Topological quantum numbers are distinguished from quantum numbers based on symmetry because they are insensitive to the imperfections of the systems in which they are observed. They have become very important in precision measurements in recent years, and provide the best measurements of voltage and electrical resistance. This book describes the theory of such quantum numbers, starting with Dirac's argument for the quantization of electric charge, and continuing with discussions on the helium superfluids, flux quantization and the Josephson effect in superconductors, the quantum Hall effect, solids and liquid crystals, and topological phase transitions. The accompanying reprints include some of the classic experimental and theoretical papers in this area.Physicists both experimental and theoretical who are interested in the topic will find this book an invaluable reference.
Preface v
1. Introduction 1(15) 1.1 Whole numbers in physics 1(2) 1.2 Quantum numbers due to symmetry and topological quantum numbers 3(1) 1.3 Topics covered in this book 4(2) 1.4 Order parameters and broken symmetry 6(4) 1.5 Homotopy classes 10(4) 1.6 Defects 14(2)
2. Quantization of Electric Charge 16(5) 2.1 Magnetic monopoles and electric charge 16(2) 2.2 Gauge invariance and the Aharonov-Bohm effect 18(3)
3. Circulation and Vortices in Superfluid 4(He) 21(14) 3.1 Theory of Bose superfluids 21(5) 3.2 Vortex lines 26(3) 3.3 Detection of quantized circulation and vortices 29(3) 3.4 The Magnus force 32(3)
4. Superconductivity and Flux Quantization 35(11) 4.1 Superfluids and superconductors 35(1) 4.2 Order parameter for superconductors 36(1) 4.3 Londons equation and flux quantization 37(2) 4.4 Types I and II superconductors 39(2) 4.5 Ginzburg-Landau theory 41(3) 4.6 Flux-line lattice 44(2)
5. Josephson Effects 46(9) 5.1 Josephson junctions and SQUIDs 46(6) 5.2 Voltage-frequency relation 52(3)
6. Superfluid 3(He) 55(13) 6.1 The nature of the order parameter 55(3) 6.2 Vortices and circulation in superfluid 3(He) 58(6) 6.3 Defects and textures 64(2) 6.4 Superfluid 3(He) in thin films and narrow channels 66(2)
7. The Quantum Hall Effect 68(21) 7.1 Introduction 68(1) 7.2 Proportionality of current density and electric field 69(2) 7.3 Blochs theorem and the Laughlin argument 71(3) 7.4 Chern numbers 74(3) 7.5 Long range order in quantum Hall systems 77(2) 7.6 Edge states in the integer quantum Hall effect 79(1) 7.7 Fractional quantum Hall effect 80(2) 7.8 Fractional quantization and degenerate ground states 82(1) 7.9 Topology of fractional quantum Hall fluids 83(2) 7.10 Coupled quantum Hall systems 85(4)
8. Solids and Liquid Crystals 89(13) 8.1 Dislocations in solids 89(3) 8.2 Order in liquid crystals 92(2) 8.3 Defects and textures 94(8)
9. Topological Phase Transitions 102(14) 9.1 Introduction 102(1) 9.2 The vortex induced transition in superfluid helium films 103(5) 9.3 Two-dimensional magnetic systems 108(2) 9.4 Topological order in solids 110(2) 9.5 Superconducting films and layered materials 112(1) 9.6 Josephson junction arrays 113(3) References 116(21) Reprinted Papers 137(280)
1. Introduction 137(16) G. Toulouse M. Kleman (1.1) Principles of a Classification of Defects in Ordered Media, J. Phys. Lett. (Paris) 37(1976)L149-51 138(3) G.E. Volovik V.P. Mineev (1.2) 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(12)
2. Quantization of Electric Charge 153(22) P.A.M. Dirac (2.1) Quantised Singularities in the Electromagnetic Field, Proc. Roy. Soc. London 133(1931)60-72 154(13) Y. Aharonov D. Bohm (2.2) Significance of Electromagnetic Potentials in the Quantum Theory, Phys. Rev. 115(1959)485-91 167(8)
3. Circulation and Vortices in Superfluid 4(He) 175(36) L. Onsager (3.1) Nuovo Cimento 6, Suppl. 2(1949)249-50 177(2) W.F. Vinen (3.2) The Detection of Single Quanta of Circulation in Liquid Helium II, Proc. Roy. Soc. London A260(1961)218-36 179(20) G.W. Rayfield F. Reif (3.3) Evidence for the Creation and Motion of Quantized Vortex Rings in Superfluid Helium, Phys. Rev. Lett. 11(1963)305-8 199(4) E.J. Yarmchuk M.J.V. Gordan R.E. Packard (3.4) Observation of Stationary Vortex Arrays in Rotating Superfluid Helium, Phys. Rev. Lett. 43(1979)214-7 203(4) D.J. Thouless P. Ao Q. Niu (3.5) Transverse Force on a Quantized Vortex in a Superfluid, Phys. Rev. Lett. 76(1996)3758-61 207(4)
4. Superconductivity and Flux Quantization 211(12) N. Byers C.N. Yang (4.1) Theoretical Considerations Concerning Quantized Magnetic Flux in Superconducting Cylinders, Phys. Rev. Lett. 7(1961)46-9 212(4) B.S. Deaver, Jr. W.M. Fairbank (4.2) Experimental Evidence for Quantized Flux in Superconducting Cylinders, Phys. Rev. Lett. 7(1961)43-6 216(4) R. Doll M. Nabauer (4.3) Experimental Proof of Magnetic Flux Quantization in a Superconducting Ring, Phys. Rev. Lett. 7(1961)51-2 220(2) 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 S. Sutton (4.4) Flux Quantization in a High-Tc Superconductor, Nature 326(1987)855 222(1)
5. Josephson Effects 223(18) B.D. Josephson (5.1) Possible New Effects in Superconductive Tunnelling, Phys. Lett. 1(1962)251-3 225(3) R.C. Jaklevic J.J. Lambe A.H. Silver J.E. Mercereau (5.2) Quantum Interference from a Static Vector Potential in a Field-Free Region, Phys. Rev. Lett. 12(1964)274-5 228(2) S. Shapiro (5.3) Josephson Currents in Superconducting Tunnelling: The Effect of Microwaves and Other Observations, Phys. Rev. Lett. 11(1963)80-2 230(3) D.N. Langenberg J.R. Schrieffer (5.4) Comments on Quantum-Electrodynamic Corrections to the Electron Charge in Metals, Phys. Rev. B3(1971)1776-8 233(3) J.S. Tsai A.K. Jain J.E. Lukens (5.5) High-Precision Test of the Universality of the Josephson Voltage-Frequency Relation, Phys. Rev. Lett. 51(1983)316-9 236(5)
6. Superfluid 3(He) 241(38) P.W. Anderson G. Toulouse (6.1) Phase Slippage without Vortex Cores: Vortex Textures in Superfluid 3(He), Phys. Rev. Lett. 38(1977)508-11 242(4) V.M.H. Ruutu U. Parts M. Krusius (6.2) NMR Signatures of Topological Objects in Rotating Superfluid 3(He)-A, J. Low. Temp. Phys. 103(1996)331-43 246(13) N.D. Mermin (6.3) Surface Singularities and Superflow in 3(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(20)
7. The Quantum Hall Effect 279(50) K.v. Klitzing G. Dorda M. Pepper (7.1) New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance, Phys. Rev. Lett. 45(1980)494-7 281(4) A. Hartland K. Jones J.M. Williams B.L. Gallagher T. Galloway (7.2) Direct Comparison of the Quantized Hall Resistance in Gallium Arsenide and Silicon, Phys. Rev. Lett. 66(1991)969-73 285(5) R.B. Laughlin (7.3) Quantized Hall Conductivity in Two Dimensions, Phys. Rev. B23(1981)5632-3 290(2) J.E. Avron R. Seiler (7.4) Quantization of the Hall Conductance for General, Multiparticle Schrodinger Hamiltonians, Phys. Rev. Lett. 54(1985)259-62 292(4) M. Kohmoto (7.5) Topological Invariant and the Quantization of the Hall Conductance, Ann. Phys. (NY) 160(1985)343-54 296(12) R.B. Laughlin (7.6) Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, Phys. Rev. Lett. 50(1983)1395-8 308(4) D.J. Thouless Y. Gefen (7.7) Fractional Quantum Hall Effect and Multiple Aharonov-Bohm Periods, Phys. Rev. Lett. 66(1991)806-9 312(4) X.G. Wen A. Zee (7.8) Classification of Abelian Quantum Hall States and Matrix Formulation of Topological Fluids, Phys. Rev. B46(1992)2290-301 316(13)
8. Solids and Liquid Crystals 329(18) M. Kleman (8.1) Relationship between Burgers Circuit, Volterra Process and Homotopy Groups, J. Phys. Lett. (Paris) 38(1977)L199-202 330(4) M. Kleman L. Michel (8.2) 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(4) V. Poenaru G. Toulouse (8.3) The Crossing of Defects in Ordered Media and the Topology of 3-Manifolds, J. Phys. 38(1977)887-95 338(9)
9. Topological Phase Transitions 347(70) J.M. Kosterlitz D.J. Thouless (9.1) Ordering, Metastability and Phase Transitions in Two-Dimensional Systems, J. Phys. C6(1973)1181-203 349(23) D.R. Nelson J.M. Kosterlitz (9.2) Universal Jump in the Superfluid Density of Two-Dimensional Superfluids, Phys. Rev. Lett. 39(1977)1201-5 372(5) J.M. Kosterlitz (9.3) The Critical Properties of the Two-Dimensional xy Model, J. Phys. C7(1974)1046-60 377(15) D.J. Bishop J.D. Reppy (9.4) Study of the Superfluid Transition in Two-Dimensional 4(He) Films, Phys. Rev. Lett. 40(1978)1727-30 392(4) B.I. Halperin D.R. Nelson (9.5) Theory of Two-Dimensional Melting, Phys. Rev. Lett. 41(1978)121-4; Errata, Phys. Rev. Lett. 41(1978)519 396(5) M.R. Beasley J.E. Mooij T.P. Orlando (9.6) Possibility of Vortex-AntiVortex Pair Dissociation in Two-Dimensional Superconductors, Phys. Rev. Lett. 42(1979)1165-8 401(4) S. Doniach B.A. Huberman (9.7) Topological Excitations in Two-Dimensional Superconductors, Phys. Rev. Lett. 42(1979)1169-72 405(4) A.F. Hebard A.T. Fiory (9.8) Critical-Exponent Measurements of a Two-Dimensional Superconductor, Phys. Rev. Lett. 50(1983)1603-6 409(4) B.A. Huberman S. Doniach (9.9) Melting of Two-Dimensional Vortex Lattices, Phys. Rev. Lett. 43(1979)950-2 413(4) Index 417
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