and manipulate quantum information in a global form that is resistant to local disturbances. A fault tolerant gate should be designed to act on this global information so that the action it performs on the encoded data remains unchanged even if we deform the gate slightly, that is even if the implementation of the gate is not perfect. This is achieved by anyonic interactions which are topological in nature and which are immune to local disturbances. The idea here is to make use of braid group properties in order to do quantum computation. One can show that the accidental, uncontrolled exchanges are rare if the exchanged particles are widely separated and further if thermal anyons are suppressed. While the original proposal of anyon based quantum computation was by Kitaev [56,49], the first concrete model was given by Ogburn and Preskill [57,53] for anyons in the group A&, the even permutations of five elements. This work has subsequently been generalized by Mochon [58]. It must be pointed out at this stage that while all these schemes are theoretically possible, it is still not clear if we will be able to build an anyon based computer. The point is that even the simplest model with group A5 has 60 elements. Physically, this would require a sixty-component spin residing at each lattice link! Over the years, it has been realized that if quantum computer has to perform interesting computations then it must employ nonabelian anyons, as only they will be able to build up complex unitary transformations by performing many particle exchanges in succession. Kitaev [56] has described a family of simple spin systems on a square lattice with local interactions in which the existence of quasi-particles with nonabelian anyons can be demonstrated. The models are sufficiently interesting and yet they give rise to highly entangled state with infinite range quantum correlations.
100
Fractional Statistics and Quantum Theory
Unfortunately, they require four-body interactions. One remarkable thing coming out of whole thing is that the gauge phenomenon can emerge as collective effect in systems with only short range interactions. Could it be that the gauge symmetries in nature have a similar origin? Another possible anyon based model for quantum computation makes use of fractionally quantized quantum Hall effect. Unfortunately, mostly abelian anyons seem to play a role here though there is speculation that in some cases even nonabelian anyons could play an important role. Few years back, Averin and Goldman have proposed a device based on topological ideas [59]. In their model, anyons group around anti-dot holes of 0.2/XTO diameter made in a two-dimensional electron sheet. The holes are separated by O.Ol^zm wide gates. Individual anyons are then moved between the anti-dots in a way that allows for controlled braiding. They have demonstrated how a two-qubit controlled-NOT gate and single qubit gates could be constructed this way, thus showing their scheme implemented universal computation. They also discussed the decoherence mechanisms that would affect their model and provided some estimates of the dissipation and decoherence rates which showed that the device would not be significantly better, in this respect, than, say, schemes based on quantum dots. Summarizing, I feel that this field is still in its infancy and it is not completely clear if anyon based quantum computation will be feasible in practice or not. In any case, it is highly unlikely that a quantum computer will be in actual operation in foreseeable future, even the optimists do not expect one in coming fifteen-twenty years! References [1] D.P. Arovas, R. Schrieffer, F. Wilczek and A. Zee, Nucl. Phys. B251 (1985) 117. [2] Handbook of Mathematical Functions, eds. M. Abramowitz and LA. Stegun (Dover, New York, 1970). [3] J.M. Leinaas and J. Myrheim, Nuovo Cim. B37 (1977) 1. [4] L.D. Landau and L. Lifschitz, Quantum Mechanics: Non-relativistic Theory (Pergamon Press, Oxford, 1977). [5] A. Comtet and A. Khare, Institute of Physics, Bhubaneswar Preprint IOPBBSR/91-11 (Unpublished). [6] R. Chitra, C.N. Kumar and D. Sen, Mod. Phys. Lett. A7 (1992) 855. [7] J. Law, M.K. Shrivastava, R.K. Bhaduri and A. Khare, J. Phys. A25 (1992) L183.
Ch 3. Quantum Mechanics of Anyons
101
[8] M.D. Johnson and C. Canright, Phys. Rev. B41 (1990) 6870. [9] A. Comtet, Y. Georglin and S. Ouvry, J. Phys. A22 (1989) 3917. [10] A. Vercin, Phys. Lett. B260 (1991) 120 ; J. Myrheim, E. Halvorsen and A. Vercin, Phys. Lett. B278 (1992) 171. [11] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [12] A. Suzuki, M.K. Srivastava, R.K. Bhaduri and J. Law, Phys. Rev. B44 (1991) 10 731. [13] R.L. Siddon and M. Schick, Phys. Rev. A9 (1974) 907. [14] W.G. Gibson, Mol. Phys. 49 (1983) 103. [15] Y.-S. Wu, Phys. Rev. Lett. 53 (1984) 111; 53 (1984) E1028. [16] C. Chou, Phys. Rev. D44 (1991) 2533 ; Phys. Rev. D45 (1992) E1433 ; Phys. Lett. A155 (1991) 245. [17] R. Basu, G. Date and M.V.N. Murthy, Phys. Rev. B46 (1992) 3139. [18] A. Polychronakos, Phys. Lett. B264 (1991) 362. [19] G. Date, M. Krishna and M.V.N. Murthy, Int. J. Mod. Phys. A9 (1994) 2545. [20] F. Calogero, J. Math. Phys. 10 (1969) 2191, 2197 ; 12 (1971) 419. [21] A. Lerda, Anyons: Quantum Mechanics of Particles with Fractional Statistics, Lecture Notes in Physics m 14 (Springer-Verlag, Berlin, 1992). [22] J.E. Kilpatrick and S.Y. Larsen, Few Body Systems 3 (1987) 75. [23] A. Khare and J. McCabe, Phys. Lett. B269 (1991) 330. [24] J. McCabe and S. Ouvry, Phys. Lett. B260 (1991) 113. [25] M. Spoore, J.J.M. Verbaarschot and I. Zahed, Nucl. Phys. B389 (1993) 645. [26] J. Grundberg, T.H. Hansson, A. Karlhede and E. Westerberg, Phys. Rev. B44 (1991) 8373. [27] D. Sen, Nucl. Phys. B360 (1991) 397. [28] M.V.N. Murthy, J. Law, M. Brack and R.K. Bhaduri, Phys. Rev. Lett. 67 (1991) 1817 ; Phys. Rev. B45 (1992) 4289. [29] M. Sporre, J.J.M. Verbaarschot, and I. Zahed, Phys. Rev. Lett. 67 (1991) 1813. [30] D. Sen, Phys. Rev. Lett. 68 (1992) 2977; Phys. Rev. D46 (1992) 1846. [31] M. Sporre, J.J.M. Verbaarschot, I. Zahed, Phys. Rev. B46 (1992) 5738. [32] F. Iluminati, F. Ravndal and J.Aa. Ruud, Phys. Lett. A161 (1992) 323; J. Aa. Ruud and F. Ravndal, Phys. Lett. B291 (1992) 137. [33] R. Chitra and D. Sen, Phys. Rev. B46 (1992) 10 923. [34] A. Khare, Phys. Lett. A 221 (1996) 365. [35] A. Khare, J. Phys. A29 (1996) L45 , 6459. [36] G.V. Dunne, A. Lerda and C.A. Trugenberger, Mod. Phys. Lett. A6 (1991) 2819; Int. J. Mod. Phys. B5 (1991) 1675; G.V. Dunne, A. Lerda, S. Sciuto and C.A. Trugenberger, Nucl. Phys. B370 (1992) 601. [37] K. Cho and C. Rim, Ann. Phys. 213 (1992) 295. [38] A. Karlhede and E. Westerberg, Int. J. Mod. Phys. B6 (1992) 1595. [39] A. Khare, J. McCabe and S. Ouvry, Phys. Rev. D46 (1992) 2714. [40] G. Date and M.V.N. Murthy, Phys. Rev. A48 (1993) 105. [41] G. Date, M.V.N. Murthy and R. Vathsan, corad-mai/0302019. [42] P.J. Richens and M.V. Berry, Physica 2D (1981) 495.
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Fractional Statistics and Quantum Theory
G. Date, S.R. Jain and M.V.N. Murthy, Phys. Rev. E51 (1995) 198. R. Feynman, Int. J. Theor. Phys. 21 (1982) 467. D. Deutsch, Proc. Roy. Soc. London A400 (1985) 97. P. Shor, Proc. 35th Annual Symposium on Foundations of Computer Sc., IEEE Computer Society Press, Los Alamitos, CA (1994). L. Grover, Phys. Rev. Lett. 79 (1997) 325. S. Lloyd, Science 273 (1996) 1072. M.H. Freedman, A. Kitaev, M.L. Larsen and Z. Wang, quant-ph/0101025. P. Shor, Phys. Rev. A52 (1995) 2493. A.M. Steane, Phys. Rev. Lett. 77 (1996) 793. P. Shor, Proc. 37th Annual Symposium on Foundations of Computer Sc, IEEE Computer Society Press, Los Alamitos, CA (1996). J. Preskill, quant-ph/9712048. J. von Neumann, in Automata studies, ed. C.E. Shannon and J. McCarthy (Princeton Univ. Press, Princeton, 1956). P. Gacz, J. Comp. Sys. Sc. 32 (1986) 15. A. Kitaev, quant-ph/'9707021. R.W. Ogburn and J. Preskill, Proc. of QCQC 98, edited by C.P. Williams (Springer-Verlag, Berlin, 1999). C. Mochon, Phys. Rev. A67 (2003) 022315. D.V. Averin and V.J. Goldman, corad-mat/0110193.
Chapter 4
Statistical Mechanics of an Ideal Anyon Gas Be patient, for the world is broad and wide
— E.A. Abbott in Flatland 4.1
Introduction
In this chapter, we shall discuss the statistical mechanics of an ideal anyon gas. By an ideal gas we mean a gas of particles with no interaction apart from the statistical interaction. It is worth emphasizing here that the study of an ideal anyon gas is a kind of bench-mark study which is a must. Without a proper understanding of this problem, no worthwhile progress can be expected in the more realistic interacting anyon systems. Recall that a similar study in the case of an ideal Bose gas and an ideal Fermi gas was done right in the early days of quantum statistical mechanics. Such a study was possible because the wave function for TV non-interacting bosons (or fermions) is merely the product of the single particle wave functions with appropriate symmetry (or antisymmetry) factors. No such equivalent rule is known for anyons. As seen in the last chapter, even the two-anyon spectrum is not related to the single particle spectrum. Of course, this is because, the non-interacting anyon gas is not really non-interacting but is equivalent to an interacting Bose (or Fermi) gas which is known to be a notoriously difficult problem. In the absence of the standard path, perhaps the only other way to attack the problem of the non-interacting anyon gas is to try to obtain the spectrum of ./V-anyons in some potential and to use this spectrum to calculate the virial coefficients and hence the equation of state of an ideal anyon gas. However, as we have seen in the last chapter, so far, the quantum mechanics of only two-anyon system has been solved 103
104
Fractional Statistics and Quantum Theory
exactly in a few potentials while not even one iV-body (N > 3) problem has been completely solved as yet. As a result, to date only the second virial coefficient of an ideal anyon gas has been analytically computed. As far as the higher virial coefficients are concerned, it has been shown that the third and the higher virial coefficients do not receive any contribution at order a either around the bosonic or the fermionic end. Further, 03(0) to ae(a) have been computed perturbatively upto O(a2) around both the bosonic and the fermionic ends. Besides, by using the numerical and the perturbative methods, 0,3(0) has been computed for any a(0 < a < 1). The plan of this chapter is the following: In Sees. 4.2 and 4.3, we first give the results for the statistical mechanics of an ideal Fermi and Bose gas respectively in two dimensions. In particular, we show that for an ideal Bose gas there is no Bose-Einstein condensation in two dimensions (unlike in three and higher dimensions), and further the specific heat of the Bose and the Fermi gas are identical in two dimensions. Analytical expressions for all the virial coefficients are also derived. In Sec. 4.4, the second virial coefficient of an ideal anyon gas is calculated in three different ways and all the approaches are shown to give an identical answer. In Sec. 4.5, computation of the third virial coefficient of an ideal anyon gas is discussed from several different angles and exact and approximate results about it are discussed at length. Finally, some results about the higher virial coefficients CLN(N > 3) are mentioned in Sec. 4.6. 4.2
Ideal Fermi Gas in Two Dimensions
In this section we briefly discuss some exact results for an ideal Fermi gas in two dimensions. While the general formalism as well as a detailed discussion about the ideal Fermi gas in three dimensions is given in several text books [1,2], not many books have discussed the exact results in two dimensions. Let us consider an ideal Fermi gas in a grand canonical ensemble, in which case the equation of state is given by PA — =ln£(z,A,T) = £ l n ( l + * e - ^ )
(4.1)
from which z is to be eliminated with the help of the equation
N = zlzmc(z,A,T) = Y,TT^
(4.2)
Ch 4-
105
Statistical Mechanics of an Ideal Anyon Gas
where 0 = 1/kT, fugacity z = e^lkT with fi being the chemical potential and k the Boltzmann constant. Here P, A, N, T denote the pressure, area, number of particles and the temperature of the gas respectively. On replacing the summations over e by corresponding integrations, the two equations become (4.3) (4.4)
where A is the mean thermal wave length of the particles of mass m i.e. (4.5)
and , .,
Z100 xn~1dx
1
,
.
On eliminating z between Eqs. (4.3) and (4.4) one then obtains the equation of state of an ideal Fermi gas in two dimensions. The internal energy U is given by
(4.7) Thus, in two dimensions one has quite generally the relationship PA = U.
(4.8)
We shall see that an ideal Bose gas (as well as an ideal classical gas) also satisfies the same relationship in two dimensions. The specific heat Cv of the gas can now be obtained from Eq. (4.7) by making use of the formulae 2^[/,,W] = / „ - ! «
C = {%) \
U 1
/ N,A
(4.9)
(4-10)
and the relationship (dz\ \dTJN,A
zfx{z) T To{z)
(4.11)
106
Fractional Statistics and Quantum Theory
One obtains C^_
h{z) h(z)
Nk ~ 2 7 ^ y - -W) •
^12>
Similarly, one can show that Cp/Cv, Helmholtz free energy AH, and the entropy 5 are given by (4.13)
AH=Nii-PA = NkT\\nz-^\]
«-^=™ [ # - 4
(4.14)
(4 15)
-
It is worth noting that by Cv we mean here (in two dimensions), specific heat at constant area. Let us first discuss the properties of the Fermi gas at low density p(= N/A) and high temperature T. In this case, it follows from Eq. (4.4) that (4.16)
i.e the gas is highly non-degenerate. Let us now consider / i (z) as given by the integral (4.6). It is a standard integral which can be analytically done giving
(4 i7)
uz)=r^ri=Hi+z)-
-
Since at low p and high T, f\{z) « 1, hence z « 1 in that case. In case z is small but not extremely small in comparison to unity then one has to eliminate z between Eqs. (4.3) and (4.4) and obtain the equation of state in the form of virial expansion i.e. PA
_ f2(z)
^
2 |
_ ,
(4.18)
where ai are the (dimension-less) virial coefficients. It is worth pointing out that in most text books on statistical mechanics, a; are usually so defined that they are dimension-full. We now show that for two dimensional Fermi
Ch 4-
107
Statistical Mechanics of an Ideal Anyon Gas
gas all the virial coefficients can be analytically calculated. To that purpose note from Eqs. (4.4) and (4.17) that p\2
= f1(z)=ln(l
(4.19)
+ z).
On using Eq. (4.9), fi{z) can then be obtained by integration i.e.
f2(z) = I £ M dz=
f-z ln(l + z) dz
(4.20)
along with the boundary condition /2(0) = 0. On making the substitution ln(l + z) = y (= p\2) it is easily shown that for y < 2w 2
M^
°°
2/+1
+ T + ^WTiy**-
(4.21)
From Eqs. (4.3), (4.4), (4.19) and (4.21) it then follows that the equation of state is
Mf =
1+
— + ttWTiyB21-
(4 22)
"
On comparing Eqs. (4.16) and (4.23), we then obtain the virial coefficients for an ideal Fermi gas in two dimensions
4 = i, 4 = \, 4n+2 = o, 4n+1 = ~f^ where B^n are the Bernoulli numbers. Some of the low lying i?2n
B
B
B
B
B
* = \' ^-k' « = k> * = -k' ™ = k-
(4.23) are
(424
-^
Since Bin are alternatively positive and negative, it follows that [n = 1,2,...) a 4n _! > 0, ain+i < 0.
(4.25)
Also notice that a^ > 0 and all higher even virial coefficients are zero. This is a special property of the two dimensional Fermi gas (as well as the Bose gas as we shall see in the next section). The specific heat Cv can now be immediately calculated from Eq. (4.12) or more simply from Eqs. (4.18), (4.10) and (4.8). We find (4.26)
108
Fractional Statistics and Quantum Theory
Remarkably enough, only the odd virial coefficients contribute to the specific heat of an ideal Fermi gas in two dimensions. Further, at any finite temperature, the value of the specific heat is smaller than its limiting value which is Nk. We shall see below that the specific heat of an ideal Fermi gas decreases monotonically as the temperature of the gas falls. The other thermodynamic quantities can now be easily calculated (note that fo(z) = z-^fi(z) = z/(l + z)). It is left as an exercise to the reader. In the next section we shall show that in two dimensions, the specific heat of an ideal Bose gas is identical to that of an ideal Fermi gas. Let us now consider the behavior of an ideal Fermi gas at low temperature and high density such that z and hence X2p » 1. As \2p —> oo, the gas is said to be completely degenerate and in this limit the various thermodynamic quantities take a particularly simple form. As is well known, in the limit T —• 0 (which corresponds to X2p —> oo), the mean occupation number of the single particle states e(p) becomes 1(0) for e(p) < no (> Mo) where JJLQ is the chemical potential of the system at T = 0. Thus at T = 0 all single particle states upto e = no are completely filled. The limiting energy /xo is commonly called the Fermi energy and denoted by EF and the corresponding Fermi momentum is denoted by pp. The denning equation for these parameters is
[Fa(e)de = N, Jo
(4.27)
where a(e) is the density of states which in two dimensions is given by
(4.29)
It is then easily shown that (p = N/A)
Similarly, the ground state (or zero-point) energy of the system turns out to be
Eo = [Fsa(e)de
=^
P
%
(4.30)
and hence the ground state energy per particle and the ground state pressure are given by (4.31)
Ch 4-
109
Statistical Mechanics of an Ideal Anyon Gas
Thus PQ OC p (and not p 5 / 3 as in three dimensions). What happens at low but finite temperatures? In this case \2p and hence z is finite and large compared to unity. One then expands fn(z) in powers of (lnz)" 1 . On using the well known Sommerfeld lemma [1], /2,/i,/o are given by / 2 (z) = i ( l n z ) 2 + ^ + O ( l / z )
(4.32)
f1{z) = laz + O(l/z)
(4.33)
/„(*) = l + O(l/z).
(4.34)
Using Eqs. (4.33), (4.29) and (4.4) we have kT\nz = n=^—=eF. (4.35) m Thus for the two dimensional ideal Fermi gas, \i — ep is not only true at zero but even at low temperatures. Similarly using Eqs. (4.3), (4.11), (4.15) and (4.32) to (4.34) one finds that
S-?N(£)V-S-=?K(£)1 ^ - ^ &-! + £( I.}* Nk~ 3 T F ' Cv ~ + 3 [TFJ
^-TT-YUV
J' M~TT>
(437)
{ A1}
(4 38)
-
where Tp is the Fermi temperature (Tp = £F/k). Note that these are all exact expressions at low T (T < < 2>) with no higher order correction terms in powers of (T/Tp). For T ~ TF one has to use the full expressions for /„ and calculate the various thermodynamic quantities numerically. From such a study it can be shown that the specific heat monotonically increases as T increases and reaches its classical value of Nk as T —> oo.
110
4.3
Fractional Statistics and Quantum Theory
Ideal Bose Gas in Two Dimensions
The equation of state for an ideal Bose gas in a grand canonical ensemble in two dimensions is given by PA —
= In C(z, A, T) = - £ ln(l - ze~^)
(4.39)
from where z is to be eliminated with the help of the equation
N = z§-zki£(z,A,T) = J2 z - i j e _ i -
( 44 °)
e
The summands appearing in Eqs. (4.39) and (4.40) diverge as z —> 1 since the term corresponding to p = 0 (and hence e = 0) diverges. On splitting off the terms in Eqs. (4.39) and (4.40) corresponding to e = 0 and replacing the rest of the sum by an integral, one obtains
§f = ^ » W - \ta(l" *>
(4-41)
where i 9n(z) = = ^ r
roc /
T-n-irfr . • \ x
r(n) yxo z-^^ - 1
(4.43)
We have not taken the lower limit of the integral to be 0, since for go and gi, the state e(= x/(3) = 0 does contribute to the integral and this contribution is quite significant (and in fact divergent as z —• 1). However, the last term in Eq. (4.41) is negligible for all values of z and may be dropped altogether in the thermodynamic limit. This is easily seen as follows. Since the quantity z/(l — z) is identically equal to A^o hence the term (—-^ ln(l — z)) is identically equal to (^ ln(7V0 + 1)) and is at most of O{N~1 inN) and hence negligible in the thermodynamic limit. As a result, the internal energy of the system is given by
Comparing Eqs. (4.41) and (4.44) we then find that in two dimensions, a la ideal Fermi gas, an ideal Bose gas also satisfies the relationship PA = U.
Ch 4-
111
Statistical Mechanics of an Ideal Any on Gas
Let us first consider the case of low density and high temperature. In this case the last term in Eq. (4.42) is negligible since this case corresponds to z « 1. Further, the lower limit of integration in gi{z) as denned in Eq. (4.43) can be safely taken to be zero since for z « 1 there is no divergence in gi(z). Once the lower limit is taken to be zero, g\ (z) can be obtained analytically
9i(z) = l
t°°
rlT
p ^ ; — T = -ln(l-2 f ).
(4.45)
We can now obtain z in terms of p\2 by using Eqs. (4.42) and (4.45). We get z = 1 - e~x2p .
(4.46)
We can now obtain 52(2) from g\{z) by integrating the relation gi(z) = z-§-z92{z) i.e.
(4.47)
g2(z) = -J^-Ml-z)
and using the boundary condition, 52(0) = 0. On using — ln(l — z) = y(= X2p), we find that for \y\ < 2ir
(4 48)
^^-T+gdkyfc-
-
On substituting the above expression for z and 52(2) hi Eq. (4.41) we obtain the equation of state in the form of a virial expansion. In particular, on comparing with Eq. (4.23), we find that in two dimensions, the virial coefficients for an ideal Bose gas are identical to those of an ideal Fermi gas except for the second virial coefficient which is equal and opposite in the two cases i.e. R
°1
=
a
l
P
=
1 > a2
P1 =
~a2
R =
R
W
T 1 a2n+2 ~ a2n+2
=
">
where Bin are the Bernoulli numbers. The specific heat Cv of an ideal Bose gas can now be computed by using U = PA and the relation (4.10) and one finds that Cv of the 2-dimensional ideal Bose gas is identical to that of the ideal Fermi gas [3], both being given by Eq. (4.26). It is worth repeating that only the second virial coefficient
112
Fractional Statistics and Quantum Theory
is different in the two cases but a2 does not contribute to the specific heat in two dimensions. One very important consequence of this fact is that unlike in three dimensions, for two dimensional ideal Bose gas, the value of the specific heat at finite temperature is smaller (and not larger) than its limiting (asymptotic) value of Nk (see Eq. (4.26)). Thus the (Cv-T) curve has a positive slope at high temperature. Further, as T —> 0, the specific heat must tend to zero. Thus unlike in the three dimensional case, Cv need not pass through a maximum in the two dimensional case. In fact we will see below that in two dimensions, an ideal Bose gas does not exhibit Bose-Einstein condensation. The other thermodynamic quantities are easily calculated in the limit of z —> 0 and is left as an exercise to the reader. In particular, note that the Helmholtz free energy AH, entropy S and Cp/Cv for an ideal two dimensional Bose gas are given by AH = N/j, - PA = NkT \\azL
~\] 9i{z)\
(4.50)
Here, go{z) = z-§-zgx{z) = z/(l - z) . Let us now consider the interesting but tricky case of the high density and low temperature. We now show that unlike the case of three and higher dimensions, in two (and lower) dimensions, there is no Bose-Einstein condensation [4]. The key to the whole argument is Eq. (4.42) in which the last term is identically equal to No/A, No being the number of particles not only in the ground state e — 0 but in fact the total occupation of particles in the states from e = 0 to e = eo- Thus this equation can also be written as
^o=f_^L_. A
Jxo z - ^ - 1
(4.53)
v
;
Clearly the integral is singular at z = 1. To see this, let us set z — 1. The integral is easily performed and we find that there is a logarithmic
Ch 4.
Statistical Mechanics of an Ideal Anyon Gas
113
divergence i.e.
*LzJ0l~]n(eo/l3).
(4.54)
This divergence can be interpreted in two ways: (i) One can choose any arbitrary £o as the lower cut-off in which case Wo is indeed macroscopic, but then it is not really a condensation into one state (e = 0) and the particles are distributed over a large number of states in the interval 0 < e < EQ. (ii) One can choose e0 -* 0 but in this case, the number of particles in the excited state is as large as possible, and is even larger than NQ. We thus conclude that unlike in three and higher dimensions (in fact unlike in 2 + 5 dimensions, 6 > 0), in two dimensions, there is no accumulation of macroscopically large number of particles in a single quantum state (e = 0), but there is a band of states near e = 0 that gets occupied. Thus there is no Bose-Einstein condensation in two and lower dimensions. This is consistent with the celebrated Mermin-Wegner-Berezinskii theorem [5] which asserts that there is no long range order, i.e. no phase transition in two dimensions for either ideal Bose gas or even for interacting Bose gas so long as the interactions are of short range. To study the thermodynamic behavior of the various quantities in the limit z —> 1, it is useful to get the expansions of 9i(z),g2(z) and go(z) in the powers of 5 = — In z which is a small positive number. An explanation is in order at this stage. Since g\(z) as defined by Eq. (4.43) diverges at z = 1 in the limit xo —* 0, hence while discussing the z —> 1 behavior of 9i(z), it is better not to separate out the contribution from the e = 0 state while defining Eqs. (4.41) and (4.42). We thus drop the last terms in both these equations and take the lower limit of integration xo to be 0 in Eq. (4.43). Prom Eq. (4.43) with XQ = 0, it is easily shown that as z —> 1 (i.e. J-»0)
ffl
(6) = - ln(l - z) = - In 5 + S- - ^ + O{5A)
(4.55)
go(6) - -§g9i(S) = I - \ + ^ + O(S3)
(4.56)
and hence
114
Fractional Statistics and Quantum Theory
while 0, the specific heat of an ideal Bose gas in two dimensions also varies as T. Similarly the entropy is given by
On the other hand Cp/Cv diverges as T —> 0 since CP
a
=
2g2(z)g0(z)
^!w
z
^ 7T2 e ^ 2
r
^o
' Y(^F "" °°-
.. ...
(464)
It is easily shown that whereas Cp/Cv — 2 as T —> oo, it is finite and > 2 at finite T and diverges as T —> 0. The adiabatic of an ideal Bose gas is also easily studied. Now an adiabatic process implies the constancy of 5 and N and hence of z and in turn of p\2. Thus we have the relationship a oc 1/T where a = A/N. Similarly, the constancy of z and p\2 implies the relationship P oc T2. Eliminating T from these two relations we then have Pa2 = constant
(4.65)
Ch 4-
Statistical Mechanics of an Ideal Anyon Gas
115
as the desired equation for the adiabatic of an ideal Bose gas in two dimensions.
4.4
Ideal Anyon Gas: Second Virial Coefficient
In the last two sections we have discussed at some length the statistical mechanics of a two dimensional ideal Bose as well as Fermi gas and obtained the temperature and density dependence of the various thermodynamic quantities. We were able to do that since the distribution functions for Bose-Einstein and Fermi-Dirac statistics are already well known. This in turn is related to the fact that the wave function for the -/V-bosons (Nfermions) is simply a symmetrized (anti-symmetrized) product of the single particle wave functions. Unfortunately, as seen in the last chapter, even the two-anyon spectrum (wave function) is not related to the corresponding single particle spectrum (wave function). Thus the path taken to study an ideal Bose or Fermi gas is not going to be useful for studying the statistical mechanics of an ideal anyon gas. However, we can look upon an ideal anyon gas as an interacting Bose or Fermi gas and use the techniques which are usually used to study the interacting Bose or Fermi systems. For example, one of the most popular method to study an interacting Bose or Fermi gas is to use the method of quantum cluster expansions as developed by Kahn [6] as well as Uhlenbeck and Beth [7]. Since this method is extensively discussed in almost all the books on statistical mechanics [1] hence we shall not discuss all the details of the method but merely summarize the key formulae required for our purpose. The cluster expansion is specially suitable for systems such as dilute gases. We shall study the low density, high temperature limit of an ideal (noninteracting) anyon gas (i.e. interacting Bose/Fermi gas) using the virial expansion i.e. the equation of state can be expressed as J^
= l+a2(p\2)
+ a3(p\2)2 + ...
(4.66)
where 02,03,... are the (dimension-less) virial coefficients while p is the density and A the thermal wave length. How does one evaluate 02,03,... for an interacting Bose/Fermi gas? One usually expresses them in terms of the cluster coefficients 62, &3, ••• • In particular, in the low density, high temperature approximation, the grand canonical partition function can be
116
Fractional Statistics and Quantum Theory
written as a cluster expansion -ln£ =^ 6 ^
(4.67)
i=i
where b\ are the cluster integrals, z is the fugacity and A is the area of the system. On the other hand, as we have seen in the last two sections, the pressure P and density p are given by kT P=— ln£
(4.68)
(469)
P
-4X^C)A,T
On inserting the cluster expansion (4.67) in Eqs. (4.68) and (4.69), we get oo
oo
P = kT^blzl,
p = ^2lhzl.
(=i
(4.70)
i=i
As a result, the equation of state (4.66) takes the form OO
OO
i
r
zl
j2biz = C£tti )U+ ^ ( 6=1
1=1
L
OO
i
->
J
a 2
/
4
OO
5>i* )+a 3 A ( J2 i )
^ 1=1
'
\
lb zl
^ (=1
/
2
-i
+••• • (4-71) J
On expanding both sides of this equation and equating the coefficients of equal powers of z, we obtain the expressions for the virial coefficients in terms of the cluster integrals ; the first few of which are a2A2 = - | , a
3
A
4
=4a^-^,....
(4.72)
On using the fact that the grand canonical function can be written as an infinite sum of canonical partition functions ZN i-e. oo
C = J2 *NZN
(4.73)
JV=O
one can express the cluster integrals 6; in terms of the canonical partition functions up to Zj. In particular, on using the fact that for small x, ln(l + x)=xx2/2 + x3/3 + ... we get from Eqs. (4.67) and (4.73) 2Z Z h 7 IA h i~ l 3Z3-3Z2Z1+Z13 h 3= 6i = Z1/A, b2 = — ^ 4 — ' " §]4 '
t4'74)
Ch 4.
117
Statistical Mechanics of an Ideal Any on Gas
It must be emphasized that all these expansions are meaningful only in the infinite area limit, so that particular care must be taken in their evaluation. Using this formalism we shall now calculate the second virial coefficient of an ideal anyon gas by treating it as an interacting Bose gas. On using Eqs. (4.72) and (4.74) we have 2
-A(2Z2-Zf) 2Z2
'
^
'
Now, since we are treating an ideal anyon gas as an interacting Bose gas hence we shall break up a2 (a) into two pieces; one from an ideal Bose gas and the rest from the interaction. In particular, we write O2 (a)
= ffl2(0) - lim -Aj[Z 2 (a) - Z2(0)}.
(4.76)
Here we have used the fact that Z\ is the same for fermions, anyons and bosons since no statistical effect is present in the one body problem. We have explicitly mentioned the limit A —> 00 to emphasize the point that all these expressions are meaningful only in the infinite area limit. Here 0 < a < 1 and with our notations, a — 0 corresponds to bosons. It may be noted that in two spatial dimensions, one has h = I/A2
(4.77)
Zx = biA = A/X2 .
(4.78)
and hence
Note that we have defined the equation of state (4.18) such that a2,O3,... are all dimension-less. We now compute a2(a) for an ideal anyon gas by different methods and show that all the approaches give the same answer which is finite. Conceptually, the fact that a2(a) is finite is a nontrivial statement since the anyonic interaction is a long range interaction and not a short range interaction as is usually assumed while deriving the cluster expansion. In particular, if the particles interact by a two-body potential, then a necessary condition for the existence of the virial expansion in the thermodynamic limit of infinite volume and constant density, is that if the potential decreases as r~n at a large distance r, then [8,9] n>d,
(4.79)
118
Fractional Statistics and Quantum Theory
where d is the configuration space dimension. Thus, naively, the virial expansion should not hold good for the non-interacting anyon gas since the statistical interaction between the anyons may be represented by a vector potential proportional to 1/r. However, it seems that the above criterion does not apply in the case of the vector potentials which is the case of anyons. Put another way, if one looks at the anyon Hamiltonian, then one finds that one has not only the two-body but also the three-body interactions coming from the non-trivial braiding effects and hence those theorems of statistical mechanics which are derived by assuming only the two-body interaction among the particles may not apply in the case of anyons. In fact there is a widespread belief that the virial expansion exists for an ideal anyon gas. The fact that a2(a) is indeed finite provides strong support to this belief. a2(a) using oscillator as a regulator In order to calculate a2(a) by using the harmonic oscillator as a regulator, we must first figure out the equivalent of ^4 —> oo limit in this approach. This is easily done by computing Z\, the one particle partition function for a particle of mass m in two dimensions in an oscillator potential of frequency w. In this case, it is well known that the energy eigenvalues are (4.80)
En = (nx + ny + l)hu> =(n + l)huj , n = nx + ny
with degeneracy being n+1. Here n, nx, ny = 0,1, 2,.... Hence the canonical partition function Z\ is given by
4sinh2(M^)
h
l
'
Taking the high temperatures limit and using Eq. (4.78), one is led to the identification A A2
\ h2f32uj2
2
2TT
.
m(3A
Thus the harmonic frequency w2 can be thought of as proportional to the inverse of the area A and hence the infinite area limit [A —• oo) becomes equivalent to the zero harmonic frequency limit (i.e. u> —> 0). This is physically understandable since putting the system in a harmonic potential confines the particles to move in a finite region; removing the harmonic force (u> —> 0) amounts to removing any restriction on the space (i.e. A —> oo).
Ch 4.
Statistical Mechanics of an Ideal Anyon Gas
119
Before we proceed with computing a2 (a) using the harmonic regulator, we must dispose off one more subtle point in connection with the harmonic regularization. For two particles, the harmonic potential is w2(r2 +
r2) =
2a;2R2 +
^
r
2
(4-g3)
where R = (ri + r 2 )/2 is the center of mass coordinate while r = i^ — r2 is the relative coordinate. Thus, in the two body problem a;2 m — 2LO2. Generalizing to the Z-body case, it is easily shown that in that case uJc.m. = ^ 2 - Since it is LJc.m. which is actually related to the area (see Eq. (4.82)), hence it follows that the two-dimensional cluster integrals h must be multiplied by I. In view of the fact that a2A2 = — 62/&1 ( s e e Eq. (4.72)) it then follows from Eqs. (4.76) and (4.78) that with harmonic regularization, 02(0) can be written as
«(.)~i-»a[*"-V (l 'i
(484)
where we have used the fact that a2(0) = af = —1/4 (in units of A2). It is convenient to separate out the contribution to Z2 due to center of mass by using Z2 = Z%m-Z2 = ZXZ2
(4.85)
so that a2 (a) takes the simple form 1 a2(a) = - - - 2 lim [Z2(a,w) - Z2(0,u)] •
(4.86)
Let us now compute the canonical two-body relative partition function Z2(a,ui) which is given by Z2(a, w) = Tre-0H^a'^
(4.87)
where H2 is the two-body relative Hamiltonian which has been extensively discussed in the last chapter. As has been shown in the last chapter, the energy eigenvalues (including degeneracy) of the two anyon relative Hamiltonian in an oscillator potential are given by En(a,uj) = (2n+l + a)froj, deg. = (n + l) = (2n + 1 - a)fno,
deg. = n
(4.88)
120
Fractional Statistics and Quantum Theory
where n — 0,1,2,.... Hence J?2(a, w) is given by oc
Z2(a,uj) = J2[(n+l)e-(2n+1+a'>hl3"
+ ne-(2n+1-a)hf3"}.
(4.89)
n=0
These sums are easily done and we obtain 2smh (hptJ) Note that both ^2(a,w) and Z2(0,u;) diverge like w~2 as w —» 0 i.e. they diverge linearly with the area A as A —> oo. However, the divergence cancels in their difference and hence 02(0) turns out to be finite and given by [10] a2(a)
= _ I ( l - 4 a + 2a 2 ).
(4.91)
Not surprisingly, for a = 0 one gets back the correct Bose value of 02 (0) = a^ = —1/4. However, what is really remarkable is the fact that even for a = 1 one gets back the correct Fermi value i.e. 02(1) = af = +1/4 thereby providing a nontrivial check on the calculations. Further a 2 (a) interpolates continuously between the bosonic value (—1/4) and the fermionic value (+1/4). We remark that 02(0) must clearly be periodic in a with period Aa = 2. In particular, if a = 2j + S (i.e. quasi bosonic case) with |-0
(4.93)
where a2{y = 0) = a2 = j . In the last chapter, we have seen that the spectrum of two-anyons in an oscillator potential is given by Enrti(v, u) = (2nr + \l-v\
+ \)hu
(4.94)
where in the fermionic basis I = ±1,±3,.... Thus the spectrum and the degeneracy in a compact form are En(i/,u) = (2n ± v)hjj , deg. = n where n{=nr + ^±i) == 1,2,3,.... Hence ^J
2sinh 2 (^o;)
(4.95)
122
Fractional Statistics and Quantum Theory
From Eqs. (4.93) and (4.96) it then follows that a2(v) = \(l-2v2)
(4.97)
which is the same as given in Eq. (4.91) since u = I — a. Notice that unlike the bosonic end, there are no cusps at the fermionic end (the linear term in v is absent). It is worth emphasizing that the harmonic oscillator potential was merely used as a regulator to discretize the two-anyon spectrum and the final answer for ai{o) is in fact independent of the regulator used. For example, instead of an oscillator, one could instead consider the two-anyons in a circular box of radius R so that the spectrum is again discrete and then take the limit R —> oo at the end of the calculation. Historically, in fact, this is how a2(a) was first calculated [11,12] and only later it was calculated by using the harmonic confinement [10] and it was shown that both the approaches give the same answer for 02(0:)- In fact 02(0;) for an ideal anyon gas has also been calculated by using the path integral approach [11] and once again one obtains the same answer. In a way this approach is more satisfactory, since, even though one is still presented with the delicacy involved with extracting the finite difference of two divergent expressions, there is no necessity to impose a finite area constraint (which is necessary in the other two approaches in order to perform the mode counting). It is highly satisfying that all the approaches give the same answer for 02(0:) thereby establishing the regularization independence of the result. We thus have obtained the remarkable result that 02(0) for a noninteracting gas is finite even though the anyons do experience a long ranged, statistical vector interaction. As mentioned before, a priori it is not at all obvious whether a virial expansion should exist in the case of noninteracting anyon gas since the statistical interaction is long ranged. What about the higher virial coefficients? Would they also be finite? While it is believed that all the higher virial coefficients would be finite, so far, there is no rigorous proof of it. It is thus clearly very important to calculate the higher virial coefficients. However, it is very difficult to obtain the full spectrum of multi-anyons (N > 2) in any external potential and hence till today no one has been able to compute the third and the higher virial coefficients exactly. We therefore discuss an alternative method for evaluating the second virial coefficient due to Uhlenbeck and Beth [7] and Kahn [6]. The advantage of this method, called the semi-classical approximation, is that it does not require the knowledge of the eigenspectrum. Thus, hope-
Ch 4-
Statistical Mechanics of an Ideal Anyon Gas
123
fully, one may be able to extend this method to evaluate the higher virial coefficients. 02(0?) Using Semi-classical Approximation We now show that the second virial coefficient of an ideal anyon gas can be analytically evaluated by using the semi-classical approximation and it in fact yields the exact quantum result [13]. As seen in the last chapter, the relative classical Hamiltonian for twoanyons is given by
H$ = ^ + {Ve ~ hf 2
(4.98)
m mr where m is the mass of an anyon and pg and pr are the canonical momenta corresponding to the coordinates 6 and r respectively. Hence the canonical classical, partition function is given by iPe h Z* = -\~ f }} drd d0dPe . 2 2 f exp f - pi £ + (2nti) J I [m mr2 J J Pr
(4.99)
The limits of integration are 0 < r < 00, — 00 < pr < 00, 0 < 0 < 2?r, — 00 < P0 < 00. On shifting the variable of integration from pg to pg — ha we see that the integral becomes independent of a thereby confirming once again that the anyonic statistics is a genuine quantum mechanical effect. Let us recall that the classical equations of motion derived from the anyon Lagrangian are in fact independent of a and hence same as for the bosonic case. To unravel the quantum effect, one recognizes that pg = HI where I = 0,±2, ±4 ... in the bosonic basis. The semi-classical partition function is thus obtained from Eq. (4.99) by replacing J dpg by ^X^> with the sum over appropriate I according to the basis used. The above integral is divergent and must be regularized so as to extract the interacting part of the partition function Z 2 (a) — Z2{0). One could regularize it either by adding an external harmonic oscillator potential or by putting the particles in a circular box of radius R so that each particle moves in an area of TTR2. Needless to say that both the methods give the same answer in the limit of R —> 00 or vanishing harmonic potential. We therefore present only the R-cutoff method [13,14] and leave the other method as an exercise to the interested reader. It is worth pointing out that even in the i?-cutoff method, 02 (a) is still given by the same expression as in the oscillator basis, i.e. by Eq. (4.86). This is because, whereas in the oscillator case one had to multiply the cluster integral 6; by I, in the i?-cutoff basis one has to take care of the fact that each particle moves in an area of nR2.
124
Fractional Statistics and Quantum Theory
On performing the trivial pr and 9 integrals and replacing / dp$ by H £]; we find that for a given partial wave
(41 0)
^a) = W2[dreA^l-a?\
°
a
In order to compute a2(a), we now have to calculate J2i(^2( ) ~ ^2(0)), where 1 = 0, ±2, ±4,... in the bosonic basis. On using the Poisson summation formula 00
J-
I—
2 2 e-« (p+*)
1-
00
-1
1 + 2 *£ cos(nnb)e~^2^2
= ^
p= — 00
L
n=l
(4.101) -I
where p is an even integer, we obtain
J2 \Zl2{a) - z£(0)] = - £ £ [ 1 - C0S(n7ra)J / rdre- 3 " 2 '- 2 /^ . A
(even
-70
n=l
(4.102) On letting R —> CXD we thus have
£ [^(oO - 3(0)] = - ^ f; / even
[1 C
- °l ( n 7 r a ) ] + O(E-).
(4.103)
n=l
This is immediately evaluated by using the summation formula (0 < x < 2TT)
E
COS KX
7TZ
7TX
X
,
A
.
We then find on using Eq. (4.86) that 02(01) is again as given by Eq. (4.91) i.e. the semi-classical approximation gives the exact quantum result for 02(0:). This is a remarkable result. One could also calculate a2(a) in the fermionic basis and show that even in that case the semi-classical approximation gives the exact result as given by Eq. (4.97). We leave it as an exercise to the interested reader. The fact that the semi-classical approximation is exact for 02(0), even though remarkable, is not that surprising! In fact, we had anticipated this result in 1989 itself when we came across an interesting paper by Comtet and Ouvry [15] in which they showed that 122(0:) is related to the axial anomaly of the (1 +1 )-dimensional fermionic field theory in the classical flux tube background. Now it is well known [16] that only one loop contributes to the axial anomaly while two and higher loops do not contribute to it, i.e. the one loop or the semi-classical approximation is exact for the axial
Ch 4'
Statistical Mechanics of an Ideal Anyon Gas
125
anomaly. Note that this result is valid in any even space-time dimensions. The fact that the semi-classical approximation is exact for the (1 + 1)dimensional axial anomaly and further that this axial anomaly is related to a2(a) then strongly suggests that the semi-classical approximation is also exact for 0,2(0:). Why is the semi-classical approximation exact for 02(0;)? While the reason is not completely clear, one possible answer could be the following. We have seen that the noninteracting anyon gas can be regarded as a noninteracting Bose/Fermi gas plus interaction which is a dimension-less scale invariant interaction. Thus the only scale in the problem is the thermal wave length A and it is known that dimensionally 02 (a) = CX2 where C is a dimension-less constant (depending only on a but not on A). Now, since the semi-classical approximation by construction must be exact as T —> 00, and since C is merely a dimension-less constant (which is T-independent), hence it follows that the semi-classical approximation must be exact for 0.2(0;). By extending this logic, we also conjecture that the semi-classical approximation may even be exact for all the higher virial coefficients. Of course it is not clear whether the higher virial coefficients can be computed within the semi-classical approximation. As an illustration of these arguments, we now discuss an example of an interacting anyon gas with a scale invariant interaction and show that in this case also the semi-classical approximation for 0.2(0:) is exact. Let us consider an interacting anyon gas when the relative interaction between the two anyons is repulsive and given by
*% = £
(4-105)
where g is a dimension-less constant. Thus the total relative Hamiltonian for the two anyons (after taking out the center of mass) is given by (see Eq. (4.98)) Hrel =
PJ+ ( P ^ M ! + _l_
(4 106)
We shall first calculate the exact second virial coefficient and then the same by using the semi-classical approximation. The exact second virial coefficient 0-2(0:, g) for this system can be easily obtained by either using the harmonic confinement or by putting the particles in a circular box or even by using path integral approach [17] using directly the relative Hamiltonian as given by Eq. (4.106). All the approaches give the same answer. As an illustration let us compute it by using the harmonic confinement.
126
Fractional Statistics and Quantum Theory
The relative spectrum of two-anyons experiencing both g2/mr2 and the harmonic interaction (\muj2r2) is easily written down by following the discussion of the last chapter
EH:l= [2n + l + y/(l-a)2+g2}tKj
(4.107)
where, n = 0,1,2,... while 1 = 0, ±2,... in the bosonic basis. In the bosonic basis, a2{a) as given by Eq. (4.86) takes the form OO
OO
"^
r
^ Ue-Q«+W»
a2(a,g)-a2(0,g)=limJ2
n=0l=-oo'-
} - exp [ - ^ 2 +