Quantum inverse scattering method and correlation functions

Front cover......Page 1
Abstract......Page 2
CAMBRIDGE MONOGRAPHS ON MATHEMATICAL PHYSICS Series......Page 3
Title page......Page 5
Date-line......Page 6
Contents......Page 7
Preface......Page 15
Introduction to Part I......Page 21
Introduction......Page 23
I.1 The coordinate Bethe Ansatz......Page 24
I.2 Periodic boundary conditions......Page 30
I.3 The thermodynamic limit at zero temperature......Page 35
I.4 Excitations at zero temperature......Page 39
I.5 Thermodynamics of the model......Page 47
I.6 The Yang-Yang equation......Page 52
I.7 Limiting cases......Page 55
I.8 Excitations and correlations at nonzero temperature......Page 57
I.9 Finite size corrections......Page 62
Conclusion......Page 65
Appendix I.1: Fermionization......Page 67
Appendix I.2: Stability of thermal equilibrium......Page 69
Appendix I.3: Fermi velocity......Page 72
Appendix I.4: Finite size corrections......Page 74
Introduction......Page 83
II.1 The Bethe equations......Page 84
II.2 The ground state......Page 88
II.3 Interaction with a magnetic field......Page 91
II.4 The $XXX$ magnet......Page 93
II.5 Finite size corrections......Page 94
II.6 Fractional charge......Page 95
Conclusion......Page 98
III.1 The Bethe Ansatz......Page 100
III.2 The ground state......Page 103
III.3 Fractional charge and repulsion beyond the cutoff......Page 105
Conclusion......Page 108
IV.1 Bethe Ansatz solution for the Hubbard model......Page 109
IV.2 Finite size corrections......Page 112
Conclusion......Page 114
Introduction to Part II......Page 117
Introduction......Page 119
V.1 The Lax representation......Page 120
V.2 The classical $r$-matrix......Page 124
V.3 Examples......Page 129
Conclusion......Page 132
Appendix V.1: Tensor notation......Page 133
Introduction......Page 135
VI.1 General scheme......Page 136
VI.2 The Yang-Baxter relation......Page 140
VI.3 Trace identities......Page 141
VI.4 Quantum field theory models......Page 144
VI.5 Fundamental spin models......Page 146
VI.6 Fundamental models of classical statistical physics......Page 150
Conclusion......Page 155
Introduction......Page 157
VII.1 The algebraic Bethe Ansatz......Page 158
VII.2 Comments on the algebraic Bethe Ansatz......Page 162
VII.3 Examples......Page 164
VII.4 The Pauli principle for one-dimensional interacting bosons......Page 168
VII.5 The shift operator......Page 171
VII.6 Classification of monodromy matrices......Page 174
VII.7 Comments on the classification of monodromy matrices......Page 177
VII.8 The quantum determinant......Page 180
VII.9 Recursion properties of the partition function......Page 182
VII.10 $Z_N$ as a determinant......Page 185
Conclusion......Page 188
Appendix VII.1: Matrix representation of quantum operators......Page 190
Appendix VII.2: Multisite model......Page 191
Introduction......Page 192
VIII.1 Classical lattice nonlinear Schrodinger equation......Page 193
VIII.2 Classical lattice sine-Gordon model......Page 196
VIII.3 Quantum lattice nonlinear Schrodinger equation......Page 198
VIII.4 Classification of quantum $\L$-operators......Page 203
VIII.5 Quantum lattice sine-Gordon model......Page 205
Conclusion......Page 207
Introduction to Part III......Page 209
Introduction......Page 213
IX.1 The scalar product......Page 214
IX.2 Properties of the coefficients $K_N$......Page 216
IX.3 The residue formula......Page 221
IX.4 The recursion formula for scalar products......Page 225
IX.5 The determinant representation of scalar products in terms of auxiliary quantum fields......Page 229
IX.6 Another proof of the determinant representation for scalar products......Page 232
IX.7 Remarks about norms......Page 237
Conclusion......Page 239
Appendix IX.1: Determinant of the sum of two matrices......Page 241
Introduction......Page 244
X.1 Generalized Gaudin hypothesis......Page 246
X.2 Properties of the Jacobian......Page 248
X.3 Proof of Gaudin's hypothesis......Page 251
X.4 Thermodynamic limit......Page 254
X.5 A special case of the scalar product......Page 257
Conclusion......Page 258
Introduction......Page 259
XI.1 Generalized two-site model......Page 260
XI.2 The determinant representation for the matrix elements of the operator $\exp\{\alpha Q_1\}$ in the generalized model......Page 266
XI.3 Mean value of $\exp\{\alpha Q_1\}$......Page 270
XI.4 Thermodynamic limit in the one-dimensional Bose gas at zero temperature......Page 274
XI.5 Temperature-dependent correlation function......Page 277
Conclusion......Page 281
XII.1 Local fields in the generalized model......Page 282
XII.2 Matrix elements of the operator $\Psi^\dagger(x)\Psi(0)$ in the generalized model......Page 285
XII.3 Auxiliary quantum fields and the determinant representation......Page 288
XII.4 Mean value of the operator $\Psi^\dagger(x)\Psi(0)$ in the NS model......Page 292
XII.5 Thermodynamic limit......Page 295
Conclusion......Page 297
Conclusion to Part III......Page 298
Introduction to Part IV......Page 301
Introduction......Page 303
XIII.1 Impenetrable bosons......Page 304
XIII.2 Generating functionals of current correlation functions......Page 307
XIII.3 Equal-time two-point correlator of fields. Lenard's formula......Page 312
XIII.4 Form factors......Page 317
XIII.5 Normalized mean value for a finite number of particles......Page 321
XIII.6 Time-dependent correlation function for impenetrable bosons......Page 324
XIII.7 Multifield correlation function for impenetrable bosons......Page 327
Conclusion......Page 330
Appendix XIII.1: Temperature correlations......Page 331
Appendix XIII.2: Thermodynamic limit of singular sums......Page 333
Introduction......Page 335
XIV.1 Integrable linear integral operators......Page 336
XIV.2 Differential equations for temperature-dependent equal-time two-point correlators for the impenetrable Bose gas......Page 339
XIV.3 Correlators as a completely integrable system......Page 346
XIV.4 Differential equations for temperature-dependent equal-time multifield correlators for the impenetrable Bose gas......Page 353
XIV.5 Differential equations for the time-dependent correlator of fields for the impenetrable Bose gas at zero temperature......Page 358
XIV.6 Temperature- and time-dependent correlation function of two fields for the impenetrable Bose gas......Page 365
XIV.7 Integro-differential equation for finite coupling constant......Page 374
Conclusion......Page 381
Appendix XIV.1: Short distance asymptotics of temperature- dependent equal-time two-point correlators for the impenetrable Bose gas......Page 383
Appendix XIV.2: Low density expansion......Page 386
Appendix XIV.3: Logarithmic derivatives of correlation function......Page 389
Appendix XIV.4: Evaluation of Predholm minor......Page 393
Appendix XIV.5: Expression for $C$ potentials......Page 395
Introduction......Page 397
XV.1 The matrix Riemann-Hilbert problem for temperature-dependent equal-time two-point correlators for the impenetrable Bose gas......Page 400
XV.2 Representation of equal-time two-point correlators in terms of the matrix Riemann-Hilbert problem......Page 407
XV.3 The matrix Riemann-Hilbert problem for the temper at ure- and time-dependent correlation function of two fields for the impenetrable Bose gas......Page 415
XV.4 Representation of the temperature- and time-dependent two-field correlator for impenetrable bosons in terms of the matrix Riemann-Hilbert problem......Page 419
XV.5 Matrix problem for the multifield equal-time correlator for the impenetrable Bose gas......Page 426
XV.6 The operator Riemann-Hilbert problem for a correlator with finite coupling constant......Page 429
Conclusion......Page 437
Appendix XV.1: Degeneration points......Page 438
Introduction......Page 440
XVI.1 Generating functional of currents for the impenetrable Bose gas......Page 441
XVI.2 The emptiness formation probability......Page 449
XVI.3 Long distance asymptotics of the temperature-dependent equal-time correlator of two fields in the case of negative chemical potential......Page 455
XVI.4 Long distance asymptotics of the temperature-dependent equal-time correlator of two fields in the case of positive chemical potential......Page 459
XVI.5 Explicit formulae for asymptotics of the temperature-dependent equal-time field correlator for the impenetrable Bose gas......Page 466
XVI.6 Time-dependent correlator of two fields at finite temperature in the case of negative chemical potential......Page 469
XVI.7 Time-dependent correlator of two fields at finite temperature in the case of positive chemical potential. Space-like region......Page 479
XVI.8 Time-dependent correlator of two fields at finite temperature in the case of positive chemical potential. Time-like region......Page 484
XVI.9 Results for the time-dependent correlator of two fields at finite temperature......Page 488
Conclusion......Page 491
Appendix XVI.1: Scalar Riemann-Hilbert problem......Page 493
Appendix XVI.2: Properties of $\alpha(\lambda_{1,2})$......Page 496
Appendix XVI.3: Proof of (4.41)......Page 499
Appendix XVI.4: Asymptotics of correlation length at large $\beta$......Page 501
Introduction......Page 503
XVII.1 Another approach to the calculation of correlators in the algebraic Bethe Ansatz......Page 504
XVII.2 The series for the correlator of currents in the nonrelativistic Bose gas at zero temperature......Page 507
XVII.3 The series for the correlator of currents in the nonrelativistic Bose gas at finite temperature......Page 511
XVII.4 Asymptotics of the correlator of currents at finite temperature......Page 514
XVII.5 Asymptotics of the emptiness formation probability......Page 517
Conclusion......Page 519
Appendix XVII.1: Analytic properties of solutions of the Yang-Yang equation......Page 520
Introduction......Page 522
XVIII.1 Finite size effects......Page 524
XVIII.2 Asymptotics of correlation functions for the Bose gas......Page 528
XVIII.3 Asymptotics of correlation functions for the Heisenberg magnet......Page 530
XVIII.4 Asymptotics of correlation functions for the one-dimensional Hubbard model......Page 533
Conclusion......Page 535
Conclusion to Part IV......Page 537
Final Conclusion......Page 538
References......Page 540
Index......Page 574
Back cover......Page 576