 | Modern theories of many-particle systems in condensed matter physics / ed. by D.C.Cabra, A.Honecker, P.Pujol. - Berlin; Heidelberg; New York: Springer, 2012. - xiii, 368 p.: ill. - (Lecture notes in physics; vol.843). - ISBN 978-3-642-10448-0; ISSN 0075-8450
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1 Quantum Phase Transitions of Antiferromagnets and the
Cuprate Superconductors ...................................... 1
1.1 Introduction ............................................ 1
1.2 Insulators .............................................. 6
1.2.1 Coupled Dimer Antiferromagnets: Bond Operators ... 6
1.2.2 Frustrated Square Lattice Antiferromagnets:
Schwinger Bosons ................................ 13
1.3 d-Wave Superconductors ................................. 30
1.3.1 Dirac Fermions .................................. 31
1.3.2 Magnetic Ordering ............................... 32
1.3.3 Ising Transitions ............................... 36
1.4 Metals ................................................. 42
1.4.1 SDW Ordering .................................... 43
1.4.2 Nematic Ordering ................................ 45
References .................................................. 46
2 Electronic Liquid Crystal Phases in Strongly Correlated
Systems ..................................................... 53
2.1 Electronic Liquid Crystal Phases ....................... 53
2.1.1 Symmetries of Electronic Liquid Crystal
Phases .......................................... 54
2.1.2 Order Parameters and Their Symmetries ........... 56
2.1.3 Electronic Liquid Crystal Phases and Strong
Correlation Physics ............................. 58
2.2 Experimental Evidence in Strongly Correlated Systems ... 60
2.2.1 Nematic Phases in the 2DEG in High Magnetic
Fields .......................................... 60
2.2.2 The Nematic Phase of Sr3Ru2O7 ................... 62
2.2.3 Stripe Phases and Nematic Phases in the
Cuprates ........................................ 63
2.2.4 Conventional CDW Materials ...................... 70
2.3 Theories of Stripe Phases .............................. 71
2.3.1 Stripe Phases in Microscopic Models ............. 71
2.3.2 Phases of Stripe States ......................... 74
2.4 Is Inhomogeneity Good or Bad for Superconductivity? .... 79
2.5 The Striped Superconductor: A Pair Density Wave
State .................................................. 81
2.5.1 Landau-Ginzburg Theory of the Pair Density
Wave ............................................ 83
2.5.2 Charge 4e SC Order and the Topological
Excitations of the PDW SC State ................. 85
2.6 Nematic Phases in Fermi Systems ........................ 87
2.6.1 The Pomeranchuk Instability ..................... 87
2.6.2 The Nematic Fermi Fluid ......................... 89
2.7 Generalizations: Unconventional Magnetism and Time
Reversal Symmetry Breaking ............................. 95
2.8 Nematic Order in the Strong Correlation Regime ......... 98
2.9 The Quantum Nematic-Smectic QCP and the Melting of
the Stripe Phase ...................................... 101
2.10 Outlook ............................................... 105
References ................................................. 106
3 Selected Topics in Graphene Physics ........................ 117
3.1 Introduction .......................................... 117
3.2 The Chemistry ......................................... 119
3.3 The Crystal and Band Structure ........................ 123
3.4 Phonons in Free Floating Graphene ..................... 129
3.5 Constrained Graphene .................................. 132
3.6 Deformed Graphene ..................................... 135
3.6.1 A Non-Trivial Example: The Scroll .............. 139
3.7 Conclusions ........................................... 142
References ................................................. 143
4 Strong Electronic Correlations: Dynamical Mean-Field
Theory and Beyond .......................................... 145
4.1 The Physics of Strongly Correlated Materials .......... 146
4.1.1 Theoretical Description ........................ 147
4.2 The Dynamical Mean-Field Theory Construction: An
Embedded Atom ......................................... 148
4.2.1 A Reminder on Classical Mean-Field Theory ...... 149
4.2.2 Generalization to the Quantum Case: Dynamical
Mean-Field Theory .............................. 150
4.2.3 The Embedded Atom as an Anderson Impurity
Model .......................................... 152
4.2.4 The Self-Consistency Loop in Practice .......... 152
4.3 The Metal to Mott Insulator Transition from a DMFT
Perspective ........................................... 153
4.4 From Models to Real Materials: Combining DMFT with
First-Principle Electronic Structure Methods .......... 156
4.4.1 Illustrations .................................. 162
4.5 Efficient Algorithms for Solving Multiorbital
Quantum Impurity Problems ............................. 164
4.5.1 Interaction Expansion Algorithm ................ 164
4.5.2 Monte Carlo Sampling ........................... 167
4.5.3 Fast Update Formulae ........................... 167
4.5.4 Measurement of Green's Function ................ 169
4.5.5 Sign Problem and α-Parameters .................. 172
4.5.6 Hybridization Expansion Algorithm .............. 172
4.5.7 Measurement of Green's Function ................ 176
4.6 Cluster Extensions of DMFT: Short-Range Non-Local
Correlations .......................................... 177
4.7 Long-Range Correlations and the Dual-Fermion
Perturbation Expansion ................................ 182
4.7.1 Formalism ...................................... 184
4.7.2 Dual Perturbation Theory ....................... 187
4.7.3 Exact Relations ................................ 188
4.7.4 Self-Consistency Condition and Relation to
DMFT ........................................... 190
4.7.5 Numerical Results .............................. 192
4.7.6 Generalizations of the Dual Fermion Approach ... 195
4.7.7 Cluster Dual Fermion Approach .................. 195
4.7.8 Calculation of Susceptibilities ................ 201
4.7.9 Ladder Approximation ........................... 204
4.7.10 Convergence Properties ......................... 207
References ................................................. 209
5 Nonequilibrium Transport and Dephasing in Coulomb-
Blockaded Quantum Dots ..................................... 215
5.1 Introduction .......................................... 215
5.2 Coulomb Blockade Phenomena in Quantum Dots ............ 216
5.2.1 Basics ......................................... 216
5.2.2 Energy Scales .................................. 218
5.2.3 Coulomb Blockade and Transport Regimes ......... 219
5.2.4 Kondo Effect in Quantum Dots ................... 221
5.3 Keldysh Approach ...................................... 224
5.3.1 Basic Idea ..................................... 224
5.3.2 Boson Mode ..................................... 226
5.3.3 Fermion Mode ................................... 230
5.4 Ambegaokar-Eckern-Schön Action Within Keldysh
Approach .............................................. 232
5.4.1 Model .......................................... 232
5.5 Nonequilibrium Dephasing .............................. 237
5.5.1 Tunneling Density of States .................... 237
5.5.2 Strong Tunneling: Open Dot ..................... 239
5.6 Conclusions ........................................... 242
References ................................................. 243
6 Many-Body Physics from a Quantum Information Perspective ... 245
6.1 Introduction .......................................... 245
6.2 Aspects of Quantum Information ........................ 246
6.2.1 Bipartite Pure States: Schmidt Decomposition ... 247
6.2.2 Bipartite Mixed States: Separable and
Entangled States ............................... 248
6.2.3 Entanglement Criteria .......................... 249
6.2.4 Partial Transposition .......................... 249
6.2.5 Entanglement Witnesses from the Hahn-Banach
Theorem ........................................ 251
6.2.6 Positive Maps and the Entanglement Problem ..... 253
6.2.7 Positive Maps and Entanglement Witnesses:
Further Characterization and Examples .......... 256
6.2.8 Entanglement Measures .......................... 258
6.3 Area Laws ............................................. 261
6.3.1 Mean Entanglement of Bipartite States .......... 261
6.3.2 Area Laws in a Nutshell ........................ 264
6.4 The Tensor Network Product World ...................... 272
6.4.1 The Tensor Network Representation of Quantum
States ......................................... 273
6.4.2 Examples ....................................... 276
6.4.3 Classical Kinetic Models ....................... 279
6.5 Quantum Kinetic Ising Models .......................... 281
6.5.1 A Two Spin Flip Model .......................... 281
6.5.2 The Single Flip Model .......................... 285
6.6 Discussion and Outlook ................................ 287
References ................................................. 289
7 Statistical Mechanics of Classical and Quantum
Computational Complexity ................................... 295
7.1 Introduction .......................................... 296
7.2 Complexity Theory for Physicists ...................... 298
7.2.1 Problems, Instances, Computers and
Algorithms ..................................... 299
7.2.2 Polynomial Reductions and Worst-Case
Behavior ....................................... 301
7.2.3 Classical: P and NP ............................ 302
7.2.4 Quantum: BQP and QMA ........................... 303
7.2.5 NP-Completeness: Cook-Levin .................... 305
7.2.6 QMA-Completeness: Kitaev ....................... 307
7.3 Physics for Complexity Theory ......................... 312
7.3.1 Typical Versus Worst-Case Complexity ........... 313
7.3.2 Classical Statistical Mechanics of k-SAT ....... 313
7.3.3 Schematic Phase Diagram of Classical Random
k-SAT .......................................... 315
7.3.4 Cavity Analysis ................................ 317
7.3.5 Adiabatic Quantum Algorithm for Classical
k-SAT .......................................... 320
7.4 Statistical Mechanics of Random k-QSAT ................ 322
7.4.1 Random k-QSAT Ensemble ......................... 322
7.4.2 Phase Diagram .................................. 323
7.4.3 Geometrization Theorem ......................... 324
7.4.4 A Few Details of Phases and Transitions ........ 325
7.4.5 Satisfying the Promise ......................... 329
7.4.6 Open Questions ................................. 330
7.5 Conclusion ............................................ 330
References ................................................. 330
8 Non-Perturbative Methods in (1+1) Dimensional Quantum
Field Theory ............................................... 333
8.1 Introduction .......................................... 333
8.2 Functional Form of the Free Energy .................... 334
8.2.1 Critical Exponents and Universal Ratios ........ 336
8.3 General Properties of the Form Factors ................ 338
8.3.1 Faddeev-Zamolodchikov Algebra .................. 339
8.3.2 Form Factors ................................... 340
8.4 Watson's Equations .................................... 341
8.5 Recursive Equations ................................... 344
8.6 The Operator Space .................................... 346
8.7 Correlation Functions ................................. 347
8.8 Form Factors of the Stress-Energy Tensor .............. 350
8.9 Ultraviolet Limit ..................................... 352
8.10 The Ising Model at T ≠ Tc ............................. 355
8.10.1 The Energy Operator ........................... 356
8.10.2 Magnetization Operators ....................... 357
8.10.3 The Painleve Equation ......................... 360
8.11 The Ising Model in a Magnetic Field ................... 362
8.12 Conclusions ........................................... 367
References ................................................. 367
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