Strongly correlated systems: theoretical methods (Heidelberg, 2012). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаStrongly correlated systems: theoretical methods / ed. by A.Avella, F.Mancini. - Heidelberg: Springer, 2012. - xxxi, 461 p. - (Springer series in solid-state sciences; 171). - Bibliogr. at the end of the art. - Ind.: p.455-461. - ISBN 978-3-642-21830-9; ISSN 0171-1873
 

Оглавление / Contents
 
Foreword ..................................................... xvii
Peter Fulde

1  Density Functional Theory: A Personal View ................... 1
   Robert O. Jones
   1.1  Introduction ............................................ 1
   1.2  The Early Years: The Density as a Basic Variable ........ 3
   1.3  Towards an "Approximate Practical Method" ............... 6
   1.4  Density Functional Formalism ............................ 8
        1.4.1  Single-Particle Description of a Many-Particle
               System ........................................... 8
        1.4.2  Approximations to Exc ............................ 9
        1.4.3  Exchange-Correlation Energy, Exc ................ 11
        1.4.4  DF Theory in Retrospect ......................... 13
   1.5  The Beryllium Dimer .................................... 15
        1.5.1  The Story to Late 1979 .......................... 15
        1.5.2  1980-1984 ....................................... 20
        1.5.3  After 1984 ...................................... 21
   1.6  Concluding Remarks ..................................... 23
   References .................................................. 27
2  Projected Wavefunctions and High Tс Superconductivity
   in Doped Mott Insulators .................................... 29
   Mohit Randeria, Rajdeep Sensarma, and Nandini Trivedi
   2.1  Introduction ........................................... 29
   2.2  Hubbard Model and Projected Wavefunctions .............. 32
   2.3  Particle-hole Asymmetry in Doped Mott Insulators ....... 35
   2.4  Gutzwiller Approximation ............................... 38
   2.5  Superconducting Ground State ........................... 41
        2.5.1  Energy Gap ...................................... 42
        2.5.2  Order Parameter ................................. 43
   2.6  Momentum Distribution .................................. 45
   2.7  Electronic Excitations and Spectral Properties ......... 46
        2.7.1  Nodal Fermi Velocity ............................ 47
        2.7.2  Spectral Function ............................... 48
        2.7.3  Sum Rules ....................................... 49
        2.7.4  Fermi Surface ................................... 50
   2.8  Superfluid Density ..................................... 51
        2.8.1  Doping Dependence of Ds ......................... 52
        2.8.2  Temperature Dependence of Ds .................... 54
   2.9  Disorder and Strong Correlations ....................... 56
   2.10 Competing Orders: Antiferromagnetism ................... 59
   2.11 Conclusion ............................................. 60
   References .................................................. 62
3  The Pseudoparticle Approach to Strongly Correlated
   Electron Systems ............................................ 65
   Raymond Frésard, Johann Kroha, and Peter Wölfle
   3.1  Introduction ........................................... 66
   3.2  Pseudoparticle Representations of Quantum Operators .... 67
        3.2.1  Spin Operators .................................. 67
        3.2.2  Electron Operators .............................. 70
   3.3  Mean-Field Approximations .............................. 77
        3.3.1  Saddle-Point Approximation to the Barnes
               Representation .................................. 77
        3.3.2  Saddle-Point Approximation to the KR
               Representation .................................. 79
   3.4  Fluctuation Corrections to the Saddle-Point
        Approximation: SRI Representation of the Hubbard
        Model .................................................. 83
        3.4.1  Magnetic and Stripe Phases ...................... 84
   3.5  Conserving Self-Consistent Approximations .............. 85
        3.5.1  General Properties .............................. 85
        3.5.2  Exact Infrared Properties of Pseudoparticle
               Propagators ..................................... 87
        3.5.3  Fock Space Projection in Saddle-Point
               Approximation ................................... 87
        3.5.4  Noncrossing Approximation (NCA) ................. 88
        3.5.5  Conserving T-Matrix Approximation (CTMA) ........ 90
   3.6  Renormalization Group Approaches ....................... 93
        3.6.1  "Poor Man's Scaling" in the Equilibrium
               Kondo Model ..................................... 94
        3.6.2  Functional RG for the Kondo Model Out of
               Equilibrium ..................................... 95
        3.6.3  RG Approach to the Kondo Model at Strong
               Coupling ........................................ 96
        3.6.4  Functional RG Approach to Frustrated
               Heisenberg Models ............................... 97
   3.7  Conclusion ............................................. 98
   References .................................................. 98
4  The Composite Operator Method (COM) ........................ 103
   Adolfo Avella and Ferdinando Mancini
   4.1  Strong Correlations and Composite Operators ........... 103
   4.2  The Composite Operator Method (COM) ................... 106
        4.2.1  Basis ψ ........................................ 106
        4.2.2  Equations of Motion of ψ ....................... 108
        4.2.3  Dyson Equation for G ........................... 110
        4.2.4  Propagator G0  ................................. 11l
        4.2.5  Residual Self-Energy Σ ......................... 114
        4.2.6  Self-Consistency ............................... 115
        4.2.7  Summary ........................................ 116
   4.3  Case Study: The Hubbard Model ......................... 116
        4.3.1  The Hamiltonian ................................ 116
        4.3.2  The Residual Self-Energy Σ(k,ω) ................ 131
        4.3.3  Four-Pole Solution ............................. 134
        4.3.4  Superconducting Solution ....................... 136
   4.4  Conclusions and Outlook ............................... 138
   References ................................................. 139
5  LDA+GTB Method for Band Structure Calculations in the
   Strongly Correlated Materials .............................. 143
   Sergey G. Ovchinnikov, Vladimir A. Gavrichkov, Maxim M.
   Korshunov, and Elena I. Shneyder
   5.1  Introduction .......................................... 144
   5.2  Quasiparticle Definition of an Electron From the
        Lehmann Representation ................................ 145
   5.3  The Main Steps of the LDA+GTB Method .................. 146
        5.3.1  Step I: LDA .................................... 148
        5.3.2  Step II: Exact Diagonalization ................. 148
   5.4  Step III: Perturbation Theory ......................... 154
        5.4.1  Perturbation Theory in the X-Operators
               Representation ................................. 154
        5.4.2  Virtual and In-Gap States ...................... 157
        5.4.3  Summary of Step III ............................ 158
   5.5  LDA+GTB Band Structure of the Hole and Electron
        Doped Cuprates ........................................ 159
        5.5.1  LDA+GTB Band Structure of the Undoped
               La2CuO4 ........................................ 159
        5.5.2  Low-Energy Effective t-t'-t"-J* Model
               and the Fermi Surface of 8m2-xСеxСuO4 ........... 161
        5.5.3  Doping-Dependent Evolution of the Fermi
               Surface and Lifshitz Quantum Phase
               Transitions in La2-xSrxCuO4 ..................... 163
   5.6  LDA+GTB Band Structure of Manganite La1-xСаxМnО3 ....... 165
   5.7  Finite-Temperature Effect on the Electronic
        Structure of LaCoO3 ................................... 166
   5.8  Conclusions ........................................... 169
6  Projection Operator Method ................................. 173
   Nikolay M. Plakida
   6.1  Introduction .......................................... 173
   6.2  Equation of Motion Method for Green Functions ......... 175
        6.2.1  General Formulation ............................ 175
        6.2.2  Projection Technique for Green Functions ....... 177
   6.3  Superconducting Pairing in the Hubbard Model .......... 180
        6.3.1  Hubbard Model .................................. 180
        6.3.2  Dyson Equation ................................. 181
        6.3.3  Mean-Field Approximation ....................... 183
        6.3.4  Self-Energy Operator ........................... 184
        6.3.5  Equation for Superconducting Gap and Tc ........ 186
   6.4  Spin-Excitation Spectrum .............................. 189
        6.4.1  Dynamical Spin Susceptibility .................. 189
        6.4.2  Spin Susceptibility in the t - J Model ......... 192
        6.4.3  Magnetic Resonance Mode ........................ 196
   6.5  Conclusion ............................................ 201
   References ................................................. 201
7  Dynamical Mean-Field Theory ................................ 203
   Dieter Vollhardt, Krzysztof Byczuk, and Marcus Kollar
   7.1  Motivation ............................................ 203
        7.1.1  Electronic Correlations ........................ 203
        7.1.2  The Hubbard Model .............................. 204
        7.1.3  Construction of Comprehensive Mean-Field
               Theories for Many-Particle Models .............. 205
   7.2  Lattice Fermions in the Limit of High Dimensions ...... 206
        7.2.1  Scaling of the Hopping Amplitude ............... 206
        7.2.2  Simplifications of the Many-Body Perturbation
               Theory ......................................... 207
        7.2.3  Interactions Beyond the On-Site Interaction .... 209
        7.2.4  Single-Particle Propagator ..................... 210
        7.2.5  Consequences of the Momentum Independence of
               the Self-Energy ................................ 210
   7.3  Dynamical Mean-Field Theory for Correlated Lattice
        Fermions .............................................. 212
        7.3.1  Construction of the DMFT as a Self-Consistent
               Single-Impurity Anderson Model ................. 212
        7.3.2  Solution of the Self-Consistency Equations
               of the DMFT .................................... 217
   7.4  The Mott-Hubbard Metal-Insulator Transition ........... 217
        7.4.1  DMFT and the Three-Peak Structure of the
               Spectral Function .............................. 218
   7.5  Theory of Electronic Correlations in Materials ........ 221
        7.5.1  The LDA+DMFT Approach .......................... 221
        7.5.2  Single-Particle Spectrum of Correlated
               Electrons in Materials ......................... 224
   7.6  Electronic Correlations and Disorder .................. 227
   7.7  DMFT for Correlated Bosons in Optical Lattices ........ 228
   7.8  DMFT for Nonequilibrium ............................... 229
   7.9  Summary and Outlook ................................... 231
   References ................................................. 232
8  Cluster Perturbation Theory ................................ 237
   David Sénéchal
   8.1  Introduction: CPT in a Nutshell ....................... 237
   8.2  Cluster Kinematics .................................... 240
   8.3  Lehmann Representation of the Green Function .......... 244
        8.3.1  The Lehmann Representation and the CPT Green
               Function ....................................... 245
   8.4  The Impurity Solver ................................... 246
        8.4.1  Coding of the Basis States ..................... 247
        8.4.2  The Lanczos Algorithm for the Ground State ..... 249
        8.4.3  The Lanczos Algorithm for the Green Function ... 250
        8.4.4  The Band Lanczos Algorithm for the Green
               Function ....................................... 252
        8.4.5  Cluster Symmetries ............................. 253
        8.4.6  Green Functions Using Cluster Symmetries ....... 255
        8.4.7  Other Solvers .................................. 256
   8.5  Periodization ......................................... 257
   8.6  Computing Physical Quantities ......................... 261
   8.7  Results on the Hubbard Model .......................... 263
   8.8  Applications to Other Models .......................... 266
        8.8.1  Multi-Band Hubbard Models ...................... 266
        8.8.2  t-J or Spin Models ............................. 267
        8.8.3  Extended Hubbard Models ........................ 268
        8.8.4  Phonons ........................................ 269
   References ................................................. 269
9  Dynamical Cluster Approximation ............................ 271
   H. Fotso, S. Yang, K. Chen, S. Pathak, J. Moreno,
   M. Jarrell, K. Mikelsons, E. Khatami, and D. Galanakis
   9.1  Introduction .......................................... 271
   9.2  The Dynamical Mean Field and Cluster
        Approximations ........................................ 273
        9.2.1  The Dynamical Mean-Field Approximation ......... 273
        9.2.2  The Dynamical Cluster Approximation ............ 276
        9.2.3  Φ Derivability ................................. 277
        9.2.4  Algorithm ...................................... 280
   9.3  Physical Quantities ................................... 281
        9.3.1  Particle-Hole Channel .......................... 281
        9.3.2  Particle-Particle Channel ...................... 283
   9.4  DCA and Quantum Criticality in the Hubbard Model ...... 285
        9.4.1  Evidence of the Quantum Critical Point at
               Optimal Doping ................................. 285
        9.4.2  Nature of the Quantum Critical Point in the
               Hubbard Model .................................. 291
        9.4.3  Relationship Between Superconductivity and
               the Quantum Critical Point ..................... 294
   9.5  Conclusion ............................................ 300
   References ................................................. 300
10 Self-Energy-Functional Theory .............................. 303
   Michael Potthoff
   10.1 Motivation ............................................ 303
   10.2 Self-Energy Functional ................................ 305
        10.2.1 Hamiltonian, Grand Potential and Self-
               Energy ......................................... 305
        10.2.2 Luttinger-Ward Functional ...................... 306
        10.2.3 Diagrammatic Derivation ........................ 308
        10.2.4 Derivation Using the Path Integral ............. 308
        10.2.5 Variational Principle .......................... 311
        10.2.6 Approximation Schemes .......................... 312
   10.3 Variational Cluster Approach .......................... 313
        10.3.1 Reference System ............................... 313
        10.3.2 Domain of the Self-Energy-Functional ........... 314
        10.3.3 Construction of Cluster Approximations ......... 315
   10.4 Consistency of Approximations ......................... 320
        10.4.1 Analytical Structure of the Green's
               Function ....................................... 320
        10.4.2 Thermodynamical Consistency .................... 321
        10.4.3 Symmetry Breaking .............................. 322
        10.4.4 Non-perturbative Conserving Approximations ..... 325
   10.5 Bath Degrees of Freedom ............................... 327
        10.5.1 Motivation and Dynamical Impurity
               Approximation .................................. 327
        10.5.2 Relation to Dynamical Mean-Field Theory ........ 328
        10.5.3 Cluster Mean-Field Approximations .............. 330
        10.5.4 Translation Symmetry ........................... 332
        10.5.1 Systematics of Approximations .................. 333
   10.7 Summary ............................................... 336
   References ................................................. 337
11 Cluster Dynamical Mean Field Theory ........................ 341
   David Sénéchal
   11.1 The CDMFT Procedure ................................... 342
        11.1.1 The Effective Hamiltonian ...................... 342
        11.1.2 The Self-Consistency Condition ................. 345
        11.1.3 The SFA Point of View .......................... 346
   11.2 The Exact Diagonalization Implementation .............. 347
        11.2.1 Working with a Small Bath System: The
               Distance Function .............................. 347
        11.2.2 Bath Parametrization ........................... 348
        11.2.3 The Distance Function .......................... 352
   11.3 Quantum Monte Carlo Solvers ........................... 355
        11.3.1 The Hirsch-Fye Method .......................... 356
        11.3.2 The Continuous-Time Method ..................... 358
   11.4 The Mott Transition ................................... 360
   11.5 Application to the Cuprates ........................... 364
   11.6 CDMFT and DCA ......................................... 367
   References ................................................. 370
12 Functional Renormalization Group for Interacting
   Many-Fermion Systems on Two-Dimensional Lattices ........... 373
   Carsten Honerkamp
   12.1 Introduction .......................................... 373
   12.2 Functional RG Schemes for Fermions: Exact Flow
        Equations and Truncations ............................. 376
        12.2.1 Basic Elements ................................. 376
        12.2.2 Functional Renormalization Group
               Differential Equations ......................... 378
        12.2.3 Choice of Flow Parameter ....................... 383
   12.3 Implementation of the Fermionic fRG for Two
        Dimensional Lattices .................................. 386
   12.4 Instabilities in Two-Dimensional Lattice Systems ...... 388
        12.4.1 Two-Dimensional Hubbard Model Near Half
               Filling ........................................ 388
        12.4.2 Iron Pnictides ................................. 395
   12.5 Remarks on the 1PI fRG Scheme ......................... 398
        12.5.1 Differences to Standard Wilsonian RG ........... 398
        12.5.2 Higher Loops ................................... 399
        12.5.3 Connection to Infinite-Order Single-Channel
               Summations ..................................... 399
        12.5.4 Symmetry-Breaking: Connection to Mean-Field
               and Eliashberg Theory .......................... 400
        12.5.5 Normal-State Self-Energy ....................... 402
        12.5.6 Refined Studies ................................ 404
   12.6  Conclusions and Outlook .............................. 405
   References ................................................. 405
13 Two-Particle-Self-Consistent Approach for the Hubbard
   Model ...................................................... 409
   André-Marie S. Tremblay
   13.1 Introduction .......................................... 409
   13.2 The Method ............................................ 412
        13.2.1 Physically Motivated Approach, Spin and
               Charge Fluctuations ............................ 413
        13.2.2 Mermin-Wagner, Kanamori-Brueckner and
               Benchmarking Spin and Charge Fluctuations ...... 416
        13.2.3 Self-Energy .................................... 420
        13.2.4 Internal Accuracy Checks ....................... 423
        13.2.5 A More Formal Derivation ....................... 425
        13.2.6 Pseudogap in the Renormalized Classical
               Regime ......................................... 429
   13.3 Case Studies .......................................... 432
        13.3.1 Pseudogap in Electron-Doped Cuprates ........... 432
        13.3.2 d-Wave Superconductivity ....................... 435
   13.4 More Insights on the Repulsive Model .................. 441
        13.4.1 Critical Behavior and Phase Transitions ........ 441
        13.4.2 Longer Range Interactions ...................... 442
        13.4.3 Frustration .................................... 443
        13.4.4 Thermodynamics, Conserving Aspects ............. 443
        13.4.5 Vertex Corrections and Conservation Laws ....... 446
   13.5 Attractive Hubbard Model .............................. 446
        13.5.1 Pseudogap from Superconductivity in
               Attractive Hubbard Model ....................... 447
   13.6 Open Problems ......................................... 448
   References ................................................. 450

Index ......................................................... 455


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