Fractional Statistics and Anyon Superconductivity

457 Pages·1990·14.63 MB·English

by  Frank Wilczek

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FRAOIONAL STATISTICS
AND ANYON
SU PERCON DUalVlTY
•

FRAalONAL STATISTICS
AND ANYON
SUPERCONDUOIVITY
Fronk Wilczek
The Institute for Advanced Study
Princeton
~
.
\\h World Scie .
,... Singapore. New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
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Tll~ er!i10[ ,,;1(1 rublisher are g:-ateful to the author,: and the following publi~hers for
!heir assi~tanee and permission tl) reproduce th ... reprinted papers found in this volume:
A.:-!'leric111 Association for the Advancement of Science (Science)
Am~rican Institute of Physics (J. }([ath. Phys.)
American Physical Society (Phys. Rev. & Phys. Rev. Lett.)
Elsevier Science Publishers B. V. (Nucl. Phys. & Phys. Le:t.)
Kluwer Academic Publishers (Lett Math. Phys.)
Societa Italiana di Fisica (n. Nuovo Omento)
FRACTIONAL STATISTICS AND ANYON SUPERCONDUCTIVITY
Copyright © 1990 by World Scientific Publishing Ce,. Pte. Ltd.
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ISBN 981-02-0048-X
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v
Introduction
Over the past few years the idea of fractional quantum statistics, and the
techniques developed to study it, has proved useful in several quite diverse areas of
physics, from cosmic strings and black holes to the fractional quantized Hall effect
and high-temperature superconductivity. While the early papers were motivated by
a spirit of exploration or simple playfulness, it now seems clear that a suprisingly
powerful and coherent set of ideas has evolved around this theme. This book is an
attempt to make what I see as the central concepts readily accessible to physicists
with diverse backgrounds. Mathematicians and chemists may also fi!ld material of
interest to them.
The book consists of two parts. The first part of the book can be regarded as a
self-contained monograph. It consists of two long articles. "Concepts in Fractional
Statistics" is mainly concer!led with the basic formulation of fractonal statistics, and
with various realizations of it. "States of Anyon Matter" is mainly concerned with
the many-body theory of anyons, and its possible application to real materials.
The second part of the book consists of a reprint collection, together with brief
introductory paragraphs designed to put the papers into a coherent context.
I have not attempted to write the ultimate anyon book. Indeed, since the
subject continues to develop at an explosive ra~e, the time is not ripe for such
an attempt. My goal has been the more modest one of making a beautiful and
exciting subject widely accessiblt: while it is still in the making. The purpose and
form of the book I hope partially justify two of its peculiarities. The first is, that
in the monograph part only a few papers closely related to specific points in the
text are referenced. The interested reader will of course find abundant entries into
the relevant literature in the reprint part. Also, there is some redundancy between
the different parts, including even repetition of some key paragraphs. I thought it
better to make the text easy to read, than to aim for formal perfection.
I have had the good fortune to work over the years with a large number of
gifted collaborators on the subject~ treated here. They include (more or less in
chronological order) Tony Zee, Dan Arovas, Bob Schrieffer, Richard MacKenzie,
Fred Goldhaber, Al Shapere, X.-G. Wen, Lawrence Krauss, Mark Alford, John
March-Russell, Yi-Hong Chen, Edward Witten, Bert Halperin, Y.-S. Wu, Steve
Giddings and Martin Greiter. Their contributions and influence permeate the sub
ject, and especially my account of it. I would also like to thank Al Shapere for a
careful reading of the monograph part and for several useful suggestions. Finally
I am happy to thank P.-H. Tham for her editorial efforts, and Maggie Best, Gisele
Murphy and Val Nowak for their help in preparing the manuscript for publication.

vii
Contents
Introduction ......... ... ........................... ...... ...................... ....... ....... .......... ... ..... .............. .......... v
Part L CONCEPTS IN FRACTIONAL STATISTICS 1
1. Transmutation and Fractionization of Angular Momentum ............................... 4
2. Fractional Statistics: Trajectory Space and the Braid Group ............................. 11
3. Fictitious Field Con3truction, and Quantum Mechanical Examples................... 17
4. Aharonov-Bohm Scattering and its Implications.................................................. 21
5. Hopf Term ................ ........... .................... ..... ........ ................... ................................. 27
6. Local Field Theory; Chern-Simons Construction .................................................. 34
7. Berry Phase and Anyons in the Quantized Hall Effec~ ..................... ................... 40
8. Induced Quantum Numbers ................................................................................... 47
9. Boundary and Coboundary ..................................................................................... 50
Part n. STATES OF ANYON MATrER 57
1. Definition of Anyons ............................... ................................................................ 60
2. Average Field Theory .............................................................................................. 62
3. Anyon Superconductivity: Basic Mechanism ....................................................... 65
4. Anyon Superconductivity: Phenomenology .......................................................... 72
5. Adiabatic Principle ............................................................................. :.................... 80
6. More Supcrconducting States ..................... ........................................................... 82
7. Quantized Hall States . ....................... .................................................................... 83
8. Anyon Metal ............................................................................................................ 87
9. Microscopic Origins ................................................................................................. 89
10. Conclusions ............ .......................................... ........................................................ 98
Part m. REPRINTED PAPERS 103
1. Classics 105
D. Finkelstein and J. Rubinstein, "Connection between Spin, Statistics,
and Kinks", J. Math. Phys. 9 (1968) 1762-1779 .......................................................... 107
Y. Aharonov and D. Bohm, "Significance of Electromagnetic Potentials
in the Quantum Theory", Phys. Rev. 115 (1959) 485-491 .......................................... 125
J. M. Lcinaas and J. Myrhcipt, "On the Theory of Identical Particles",
n
Nuovo Cimento 37 (1977) 1-23 .................................................................................. 132
2. The Last Bastion of Rationality 157
J. Goldstone and F. Wilczek, "Fractional Quantum Numbers on Solitons",
Phys. Rev. utt. 47 (1981) 986-989 ............................................................................... 159
viii
F. Wilczek, "Magnetic Flux, Angular Momentum, and Statistics",
Phys. Rev. utt. 48 (1982) 1144-1146 ........................................................................... 163
F. Wilczek, "Quantum Mechanics of Fractional-Spin Particles",
Phys. Rev. utt. 49 (1982) 957-959 ............................................................................... 166
3. Foundations 171
D. P. Arovas, R. Schri6ffer. F. \Vilczck and A. Zee. "Statistical Mechanics
l)f Apyl)!1~··.lVILcl /-'hy.C!. B2:-il (1~·m5~ 1 U-l~1) ............... -............................................ 173
i. S. Wl!.. "G~ner,:jl Theory f()~ Quaf'tuM Stati3tics in Two Dimensions",
Phys. Rev. Lett. 52 C.984) 2103-2106 ........................................................................... 183
A. S. Goldhaber and R. Mackenzie, "Arc Cyons Really Anyons?",
Phys. utt. B214 (1988) 471-474 .................................................................................. 187
A. S. Goldhaber, R. Mackenzie and F. Wilczek, "Field Corrections to
.~. Ind~c.ed Statistics", Mod. Phys. Lett. A4 (1989) 21-31 ................................................ 191
J. FrOhlich and P.-A. Marchetti, "Quantum Field Theory of Anyons",
Lett. Math. Phys. 16(1988) 347-358 ............................................................................. 202
A. M. Polyakov, "Fermi-Bose Transmutations Induced by Gauge Fields",
Mod. Phys. Lett. A3 (1988) 325-328 ............................................................................. 214
4. AnYOf1S in Model Field Theories 221
F. Wilczek and A. Zee, "Linking Numbers, Spin, and Statistics of Solitons",
Phys. Rev. utt. 51 (1983) 2250-2252 ........................................................................... 222
A. N. Redlich, "Parity Violation and Gauge Noninvariance of the
Effective Gauge Field Action in Three Dimensions", Phys. Rev. D29
(1984) 2366-2374 .......................................................................................................... 225
Y.-H. Chen and F. Wilczek, "Induced Quantum Numbers in some 2 + 1
Dimensional Models", Int. J. Mod. Phys. B3 (1989) 117-128 ..................................... 234
5. Anyons in the Quantized Hall Effect 249
B. 1. Halperin, "Statistics of Quasiparticles and the Hierarchy of
Fractional Quantized Hall States", Phys. Rev. Lett. 52 (1984) 1583-1586
(Erratum: Phys. Rev. Lett. 52 (1984) 2390) ................................................................. 251
D. Arovas, J. R. Schrieffer and F. Wilczek, "Fractional Statistics and the
Quantum Hall Effect", Phys. Rev. Lett. .5 3 (1984) 722-723 ........................................ 256
ix
S. M. Girvin and A. H. MacDonald. "Off-Diagonal Long-Range Order,
Oblique Confinement. and the Fractional Quantum Hall Effcct",
Phys. Reu. utt. 58 (1987) 1252-1255 ........................................................................... 258
R. B. Laughlin. "Fractional Statistics in the Quantum Hall Effcct" .......................... 262
6. Chiral Spin States 307
V. Kalmeyer and R B. Laughlin. "Equivalence of the Rcsonating-Valence-&nd
and Fractional Quantum Hall States", Phys. Reu. utt. 59 (1987) 2095-2098 .......... 308
X.-G. Wen. F. Wilczek and A. Zoe, "Chiral Spin Statcp and
Superconductivity", Phy.'). Rev. B39 (1989) 11~13-11423 .......................................... 312
7. Anyon Superconductivity 325
R. B. Laughlin. 'crrhe Relationship between High-Temperature
Superconductivity and the Fractional Quantum Hall Effect",
Science 242 (1988) 525-533 .......................................................................................... 326
R. B. Laughlin, "Supcrconducting Ground State of Noninteracting
Particles Obeying Fractional Statistics", Phys. Reu. utt. 60 (1988)
2677-2680 ........................ ..... ...... .. ............................................................................... 335
A. L. Fetter, C. B. Hanna and R. B. Laughlin, "Random-Phase Approximation
in the Fractional-Statistics Gas", Phys. Reu. B39 (1989) 9679-9681 ........................ 339
Y.-H. Chen, F. Wilczek, E. Witten and B. I. Halperin, "On Anyon
Superconductivity", Int. J. Mod. Phys. B3 (1989) 1001-1067 ..................................... 342
8. Some Recent Directions 411
G. S. Canright and S. M. Girvin, "Anyons, the Quantum Hall Effect, and
Two-.Dimensional Superconductivity", Int. J. Mod. Phys. B3 (1989)
1943-1963 ...................................................................................................................... 413
D.-H. Lee and M. P. A. Fisher, "Anyon Superconductivity and the Fractional
Quantum Hall Effect", Phys. Reu. utt. 63 (1989) 903-906
(Erratum: Phys. Reu.ll!tt. 63 (1989) ................................................................... 434
D.-H. Lee and C. L. Kane, "Boson-Vortex-Skyrmion Duality, Spin-Singlet
Fractional Quantum Hall Effect, and Spin-1I2 Anyon SuperconductivitY',
Phys. Reu. utt. 64 (1990) 1313-1317 ........................................................................... 439
T. Einarsson, "Fractional Statistics on a Torus", Phys. Rev. utt.
84 (1990) 1995-1998 ...................................................................................................... 444