Gauge fields in condensed matter

INTRODUCTION ......Page 2
Part I: Fluctuating Fields and Random Chains ......Page 6
+ Classical statistics ......Page 8
+ Classical fields ......Page 11
+ Free fields ......Page 13
+ Field correlation functions ......Page 19
+ Wick's theorem ......Page 24
+ Quantum statistics ......Page 26
+ General formalism ......Page 32
+ Vacuum contribution Z[0] ......Page 35
+ Two-point function and self-energy graphs ......Page 40
+ Four-point and vertex functions ......Page 43
+ Special features of local theories ......Page 46
+ Momentum space diagrams ......Page 48
+ Relation between vacuum diagrams and correlation functions ......Page 54
+ Free magnetic fields ......Page 56
+ Gauge-fixing factors ......Page 63
+ Charged scalar fields ......Page 67
+ Feynman diagrams ......Page 69
+ Non-Abelian gauge theories ......Page 73
+ Non-Abelian field theories involving the space group ......Page 84
+ Extremal field configurations ......Page 93
+ Saddle-point approximation of integrals ......Page 96
+ Saddle-point approximation for fluctuating fields ......Page 98
+ Connected correlation functions ......Page 102
+ Symmetry properties of extremal field configurations: currents and conservation laws ......Page 106
+ Appendix 4A. Vacuum diagrams ......Page 111
+ Extremal principle ......Page 115
+ Tree graphs and vertex functions ......Page 117
+ Graphical content of the effective energy ......Page 121
+ The one-particle irreducibility of Gamma ......Page 125
+ Proper self-energy ......Page 129
+ Appendix 5A: Graphs contributing to (1/T) W[j] ......Page 131
+ General remarks ......Page 134
+ Single random walk ......Page 137
+ Material random chains in thermal equilibrium ......Page 146
+ Grand canonical ensemble of closed oriented random chains and disorder fields ......Page 150
+ Critical region ......Page 152
+ Closed random chains and loop diagrams ......Page 154
+ Open random chains and lines in Feynman diagrams ......Page 155
+ Field fluctuations versus line fluctuations ......Page 157
+ Ensembles of fluctuating lines in the critical region ......Page 163
+ General properties of lattice Green functions ......Page 165
+ Calculation of lattice Green functions via the hopping expansion ......Page 168
+ Improving the convergence of hopping expansion ......Page 181
+ Exact one- and two-dimensional Coulomb Green function on a square lattice ......Page 194
+ Fluctuation determinants ......Page 205
+ Finite-size effects on the fluctuation determinant and quantum statistics of a free field ......Page 213
+ Short-range steric interactions between chains versus field interactions ......Page 221
+ Incorporating long-range forces ......Page 227
+ Interacting random chains and Feynman diagrams ......Page 232
+ Appendix 6A. Green functions and fluctuation determinants for non-cubic lattices ......Page 233
Part II: Gauge Fields in Superfluid Helium ......Page 254
+ Basic properties ......Page 256
+ Field description of superfluid phase transition ......Page 259
+ Nambu-Goldstone modes and superfluid velocity ......Page 267
+ Straight vortex line ......Page 268
+ Vortex lines of arbitrary shape, vortex density ......Page 276
+ Branching vortex lines ......Page 281
+ Hydrodynamic energy of vortex lines ......Page 283
+ Vector potential versus phase angle ......Page 291
+ Disorder fields for vortex lines ......Page 301
+ Thermodynamic quantities ......Page 303
+ Inclusion of hydrodynamic forces ......Page 305
+ London equations ......Page 308
+ Order parameter ......Page 310
+ Order field ......Page 311
+ Ginzburg-Landau energy ......Page 313
+ Gorkov's derivation ......Page 314
+ Ginzburg-Landau equations ......Page 317
+ Critical magnetic field ......Page 320
+ The two length scales and type I versus type II superconductivity ......Page 321
+ Single vortex line and critical field H_c1> ......Page 325
+ The field H_c2 where superconductivity is destroyed ......Page 331
+ Fluctuation corrections ......Page 333
+ Phase transition in disorder field theory and disorder version of the Meissner effect ......Page 342
+ Motivation ......Page 347
+ The XY model ......Page 349
+ High temperature expansion ......Page 353
+ Physical interpretation ......Page 393
+ Estimate of critical properties via the ratio test ......Page 394
+ Appendix 4A. Graphs of order beta^{10}> ......Page 402
+ Variational estimates of thermodynamic properties ......Page 410
+ Equivalent field formulation of the XY model ......Page 418
+ Fluctuation modes ......Page 424
+ One-loop correction to the mean-field energy ......Page 431
+ Local formulation with two complex fields ......Page 437
+ Correlation functions of the XY model ......Page 441
+ Improvement of the mean-field-plus-one-loop approximation ......Page 443
+ Effective energy approach ......Page 450
+ Appendix 5A. Generalization to classical O(n) spin models ......Page 463
+ Integer-valued gauge fields ......Page 467
+ Vortex lines ......Page 475
+ Formal low temperature limit of the vortex-line representation ......Page 481
+ Improved treatment of the low temperature limit ......Page 486
+ The Villain approximation ......Page 490
+ The Villain model ......Page 499
+ High and low temperature expansions of the Villain model ......Page 500
+ Monte Carlo simulation of the Villain model ......Page 519
+ Comparison of the XY model with its Villain approximation ......Page 521
+ The XY model versus Villain model of superfluidity ......Page 536
+ Perturbation expansions of the XY model (1/beta-expansion) ......Page 538
+ Hartree-Fock resummation ......Page 541
+ Comparison with the mean-field-plus-loop approach ......Page 545
+ Vortex corrections ......Page 547
+ Appendix 7A. The low temperature expansion of the Villain approximation ......Page 551
+ Vortex-gauge field and vortex-gauge invariance ......Page 554
+ Physical meaning of vortex-gauge invariance ......Page 555
+ Continuum description ......Page 558
+ Vortex gauge theory for the Villain model -- double gauge theory ......Page 560
+ Derivation of energy of vortex lines from vortex-gauge fields ......Page 564
+ Continuum limit ......Page 566
+ Path-integral representation ......Page 569
+ Monopole gauge field -- triple gauge theory ......Page 571
+ Other representation of correlation functions in the Villain model ......Page 574
+ High temperature expansion for the two-point correlation function and susceptibility ......Page 579
+ Ratio test for the susceptibility ......Page 591
+ Isotropy of high temperature correlation functions at large distances ......Page 592
+ Low temperature expansion of the correlation function in the Villain model ......Page 596
o Gas of Monopoles ......Page 599
+ Z_n Villain model ......Page 600
+ Villain model in a generalized external magnetic field ......Page 602
+ Relation to lattice gauge theory ......Page 603
+ High temperature series of Z_n models ......Page 609
+ Z_2 (Ising) model, Z_3 (three-state Potts) model, Z_4 (Ashkin-Teller) model ......Page 613
+ Films of superfluid helium ......Page 619
+ High temperature expansion of D=2 XY model ......Page 620
+ Vortex representation in two dimensions ......Page 621
+ Low temperature expansion of the Villain model ......Page 630
+ Determination of the transition point ......Page 640
+ Villain approximation for D=2 ......Page 642
+ Vortex corrections in D=2 XY model ......Page 644
+ Correlation functions in the 2-dimensional Villain model ......Page 654
+ Explicit calculation of the two-point correlation function at low temperatures ......Page 659
+ The renormalization group ......Page 664
+ Coherence length ......Page 670
+ Universality of the critical stiffness ......Page 673
+ Exact formula for the renormalization of superfluid density by vortices ......Page 676
+ Equivalence of the Villain model to the limiting case of the sine-Gordon model ......Page 680
+ Estimate of the critical temperature in the sine-Gordon model ......Page 682
+ Self-consistent approximation to the coherence length of the XY and sine-Gordon models ......Page 686
+ Surface roughening and crystal growth ......Page 689
+ Equivalence of the XY model to a lattice superconductor ......Page 695
+ From lattice superconductor to Ginzburg-Landau theory ......Page 698
+ Comparison with approximation disorder theory ......Page 702
+ Natural core energies ......Page 704
+ Type-I or type-II disorder theory ......Page 708
+ Quantum vortex dynamics in films of superfluid helium ......Page 710
+ General discussion ......Page 717
+ Disorder field theory of vortex lines in lattice superconductors ......Page 718
+ Limiting case ......Page 727
+ Phase transitions in lattice superconductors ......Page 730
+ Tricritical point in a superconductor ......Page 737