Topological Quantum Numbers in Nonrelativistic Physics

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

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Contents

Preface

v

1.

1

Introduction 1.1 1.2 1.3 1.4 1.5 1.6

Whole numbers in physics Quantum numbers due to symmetry and topological quantum numbers Topics covered in this book Order parameters and broken symmetry Homotopy classes Defects

1 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 and Vortices in Superfluid 4 He 3.1 Theory of Bose superfluids 3.2 Vortex lines

21 21 26

3.3 3.4

29 32

4.

5.

Detection of quantized circulation and vortices The Magnus force

Superconductivity and Flux Quantization

35

4.1 4.2 4.3 4.4 4.5

Superfluids and superconductors Order parameter for superconductors London's equation and flux quantization Types I and II superconductors Ginzburg-Landau theory

35 36 37 39 41

4.6

Flux-line lattice

44

Josephson Effects

46

5.1 5.2

46 52

Josephson junctions and SQUIDs Voltage-frequency relation

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6.

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

55 55 58 64 66

7.

The Q u a n t u m Hall Effect 7.1 Introduction 7.2 Proportionality of current density and electric

68 68 69

7.3 7.4 7.5 7.6 7.7 7.8

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

7.9 Topology of fractional quantum Hall 7.10 Coupled quantum Hall systems

71 74 77 79 80 82 fluids

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 Introduction 9.2 The vortex induced transition in superfluid helium films 9.3 Two-dimensional magnetic systems 9.4 Topological order in solids 9.5 Superconducting films and layered materials 9.6 Josephson junction arrays

83 85 89 89 92 94

References

102 102 103 108 110 112 113 116

Reprinted Papers 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 4 He

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, C M . 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

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)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 3 He

241

[6.1] P.W. Anderson and G. Toulouse, "Phase Slippage without Vortex Cores: Vortex Textures in Superfluid 3 He", 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 3 He-A", J. Low. Temp. Phys. 103(1996)331-43

246

3

[6.3] N.D. Mermin, "Surface Singularities and Superflow in He-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 4 He 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