The Statistical Mechanics of Quantum Lattice Systems (Ems Tracts in Mathematics)

Preface......Page 7
Contents......Page 11
Introduction......Page 15
I Mathematical Background......Page 27
The Model......Page 29
Linear Operators in Hilbert Spaces......Page 32
Quantum Oscillators......Page 50
Local Gibbs States......Page 76
Local Hamiltonians and Gibbs States......Page 77
Von Neumann Algebras and Normal States......Page 81
Høegh-Krohn\'s Theorem......Page 86
Green and Matsubara Functions......Page 92
Gibbs States of Noninteracting Harmonic Oscillators......Page 101
Beginnings......Page 107
Gaussian Processes......Page 109
Stochastic Processes in General......Page 114
Probability Measures on Polish Spaces......Page 126
Probability Measures on Hilbert Spaces......Page 132
Properties of the Høegh-Krohn Process......Page 139
Harmonic Oscillators and the Høegh-Krohn Process......Page 149
Local Euclidean Gibbs States......Page 151
Matsubara Functions for Unbounded Operators......Page 157
Periodic Local Gibbs States......Page 161
Analytic Properties of Local Gibbs States......Page 162
Comments and Bibliographic Notes......Page 164
Lattice Approximation......Page 169
Basic Correlation Inequalities......Page 176
The Logarithmic Sobolev Inequality......Page 182
The Lee–Yang Property......Page 185
More Inequalities......Page 188
The Vector Case: Scalar Domination......Page 189
 Zero Field Domination......Page 193
Estimates of Moments and Correlation Functions......Page 196
Estimates for Ursell Functions......Page 199
Comments and Bibliographic Notes......Page 202
3 Euclidean Gibbs Measures......Page 205
Thermodynamic Limit......Page 206
Gibbs States......Page 208
Configuration Spaces......Page 209
Tempered Euclidean Gibbs Measures......Page 214
Exponential Moment Estimates......Page 222
Weak Convergence of Tempered Measures......Page 226
Properties of Euclidean Gibbs Measures......Page 228
Flow Description of Euclidean Gibbs Measures......Page 236
Integration by Parts......Page 240
The Case of Local Interactions......Page 244
Periodic Euclidean Gibbs Measures......Page 246
FKG order......Page 252
Comments and Bibliographic Notes......Page 257
II Physical Applications......Page 261
Modeling Physical Substances......Page 263
The Classical Limit......Page 266
The High-Temperature Uniqueness......Page 269
Comments and Bibliographic Notes......Page 273
The Existence of Thermodynamic Pressure......Page 276
Dependence on the External Field......Page 283
Uniqueness Criteria......Page 289
Comments and Bibliographic Notes......Page 291
6 Phase Transitions......Page 294
Phase Transitions and Order Parameters......Page 295
Reflection positive functionals......Page 305
Gaussian Domination......Page 309
Infrared Bound......Page 314
Phase Transitions in Rotation Invariant Models......Page 318
Phase Transitions in Symmetric Scalar Models......Page 323
Phase Transition in Asymmetric Scalar Models......Page 325
Critical Point of a Hierarchical Model......Page 328
The Model and the Main Result......Page 329
Preliminary Statements......Page 331
Proof of Theorem 6.4.1......Page 335
Proof of Lemma 6.4.7......Page 337
Comments and Bibliographic Notes......Page 346
The Stability of Quantum Crystals......Page 350
The Main Statement......Page 354
Decay of Correlations in Gibbs States......Page 358
The Uniqueness for Scalar Ferromagnets......Page 360
Normality of Fluctuations in Vector Ferromagnets......Page 363
Comments and Bibliographic Notes......Page 366
Bibliography......Page 369
List of Symbols......Page 387
Index......Page 391