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H.Kleinert

GAUGE FIELDS IN CONDENSED MATTER

Vol I: SUPERFLOW AND VORTEX LINES       Pages 1--742
World Scientific, Singapore 1989
[ISBN 9971-50-210-0, 9971-50-210-9 (paperback)]

• PREFACE                                                                         vii

  • INTRODUCTION                                                       1

  • Part I: Fluctuating Fields and Random Chains       5

    • Statistical Physics                 7
      • Classical statistics                 7
      • Classical fields                 10
      • Free fields                 12
      • Field correlation functions                 18
      • Wick's theorem                 23
      • Quantum statistics                 25
    • Perturbation Theory                 31
      • General formalism                 31
      • Vacuum contribution Z[0]                 34
      • Two-point function and self-energy graphs                 39
      • Four-point and vertex functions                 42
      • Special features of local theories                 45
      • Momentum space diagrams                 47
      • Relation between vacuum diagrams and correlation functions                 53
    • Gauge Invariant Field Theories                 55
      • Free magnetic fields                 55
      • Gauge-fixing factors                 62
      • Charged scalar fields                 66
      • Feynman diagrams                 68
      • Non-Abelian gauge theories                 72
      • Non-Abelian field theories involving the space group                 83
    • Low Temperature Expansion Around Extrema                 92
      • Extremal field configurations                 92
      • Saddle-point approximation of integrals                 95
      • Saddle-point approximation for fluctuating fields                 97
      • Connected correlation functions                 101
      • Symmetry properties of extremal field configurations: currents and conservation laws             105
      • Appendix 4A. Vacuum diagrams                 110
    • Effective Energy                 114
      • Extremal principle                 114
      • Tree graphs and vertex functions                 116
      • Graphical content of the effective energy                 120
      • The one-particle irreducibility of Gamma[0]
      • Proper self-energy                 128
      • Appendix 5A: Graphs contributing to (1/T) W[j]                 130
    • Feynman Diagrams and Random Walks                 133
      • General remarks                 133
      • Single random walk                 136
      • Material random chains in thermal equilibrium                 145
      • Grand canonical ensemble of closed oriented random chains and disorder fields             149
      • Critical region                 151
      • Closed random chains and loop diagrams                 153
      • Open random chains and lines in Feynman diagrams                 154
      • Field fluctuations versus line fluctuations                 156
      • Ensembles of fluctuating lines in the critical region                 162
      • General properties of lattice Green functions                 164
      • Calculation of lattice Green functions via the hopping expansion                 167
      • Improving the convergence of hopping expansion                 180
      • Exact one- and two-dimensional Coulomb Green function on a square lattice                 193
      • Fluctuation determinants                 204
      • Finite-size effects on the fluctuation determinant and quantum statistics of a free field                 212
      • Short-range steric interactions between chains versus field interactions                 220
      • Incorporating long-range forces                 226
      • Interacting random chains and Feynman diagrams                 231
      • Appendix 6A. Green functions and fluctuation determinants for non-cubic lattices                 232
    
    

  • Part II: Gauge Fields in Superfluid Helium                 253

    • Superfluidity                 255
      • Basic properties                 255
      • Field description of superfluid phase transition                 258
      • Nambu-Goldstone modes and superfluid velocity                 266
      • Straight vortex line                 267
      • Vortex lines of arbitrary shape, vortex density                 275
      • Branching vortex lines                 280
      • Hydrodynamic energy of vortex lines                 282
      • Vector potential versus phase angle                 290
    • Grand Canonical Ensembles of Interacting Vortex Lines                 300
      • Disorder fields for vortex lines                 300
      • Thermodynamic quantities                 302
      • Nambu-Goldstone modes of the disorder field                 304
      • Inclusion of hydrodynamic forces                 304
    • Ginzburg-Landau Theory of Superconductivity                 307
      • London equations                 307
      • Order parameter                 309
      • Order field                 310
      • Ginzburg-Landau energy                 312
      • Gorkov's derivation                 313
      • Ginzburg-Landau equations                 316
      • Critical magnetic field                 319
      • The two length scales and type I versus type II superconductivity                 320
      • Single vortex line and critical field H_c1>                 324
      • The field H_c2 where superconductivity is destroyed                 330
      • Fluctuation corrections                 332
      • Phase transition in disorder field theory and disorder version of the Meissner effect                 341
    • Lattice Model of the Superfluid Phase Transition                 346
      • Motivation                 346
      • The XY model                 348
      • High temperature expansion                 352
      • Physical interpretation                 392
      • Estimate of critical properties via the ratio test                 393
      • Appendix 4A. Graphs of order beta^{10}>                 401
    • Mean Field Methods for Lattice Models                 409
      • Variational estimates of thermodynamic properties                 409
      • Equivalent field formulation of the XY model                 417
      • Fluctuation modes                 423
      • One-loop correction to the mean-field energy                 430
      • Local formulation with two complex fields                 436
      • Correlation functions of the XY model                 440
      • Improvement of the mean-field-plus-one-loop approximation
      • Effective energy approach                 449
      • Appendix 5A. Generalization to classical O(n) spin models                 462
    • Vortex Lines in the XY Model                 466
      • Integer-valued gauge fields                 466
      • Vortex lines                 474
      • Formal low temperature limit of the vortex-line representation                 480
      • Improved treatment of the low temperature limit                 485
    • The Villain Approximation and the Villain Model                 489
      • The Villain approximation                 489
      • The Villain model                 498
      • High and low temperature expansions of the Villain model                 499
      • Monte Carlo simulation of the Villain model                 518
      • Comparison of the XY model with its Villain approximation                 520
      • The XY model versus Villain model of superfluidity                 535
      • Perturbation expansions of the XY model (1/beta-expansion)                 537
      • Hartree-Fock resummation                 540
      • Comparison with the mean-field-plus-loop approach                 544
      • Vortex corrections                 546
      • Appendix 7A. The low temperature expansion of the Villain approximation                 550
    • Gauge Fields of Vortex Lines                 553
      • Vortex-gauge field and vortex-gauge invariance                 553
      • Physical meaning of vortex-gauge invariance                 554
      • Continuum description                 557
      • Vortex gauge theory for the Villain model -- double gauge theory                 559
      • Derivation of energy of vortex lines from vortex-gauge fields                 563
      • Continuum limit                 565
    • Correlation Functions                 568
      • Path-integral representation                 568
      • Monopole gauge field -- triple gauge theory                 570
      • Other representation of correlation functions in the Villain model                 573
      • High temperature expansion for the two-point correlation function and susceptibility                 578
      • Ratio test for the susceptibility                 590
      • Isotropy of high temperature correlation functions at large distances                 591
      • Low temperature expansion of the correlation function in the Villain model                 595
    • Gas of Monopoles                 598
      • Z_n Villain model                 599
      • Villain model in a generalized external magnetic field                 601
      • Relation to lattice gauge theory                 602
      • Z_n cosine models                 608
      • High temperature series of Z_n models             608
      • Z_2 (Ising) model, Z_3 (three-state Potts) model, Z_4 (Ashkin-Teller) model             612
    • Two-Dimensional Considerations                 618
      • Films of superfluid helium                 618
      • High temperature expansion of D=2 XY model                 619
      • Vortex representation in two dimensions                 620
      • Low temperature expansion of the Villain model                 629
      • Determination of the transition point                 639
      • Villain approximation for D=2                 641
      • Vortex corrections in D=2 XY model                 643
      • Correlation functions in the 2-dimensional Villain model                 653
      • Explicit calculation of the two-point correlation function at low temperatures             658
      • The renormalization group                 663
      • Coherence length                 669
      • Universality of the critical stiffness                 672
      • Exact formula for the renormalization of superfluid density by vortices             675
      • Equivalence of the Villain model to the limiting case of the sine-Gordon model             679
      • Estimate of the critical temperature in the sine-Gordon model                 681
      • Self-consistent approximation to the coherence length of the XY and sine-Gordon models             685
      • Surface roughening and crystal growth                 688
    • Lattice Derivation of Disorder Field Theory                 694
      • Equivalence of the XY model to a lattice superconductor                 694
      • From lattice superconductor to Ginzburg-Landau theory                 697
      • Comparison with approximation disorder theory                 701
      • Natural core energies                 703
      • Type-I or type-II disorder theory                 707
      • Quantum vortex dynamics in films of superfluid helium                 709
    • Vortex Lines in the Ginzburg-Landau Theory and Their Disorder Field Theory                 716
      • General discussion                 716
      • Vortex-line representation of a general lattice superconductor                 717
      • Disorder field theory of vortex lines in lattice superconductors                 717
      • Limiting case                 726
      • Phase transitions in lattice superconductors                 729
      • Tricritical point in a superconductor                 736
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  Vol II: STRESSES AND DEFECTS