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
|
