Non relativistic. Topological Quantum Numbers in. Physics

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Topological Quantum Numbers in

Nonrelativistic

Physics

Topological Quantum Numbers in Nonrelativistic Physics Downloaded from www.worldscientific.com by 37.44.207.38 on 01/17/17. For personal use only.

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Topological Quantum Numbers in Nonrelativistic Physics Downloaded from www.worldscientific.com by 37.44.207.38 on 01/17/17. For personal use only.

Topological Quantum Numbers in

Nonrelativistic Physics

David J. Thouless Department of Physics University of Washington, Seattle

World Scientific Singapore New Jersey. London Hong Kong

Published by

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World Scientific Publishing Co. Pte. Ltd. P 0 Box 128, Farrer Road, Singapore 912805 USA ofice: Suite lB, 1060 Main Street, River Edge, NJ 07661 UK ofsice: 57 Shelton Street, Covent Garden, London WC2H 9HE

The author and publisher are grateful to the authors and the following publishers for their assistance and permission to reproduce the reprinted papers found in this volume: Academic Press (Ann. Phys. (NY)) American Institute of Physics (Sov. Phys. JETP) American Physical Society (Phys. Rev., Phys. Rev. Lett.) Les Editions de Physique (J. Phys. (Paris),J. Phys. Lett. (Paris)) Elsevier Science Ltd (Phys. Lett.) Institute of Physics Publishing (J. Phys. C) Macmillan Magazines Ltd. (Nature) Plenum Publishing Corporation (J. Low Temp. Phys., Quantum Fluids and Solids) The Royal Society (Proc. Roy. Soc. London)

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

TOPOLOGICAL QUANTUM NUMBERS IN NONRELATIVISTIC PHYSICS Copyright 0 1998 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, orparts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-2900-3 ISBN 981-02-3025-7 (pbk)

Printed in Singapore by UtePrint

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Preface

Topological quantum numbers crept up on the physics community before the community was aware of them. I did not think in these terms until I started working on the topological aspects of long range order in the early 1970s, although I had been working on aspects of superfluidity that are now regarded as topological for several years before that. I should have known earlier of the importance of topology, as I was then a colleague of Tony Skyrme, whose pioneering work on topological quantum numbers is now so well known. It was around this time that there began to be a wide awarenesss of the importance of topology both amongst elementary particle theorists and field theorists, and amongst people who worked on superfluids and liquid crystals. The issue was brought sharply into focus for me in 1980, when Hans Dehmelt asked me about how the quantum Hall effect could possibly be used to determine the fine-structure constant when so little was known about the details of the devices used and so little understood about the theory. Dehmelt’s question is one of the unifying themes of this book, particularly in Chapters 2 to 5 and in Chapter 7. The answer is not entirely simple, since, although topological quantum numbers can provide a correspondence between countable integer quantities and physical observables, this correspondence is not usually exact, and corrections may be more or less important. A second theme, provided by the work on liquid crystals, and on the A phase of superfluid 3He, is the use of topological quantum numbers to classify defects, in situations where the relevant group is finite, rather than isomorphic to the infinite group of integers. The third theme, covered in the last chapter, is the importance of topological concepts in the theory of phase transitions in two dimensions. I have tried in this book to give enough background material to make it accessible to people whose knowledge of quantum mechanics and statistical mechanics is at the level expected in the second year of a US. graduate program in physics. For Chapters 6 and 8 a little knowledge of the theory of finite groups is also necessary. I have not assumed any previous knowledge of topology.

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vi

This book developed from a series of lectures given at the University of Washington in the winter of 1994, and I am particularly grateful to the people who attended those lectures and provided lively discussion of the material presented. Among the people who have particularly helped to sharpen my views of this material are my colleagues and former colleagues Ian Aitchison, Ping Ao, Michael Geller, Qian Niu and Boris Spivak, and my students Junghoon Han, Kiril Tsemekhman, Vadim Tsemekhman and Carlos Wexler. Carlos Wexler and Kiril Tsemekhman also helped by turning many of my figures into postscript files, and all of my students have provided me with instruction in the intelligent use of Latex. I supplemented my lectures by providing copies of classic papers on the subject, and I have done the same in this book. There is a selection of relevant papers, some very old and a few quite new, in the second part of this book. I am grateful to the publishers and authors who have given permission for the reprinting of these papers, and particularly grateful to those authors who have supplied the Publisher with reprints of their papers, which reproduce much better than photocopies of bound periodicals. I am also grateful to members of the Theory of Condensed Matter group at the Cavendish Laboratory for allowing me to copy from their extensive collection of unbound periodicals. The book has been written slowly because I have been much involved in other things during the past four years. Some of these things, concerned with the properties of quantized vortices, have made their appearance in Chapter 3. I am grateful for the help that I have had from my colleagues at the University of Washington, and for the hospitality I have received from those institutions to which I have escaped from my normal responsibilities, the Institute for Theoretical Physics at Santa Barbara, the Aspen Center for Physics, and the Isaac Newton Institute at Cambridge. The National Science Foundation has encouraged this activity by financial support provided through grant number DMR-9528345. Finally I wish to thank my wife Margaret for her support throughout the years, and for tolerating my absences listed above. David Thouless

December 1997

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Contents

Preface

V

1. Introduction

1 1

1.1 Whole numbers in physics 1.2 Quantum numbers due to symmetry and topological quantum numbers 1.3 1.4 1.5 1.6

Topics covered in this book Order parameters and broken symmetry Homotopy classes Defects

3 4 6 10 14

2.

Quantization of Electric Charge 2.1 Magnetic monopoles and electric charge 2.2 Gauge invariance and the Aharonov-Bohm effect

16 16 18

3.

Circulation an d Vortices in Superfluid *He 3.1 Theory of Bose superfluids 3.2 Vortex lines 3.3 Detection of quantized circulation and vortices 3.4 The Magnus force

21 21 26 29 32

4. Superconductivity and Flux Quantization 4.1 Superfluids and superconductors 4.2 Order parameter for superconductors 4.3 4.4 4.5 4.6

London’s equation and flux quantization Types I and I1 superconductors Ginzburg-Landau theory Flux-line lattice

5. Josephson Effects 5.1 Josephson junctions and SQUIDS 5.2 Voltage-frequency relation

35 35 36 37 39 41 44 46 46 52

6. Superfluid 3He 6.1 The nature of the order parameter 6.2 Vortices and circulation in superfluid 3He 6.3 Defects and textures 6.4 Superfluid 3He in thin films and narrow channels

7. The Quantum Hall Effect Topological Quantum Numbers in Nonrelativistic Physics Downloaded from www.worldscientific.com by 37.44.207.38 on 01/17/17. For personal use only.

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

Introduction Proportionality of current density and electric field Bloch’s theorem and the Laughlin argument Chern numbers Long range order in quantum Hall systems Edge states in the integer quantum Hall effect Fractional quantum Hall effect Fractional quantization and degenerate ground states Topology of fractional quantum Hall fluids Coupled quantum Hall systems

68 68 69 71 74 77 79 80 82 83 85 89 89 92 94

8. Solids and Liquid Crystals 8.1 Dislocations in solids 8.2 Order in liquid crystals 8.3 Defects and textures

9. Topological Phase Transitions 9.1 9.2 9.3 9.4 9.5 9.6

55 55 58 64 66

Introduction The vortex induced transition in superfluid helium films Two-dimensional magnetic systems Topological order in solids Superconducting films and layered materials Josephson junction arrays

102 102 103 108 110 112 113 116

References

Reprinted Papers 1. Introduction

137

[1.1] G. Toulouse and M. Klkman, “Principles of a Classification of Defects in Ordered Media”, J . Phys. Lett. (Paris) 37( 1976)L149-51

138

Topological Quantum Numbers in Nonrelativistic Physics Downloaded from www.worldscientific.com by 37.44.207.38 on 01/17/17. For personal use only.

ix [1.2] G.E. Volovik and V.P. Mineev, “Investigation of Singularities in Superfluid He and Liquid Crystals by Homotopic Topology Methods”, Zhur. Eksp. Teor. Fzz. 72, 2256 [Sou. Phys. JETP 45(1977)1186-961

141

2. Quantization of Electric Charge

153

(2.11 P.A.M. Dirac, “Quantised Singularities in the Electromagnetic Field”, Proc. Roy. Soc. London 133(1931)60-72

154

[2.2] Y. Aharonov and D. Bohm, “Significance of Electromagnetic Potentials in the Quantum Theory”, Phys. Rev. 115(1959)485-91

167

3. Circulation and Vortices in Superfluid 4He

175

[3.1] L. Onsager, Nuovo Czmento 6, Suppl. 2(1949)249-50

177

f3.21 W.F. Vinen, “The Detection of Single Quanta of Circulation in Liquid Helium 11”, Proc. Roy. Soc. London A260( 1961)218-36

179

[3.3] G.W. Rayfield and F. Reif, “Evidence for the Creation and Motion of Quantized Vortex Rings in Superfluid Helium”, Phys. Rev. Lett. 11(1963)305-8

199

[3.4] E.J. Yarmchuk, M.J.V. Gordan and R.E. Packard, “Observation of Stationary Vortex Arrays in Rotating Superfluid Helium”, Phys. Rev. Lett. 43(1979)214-7

203

[3.5] D.J. Thouless, P. Ao and Q. Niu, “Transverse Force on a Quantized Vortex in a Superfluid”, Phys. Rev. Lett. 76( 1996)3758-61

207

4. Superconductivity and Flux Quantization

211

[4.1] N. Byers and C.N. Yang, “Theoretical Considerations Concerning Quantized Magnetic Flux in Superconducting Cylinders”, Phys. Rev. Lett. 7( 1961)46-9

212

f4.21 B.S. Deaver, Jr. and W.M. Fairbank, “Experimental Evidence for Quantized Flux in Superconducting Cylinders”, Phys. Rev. Lett. 7( 1961)43-6

216

[4.3] R. Doll and M. Nabauer, “Experimental Proof of Magnetic Flux Quantization in a Superconducting Ring”, Phys. Rev. Lett. 7( 1961)51-2

220

[4.4] C.E. Gough, M.S. Colclough, E.M. Forgan, R.G. Jordan, M. Keene, C.M. Muirhead, A.I.M. Rae, N. Thomas, J.S. Abell, and S. Sutton, “Flux Quantization in a High-Tc Superconductor”, Nature 326( 1987)855

222

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X

5. Josephson Effects

223

[5.1] B.D. Josephson, “Possible New Effects in Superconductive Tunnelling”, Phys. Lett. 1(1962)251-3

225

[5.2] R.C. Jaklevic, J.J. Lambe, A.H. Silver, and J.E. Mercereau, “Quantum Interference from a Static Vector Potential in a Field-Free Region”, Phys. Rev. Lett. 12(1964)274-5

228

[5.3] S. Shapiro, “Josephson Currents in Superconducting Tunnelling: The Effect of Microwaves and Other Observations”, Phys. Rev. Lett. 11(1963)8&2

230

(5.41 D.N. Langenberg and J.R. Schrieffer, “Comments on Quantum-Electrodynamic Corrections to the Electron Charge in Metals”, Phys. Rev. B3(1971)1776-8

233

[5.5] J. S. Tsai, A.K. Jain, and J.E. Lukens, “High-Precision Test of the Universality of the Josephson Voltage-Frequency Relation”, Phys. Rev. Lett. 51(1983)316-9

236

6. Superfluid 3He

24 1

[6.1] P.W. Anderson and G. Toulouse, “Phase Slippage without Vortex Cores: Vortex Textures in Superfluid 3He”, Phys. Rev. Lett. 38( 1977)508-11

242

[6.2] V.M.H. Ruutu, U. Parts, and M. Krusius, “NMR Signatures of Topological Objects in Rotating Superfluid 3He-A”, J. Low. Temp. Phys. 103(1996)331-43

246

[6.3] N.D. Mermin, “Surface Singularities and Superflow in 3He-A”, in Quantum Fluids and Solids, edited by S.M. Rickey, E.D. Adams, and J.W. Dufty (Plenum, New York, 1977), pp. 3-22

259

7. The Quantum Hall Effect

279

[7.1] K.v. Klitzing, G. Dorda and M. Pepper, “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance”, Phys. Rev. Lett. 45( 1980)494-7

281

[7.2] A. Hartland, K. Jones, J.M. Williams, B.L. Gallagher, and T. Galloway, “Direct Comparison of the Quantized Hall Resistance in Gallium Arsenide and Silicon”, Phys. Rev. Lett. 66( 1991)969-73

285

[7.3] R.B. Laughlin, “Quantized Hall Conductivity in Two Dimensions”, Phys. Rev. B23 ( 1981)5632-3

290

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xi

[7.4] J.E. Avron and R. Seiler, ‘(Quantization of the Hall Conductance for General, Multiparticle Schrodinger Hamiltonians” , Phys. Rev. Lett. 54(1985)259-62

292

[7.5] M. Kohmoto, “Topological Invariant and the Quantization of the Hall Conductance”, Ann. Phys. ( N Y ) 160(1985)343-54

296

[7.6] R.B. Laughlin, “Anomalous Quantum Hall Efffect: An Incompressible Quantum Fluid with F’ractionally Charged Excitations” Phys. Rev. Lett. 50( 1983)1395-8

308

17.71 D.J. Thouless and Y.Ge€en, “Fkactional Quantum Hall Effect and Multiple Aharonov-Bohm Periods”, Phys. Rev. Lett. 66( 1991)806-9

312

[7.8] X.G.Wen and A. Zee, “Classification of Abelian Quantum Hall States and Matrix Formulation of Topological Fluids”, Phys. Rev. B46(1992)229O-301

316

8. Solids and Liquid Crystals

329

[8.1] M. KlBman, “Relationship between Burgers Circuit, Volterra Process and Homotopy Groups” J. Phys. Lett. (Paris) 38 (1977)L199-202

330

[8.2] M. Kl6man and L. Michel, “Spontaneous Breaking of Euclidean Invariance and Classification of Topologically Stable Defects and Configurations of Crystals and Liquid Crystals” , Phys. Rev. Lett. 40( 1978)1387-90

334

[8.3] V. PoBnaru and G.Toulouse, “The Crossing of Defects in Ordered Media and the Topology of 3-Manifolds”, J. Phys. 38( 1977)887-95

338

9. Topological Phase Transitions

347

[9.1] J.M. Kosterlitz and D.J. Thouless, “Ordering, Metastability and Phase Transitions in Two-Dimensional Systems”, J. Phys. C6(1973)1181-203

349

[9.2] D.R. Nelson and J.M. Kosterlitz, “Universal Jump in the Superfluid Density of Two-Dimensional Superfluids” , Phys. Rev. Lett. 39(1977)1201-5

372

[9.3] J.M. Kosterlitz, “The Critical Properties of the Two-Dimensional x y Model”, J. Phys. C7( 1974)1046-60

377

[9.4] D.J. Bishop and J.D. Reppy, “Study of the Superfluid Transition in Two-Dimensional *He Films”, Phys. Rev. Lett. 40(1978)1727-30

392

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xii

[9.5] B.I. Halperin and D.R. Nelson, “Theory of Two-Dimensional Melting”, Phys. Rev. Lett. 41( 1978)121-4; Errata, Phys. Rev. Lett. 41(1978)519

396

[9.6] M.R. Beasley, J.E. Mooij, and T.P. Orlando, “Possibility of Vortex-Antivortex Pair Dissociation in Two-Dimensional Superconductors”, Phys. Rev. Lett. 42( 1979)1165-8

40 1

[9.7] S. Doniach and B.A. Huberman, “Topological Excitations in Two-Dimensional Superconductors”, Phys. Rev. Lett. 42( 1979)1169-72

405

[9.8] A.F. Hebard and A.T. Fiory, “Critical-Exponent Measurements of a Two-Dimensional Superconductor”, Phys. Rev. Lett. 50(1983)1603-6

409

[9.9] B.A. Huberman and S. Doniach, “Melting of Two-Dimensional Vortex Lattices”, Phys. Rev. Lett. 43( 1979)950-2

413

Index

417