Gauge fields in condensed matter

Part III: Gauge Fields in solids ......Page 2
o Displacement and strain ......Page 4
o Elastic energy ......Page 5
o Stress ......Page 13
o External body forces ......Page 17
o Elastic Green function ......Page 21
o Two-dimensional elasticity ......Page 27
o Appendix 1A. The symmetry classes of the elastic matrix ......Page 31
o General remarks ......Page 36
o Dislocation lines and Burgers vector ......Page 41
o Disclinations and the Frank vector ......Page 46
o Interdependence of dislocations and disclinations ......Page 49
o Defect lines with infinitesimal discontinuities in continuous media ......Page 50
o Multivaluedness of the displacement field ......Page 51
o Smoothness properties of the displacement field and Weingarten's theorem ......Page 53
o Integrability considerations ......Page 57
o Dislocation and disclination densities ......Page 60
o Mnemonic procedures for constructing defect densities ......Page 63
o Branching defect lines ......Page 66
o Defect density and incompatibility ......Page 67
o Defects in two dimensions ......Page 73
o Strain and stress around dislocation lines ......Page 77
o Elastic interaction energy between two dislocation lines ......Page 84
o Elastic partition function ......Page 87
o Helicity decomposition of a vector field ......Page 89
o Helicity decomposition of a tensor field ......Page 94
o Helicity form of the magnetic energy ......Page 99
o Helicity form of the stress energy ......Page 101
o The two-dimensional case ......Page 109
o The symmetric stress gauge field ......Page 113
o Elastic partition function for a fixed general defect distribution ......Page 118
o Two-dimensional defects ......Page 121
o Glide ......Page 124
o Kinks ......Page 125
o General conservation motion ......Page 126
o Cross slip ......Page 127
o Dislocation sources ......Page 128
o Intersecting lines and jogs ......Page 130
o Basic energetic considerations of branching of dislocation lines ......Page 131
o Anchored branch points ......Page 133
* Some General Properties of the Melting Process ......Page 134
o Historical notes ......Page 135
o The Lindemann criterion ......Page 137
o Review of Debye's theory of specific heat ......Page 141
o Quantum corrections to the Lindemann parameter ......Page 152
o Classical melting ......Page 159
o Lattice expansion up to the melting transition ......Page 161
o Softening of elastic constants ......Page 172
o Two-dimensional crystals ......Page 183
o Appendix 7A. Some lattice properties ......Page 189
o Appendix 7B. Frequency distributions ......Page 191
* First Attempt at a Disordered Field Theory of Defect Melting ......Page 197
o Disorder fields of dislocation lines ......Page 198
o Fluctuation induced first-order transition ......Page 199
o Inclusion of stress and the Meissner effect ......Page 211
o Other possible mechanics to make a transition first order ......Page 217
o Disorder fields for disclination lines ......Page 220
o Setting up the model ......Page 230
o Defect representation of lattice model ......Page 235
o An XY type model of defect melting ......Page 258
o Appendix 9A. Derivation of defect energy (9.109) from stress energy (9.90) ......Page 267
* Defect Gauge Fields ......Page 272
o Gauge fixing ......Page 273
o Physical content of integer-valued defect gauge invariance ......Page 279
o Interaction energy between defect lines from the defect gauge field ......Page 284
o The defect model as an approximation to a first-principle N-body partition function ......Page 288
o High temperature expansion ......Page 293
o Low temperature expansion ......Page 307
o Appendix 11A. Calculation of the Green function ......Page 316
o Lowest order results for D=2 ......Page 335
o Lowest order results for D = 3 ......Page 337
o Stress and defect corrections in isotropic materials ......Page 339
o Anisotropic cubic materials ......Page 342
o Monte Carlo study of the melting model (Villain type) ......Page 346
* The Melting Model of the Cosine Type ......Page 372
o Inequality for free energy and the mean-field approximation ......Page 373
o Fluctuations around the mean-field solution ......Page 380
o One-loop correction to the mean-field energy ......Page 386
o High temperature expansion of the cosine melting model ......Page 391
o Pair corrections in the disordered phase ......Page 411
o Dissociation of dislocation pairs ......Page 421
o Renormalization group equations ......Page 426
o Triangular lattice ......Page 432
o Calculation of critical temperature ......Page 438
o The critical behavior of the coherence length ......Page 442
o Two-step melting ......Page 448
o Experimental evidences for and against a hexatic phase ......Page 449
o Comparison with molecular dynamics computer simulations ......Page 454
o Universal stiffness ......Page 457
o The Wigner electron lattice ......Page 461
o First order versus continuous KTHNY transitions ......Page 467
o Direct simulation of a gas of dislocations ......Page 471
o Disorder lattice model for three-dimensional defect configurations ......Page 477
o Coupling the stress gauge field ......Page 479
o Disorder field theory of interacting defects ......Page 481
o Defect densities on a lattice ......Page 484
o Interdependence of dislocations and disclinations ......Page 487
o Degenerate defect configurations in linear elasticity ......Page 490
o Extending the defect sum to the lattice ......Page 492
o Two-dimensional considerations ......Page 497
o Torque stresses ......Page 499
o General form of the elastic energy ......Page 501
o Canonical formalism for higher gradient theories ......Page 504
o Second-gradient elasticity ......Page 517
o Canonical formalism for second-gradient elasticity ......Page 521
o Elastic energy of plastic deformations ......Page 524
o Canonical form of the stress partition function ......Page 527
o Lattice model of defect melting with second-gradient elasticity ......Page 528
o Calculation of the interaction energy of defects via stress-gauge fields ......Page 533
o Second-gradient interaction energy derived from defect gauge fields ......Page 541
o Second-gradient elasticity and the partition function of two-dimensional defects ......Page 545
o Two successive melting transitions at large rotational stiffness ......Page 553
o Application of criterion to Lennard-Jones and Wigner lattices ......Page 564
o The partition function of general defect lines in three dimensions ......Page 578
o Cosine form of the partition function ......Page 580
o Disorder fields for dislocations and disclinations ......Page 581
o Towards a quantum defect dynamics of moving defects in two dimensions ......Page 584
Part IV Differential Geometry of Defects and Gravity with Torsion ......Page 590
* Introduction ......Page 592
o Gravity and Geometry ......Page 594
o Minkowski geometry formulated in general coordinates ......Page 597
o Torsion tensor ......Page 608
o Curvature tensor as a covariant curl of the connection ......Page 609
o Torsion and curvature from defects ......Page 615
o Differential geometric properties of metric-affine spaces with curvature and torsion ......Page 622
o Circuit integrals in metric-affine spaces with curvature and torsion ......Page 629
o Some examples of coordinate systems with defects ......Page 633
o Identities for curvature and torsion tensors ......Page 636
o Curvature from embedding ......Page 639
o Geodesic coordinates in curved space ......Page 641
o Invariant action ......Page 644
o Energy-momentum tensor and spin density ......Page 646
o Symmetric energy-momentum tensor of the gravitational field and defect density ......Page 654
o Local Lorentz invariance and non-holonomic coordinates ......Page 656
o Field equations with gravitational spinning matter ......Page 667
* Covariant Conservation Law ......Page 672
o Spin density ......Page 673
o Energy-momentum density ......Page 675
o Covariant derivation of conservation laws ......Page 679
o Matter with integer spin ......Page 680
o Relation between conservation laws and fundamental identities ......Page 684
o Local translations ......Page 686
o Local four-fermion interaction due to torsion ......Page 690
o Classical elasticity ......Page 694
o Second gradient elasticity ......Page 697
o Summing over defect configurations ......Page 700
* Summary and Outlook ......Page 702