Topological Quantum

Numbers in

Nonrelativistic

Physics

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

V t b World Scientific » • Sinaapore • New Jersey London • Singapore • New Jersey • London • Hong Kong

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Contents

Preface v

1. Introduction 1

1.1 Whole numbers in physics 1

1.2 Quantum numbers due to symmetry and topological quantum numbers 3

1.3 Topics covered in this book 4

1.4 Order parameters and broken symmetry 6

1.5 Homotopy classes 10

1.6 Defects 14

2. Quantization of Electric Charge 16

2.1 Magnetic monopoles and electric charge 16

2.2 Gauge invariance and the Aharonov-Bohm effect 18

3. Circulation and Vortices in Superfluid 4He 21

3.1 Theory of Bose superfluids 21

3.2 Vortex lines 26

3.3 Detection of quantized circulation and vortices 29

3.4 The Magnus force 32

4. Superconductivity and Flux Quantization 35

4.1 Superfluids and superconductors 35

4.2 Order parameter for superconductors 36

4.3 London's equation and flux quantization 37

4.4 Types I and II superconductors 39

4.5 Ginzburg-Landau theory 41

4.6 Flux-line lattice 44

5. Josephson Effects 46

5.1 Josephson junctions and SQUIDs 46

5.2 Voltage-frequency relation 52

V l l l

6. Superfluid 3He 55

6.1 The nature of the order parameter 55

6.2 Vortices and circulation in superfluid 3He 58

6.3 Defects and textures 64

6.4 Superfluid 3He in thin films and narrow channels 66

7. The Quantum Hall Effect 68

7.1 Introduction 68

7.2 Proportionality of current density and electric field 69

7.3 Bloch's theorem and the Laughlin argument 71

7.4 Chern numbers 74

7.5 Long range order in quantum Hall systems 77

7.6 Edge states in the integer quantum Hall effect 79

7.7 Fractional quantum Hall effect 80

7.8 Fractional quantization and degenerate ground states 82

7.9 Topology of fractional quantum Hall fluids 83

7.10 Coupled quantum Hall systems 85

8. Solids and Liquid Crystals 89

8.1 Dislocations in solids 89

8.2 Order in liquid crystals 92

8.3 Defects and textures 94

9. Topological Phase Transitions 102

9.1 Introduction 102

9.2 The vortex induced transition in superfluid helium films 103

9.3 Two-dimensional magnetic systems 108

9.4 Topological order in solids 110

9.5 Superconducting films and layered materials 112

9.6 Josephson junction arrays 113

References 116

R e p r i n t e d P a p e r s

1. Introduction 137

[1.1] G. Toulouse and M. Kleman, "Principles of a Classification of

Defects in Ordered Media", J. Phys. Lett. (Paris) 37(1976)L149-51 138

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. Fiz. 72, 2256 [Sov. Phys. JETP 45(1977)1186-96] 141

2. Quantization of Electric Charge 153

[2.1] 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. Böhm, "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 Cimento 6, Suppl. 2(1949)249-50 177

[3.2] W.F. Vinen, "The Detection of Single Quanta of Circulation in Liquid Helium II", 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

[4.2] 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. Näbauer, "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, CM. Muirhead, A.I.M. Rae, N. Thomas, J.S. Ab ell, and S. Sutton, "Flux Quantization in a High-Tc Superconductor", Nature 326(1987)855 222

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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)80-2 230

[5.4] 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 241

[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, Ü. 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. Trickey, 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, К. 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

XI

[7.4] J.E. Avron and R. Seiler, "Quantization of the Hall Conductance for General, Multiparticle Schrödinger Hamiltonians", Phys. Rev. Lett. 54(1985)259-62 292

[7.5] M. Kohmoto, "Topological Invariant and the Quantization of the Hall Conductance", Ann. Phys. (NY) 160(1985)343-54 296

[7.6] R.B. Laughlin, "Anomalous Quantum Hall Efffect: An Incompressible Quantum Fluid with Fractionally Charged Excitations", Phys. Rev. Lett. 50(1983)1395-8 308

[7.7] D.J. Thouless and Y.Gefen, "Fractional 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)2290-301 316

8. Solids and Liquid Crystals 329

[8.1] M. Kleman, "Relationship between Burgers Circuit, Volterra Process and Homotopy Groups", J. Phys. Lett. (Paris) 38(1977)L199-202 330

[8.2] M. Kleman 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. Poenaru 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 xy 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 4He Films", Phys. Rev. Lett. 40(1978)1727-30 392

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)1165-8 401

[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