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The study of the correlated motion of electrons in solids is of increasing importance in condensed matter physics. In the past few years, the discovery of high-temperature superconductors has stimulated an enormous theoretical effort in this area, building on earlier theories of heavy-fermion and organic superconductors, and magnetic insulators. In a separate development the discovery of the fractional quantum Hall effect stimulated research into the behavior of the two-dimensional electron gas in a strong transverse magnetic field.The lectures at this school gave a systematic presentation of the current status of the theory in these areas. They covered the fractional quantum Hall effect and the many-body physics of the Hubbard model and its extensions, paying particular attention to the properties of doped insulators which are relevant for high-temperature superconductivity. There were detailed discussions of situations for which controlled calculations may be carried out -- specifically infinite dimensions, one dimension, and generalized models in which the fermions have N components and N → -.

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CONTENTS
Preface = ⅴ
Charge Fluctuation Models of Superconductivity / P.B. Littlewood = 1
  1. Introduction = 1
  2. Models for Cuprate Superconductors = 3
    2.1. Transition metals and oxides = 3
    2.2. Model Hamiltonians = 8
  3. The Dielectric Function and Effective Interaction = 14
    3.1. Definitions = 15
    3.2. Stability criteria = 17
    3.3. Dynamics and collective modes = 20
    3.4. RPA = 21
    3.5. Density response function = 22
    3.6. Pairing interaction = 26
  4. Charge Fluctuation Models for Superconductivity = 29
    4.1. Excitons and plasmons = 29
    4.2. Charge transfer resonance = 34
      4.2.1. Hartree-Fock approximation = 36
      4.2.2. Generalised RPA = 39
      4.2.3. Collective modes = 40
      4.2.4. Pairing instabilities = 45
  5. Conclusion = 51
  References = 52
Investigation of Correlated Electron System Using the Limit of High Dimensions / D. Volhardt = 57
  1. The High-d Limit : General Properties and Weak Coupling Perturbation Theory = 57
    1.1. General remarks = 57
    1.2. Motivation for the large dimension limit = 58
    1.3. Classical spin models = 58
    1.4. Itinerant quantum-mechanical models = 59
    1.5. Simplifications in the limit Z → ∞ = 62
    1.6. Interactions beyond the on-site interaction = 65
    1.7. One-particle and two-particle propagators in d = ∞ = 66
    1.8. Application Ⅰ : The weak coupling correlation energy for the Hubbard model = 67
    1.9. Consequences of the k-independence of ∑(w) = 69
    1.10. Application Ⅱ : Density of states in 2. order perturbation theory = 70
    1.11. Application Ⅲ : Landau parameters in 2. order perturbation theory = 73
    1.12. Extension to finite dimensions d = 74
    1.13. Application Ⅳ : The periodic Anderson model in 2. order perturbation theory = 75
  2. Hole Motion in the t-J Model = 77
    2.1. One-particle properties = 79
    2.2. Background-restoring paths in d = ∞ = 80
    2.3. The dynamical conductivity σw) = 82
    2.4. Extension to J ＞ 0 = 84
  3. Variational Wave Functions = 84
    3.1. The Gutzwiller wave function = 85
    3.3. The Gutzwiller approximation = 86
    3.3. Connection to Fermi liquid theory = 87
    3.4. Application to normal-liquid Helium 3 = 89
    3.5. Derivation of the Gutzwiller approximation in d = ∞ = 92
    3.6. The optimal form of Gutzwiller-correlated wave functions = 94
    3.7. Application to the periodic Anderson model(PAM) = 95
  4. Exact Solution of Fermionic Lattice Models in d = ∞ : The Construction of Controlled Mean Field Theories = 96
    4.1. Hartree-approximation for Hubbard-type model = 97
    4.2. Coherent potential approximation for disordered systems = 98
    4.3. CPA and the limit d → ∞ = 101
    4.4. Alternative derivation of CPA = 101
    4.5. Generalization of the CPA approach to interaction systems = 104
    4.6. The exact free energy functional for the Hubbard model in d = ∞ = 105
    4.7. The simplified Hubbard model = 108
    4.8. Towards the exact solution of the Hubbard model in d = ∞ = 110
    4.9. Discussion = 113
  References = 114
The Large N Expansion in the Strong Correlation Problem / G. Kotliar = 118
  1. Introduction = 118
  2. The Single Impurity Anderson Model = 120
  3. The Extended Hubbard Model : The Metal Charge Transfer Insulator Transition = 126
  4. Heavy Fermions and High Tc Superconductors = 140
  5. The Valence and the Charge Transfer Instabilities = 145
  6. Conclusions = 151
  References = 152
The Semiclassical Expansion of the T-J Model / A. Auerbach = 156
  1. Introduction = 156
  2. Spin-Hole Coherent States = 158
  3. Results = 161
  References = 165
The Many-Body Problem in One Dimension / V.J. Emery = 166
  1. Introduction = 166
  2. Infinite Onsite Interactions = 170
  3. The Harmonic Chain = 174
    3.1. The discrete model = 174
    3.2. The continuum model = 177
    3.3. Determination of the model parameters = 180
    3.4. The Harmonic chain on a lattice = 181
  4. Naive Continuum Limit = 183
  5. Bosonization = 188
  6. The Spin Chain = 191
  7. Conclusion = 193
Appendices
  A1. Correlation Functions = 194
  A2. Commutation Relations of Bose Fields = 195
  A3. Anticommutation Relations of Fermi Fields = 195
References = 196
Interacting Fermions in One Dimension : From Weak to Strong Correlation / H.J. Schulz = 199
  1. Introduction = 199
  2. Weak Couping Case = 200
    2.1. The model = 200
    2.2. Mean-field theory = 202
    2.3. Renormalization group : Coupling constants = 205
    2.4. Renormalization group : Correlation functions = 208
  3. Bosonization, Spin-Charge Separation, Luttinger Liquid = 211
    3.1. Bosonization formalism = 211
    3.2. Spin-charge separation = 213
    3.3. Luttinger liquid = 213
  4. The Hubbard Model in One Dimension = 216
    4.1. The Hamiltonian and its symmetries = 216
    4.2. The exact solution : A brief introduction = 217
      4.2.1. Solutions of the Bethe ansatz equations = 219
      4.2.2. Limiting cases = 227
    4.3. Low energy properties of the Hubbard model = 229
      4.3.1. Luttinger liquid parameters = 229
      4.3.2. Transport properties = 233
      4.3.3. Spin-charge separation = 235
      4.3.4. The metal-insulator transition = 237
      4.3.5. Other models = 238
  5. Conclusion = 239
  References = 240
The Quantum Hall Effect : The Article / A. Karlhede ; S.A. Kivelson ; S.L. Sondhi = 242
  Introduction = 242
  1. Review of Basic Phenomena = 243
  2. Phases of the 2DEG at T = 0 = 250
    2.1. Stable phases = 250
    2.2. Critical states and phase transitions = 254
    2.3. Unstable states = 256
  3. The Integer Effect = 256
    3.1. The Hall effect in a translationally invariant system = 257
    3.2. Independent electron approximation = 258
    3.3. Perturabative effects of interactions = 261
    3.4. Exact quantization of the Hall conductance = 263
  4. The Fractional Effect = 266
    4.1. The Laughlin fractions : v = 1/(2k ＋ 1) = 266
    4.2. Fractional statistics and the quasiparticles = 271
    4.3. Hierarchies = 275
    4.4. Spin effects = 279
  5. Is Fractional Charge Observable? = 281
  6. Scaling Theories = 285
    6.1. General considerations = 285
    6.2. Model calculations = 287
  7. Landau-Ginzburg Theory = 289
    7.1. The Landau-Ginzburg theory for the QHE = 289
    7.2. Superconductivity and the QHE = 293
    7.3. Mean field solution plus Gaussian fluctuations = 295
    7.4. Fermionic Chern-Simons field theory = 296
  8. Collective Excitations = 297
    8.1. Spinwaves and skyrmions = 297
    8.2. Other bulk excitations = 301
    8.3. Edge states = 303
  9. The Global Phase Diagram = 306
    9.1. Law of corresponding states = 306
      9.1.1. Meaning of the law of corresponding states = 307
      9.1.2. Intuitive motivation for the law of corresponding states = 308
    9.2. The global phase diagram and selection rules = 309
    9.3. Landau-Ginzburg theory and the law of corresponding states = 314
      9.3.1. Electromagnetic response = 317
      9.3.2. The Hall insulator = 321
      9.3.3. Critical conductivitios = 322
  10. Open Problems = 322
Appendices
  A1. The Induccd Current = 326
  A2. The Plasma Analogy = 327
  A3. Landau-Ginzburg Theory, Ψ1/m and ODLRO = 329
  A4. Duality = 331
Notes = 334
References = 338