One-dimensional quantum systems show fascinating properties beyond the scope of the mean-field approximation. However, the complicated mathematics involved is a high barrier to non-specialists. Written for graduate students and researchers new to the field, this book is a self-contained account of how to derive the exotic quasi-particle picture from the exact solution of models with inverse-square interparticle interactions. The book provides readers with an intuitive understanding of exact dynamical properties in terms of exotic quasi-particles which are neither bosons nor fermions. Powerful concepts, such as the Yangian symmetry in the Sutherland model and its lattice versions, are explained. A self-contained account of non-symmetric and symmetric Jack polynomials is also given. Derivations of dynamics are made easier, and are more concise than in the original papers, so readers can learn the physics of one-dimensional quantum systems through the simplest model.
Author(s): Yoshio Kuramoto, Yusuke Kato
Edition: 1
Year: 2009
Language: English
Pages: 486
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 13
1.1 Motivation......Page 15
1.2 One-dimensional interaction as a disguise......Page 17
1.3 Two-body problem with 1/r2 interaction......Page 18
1.4 Freezing spatial motion......Page 24
1.5 From spin permutation to graded permutation......Page 25
1.6 Variants of 1/r2 systems......Page 27
1.7 Contents of the book......Page 30
Part I Physical properties......Page 33
2 Single-component Sutherland model......Page 35
2.1.1 Jastrow-type wave functions......Page 36
2.1.2 Triangular matrix for Hamiltonian......Page 38
2.1.3 Ordering of basis functions......Page 44
2.2.1 Interacting boson description......Page 46
2.2.3 Exclusion statistics......Page 48
2.3 Elementary excitations......Page 50
2.3.1 Partitions......Page 51
2.3.2 Quasi-particles......Page 54
2.3.3 Quasi-holes......Page 57
2.3.4 Neutral excitations......Page 59
2.4 Thermodynamics......Page 61
2.4.1 Interacting boson picture......Page 62
2.4.2 Free anyon picture......Page 64
2.4.3 Exclusion statistics and duality......Page 65
2.4.4 Elementary excitation picture......Page 68
2.5 Introduction to Jack polynomials......Page 69
2.6 Dynamics in thermodynamic limit......Page 75
2.6.1 Hole propagator h x,......Page 76
2.6.2 Particle propagator h x,......Page 79
2.6.3 Density correlation function......Page 81
2.7 Derivation of dynamics for finite-sized systems......Page 84
2.7.1 Hole propagator......Page 85
2.7.2 Particle propagator......Page 91
2.7.3 Density correlation function......Page 100
2.8 Reduction to Tomonaga–Luttinger liquid......Page 104
2.8.1 Asymptotic behavior of correlation functions......Page 105
2.8.2 Finite-size corrections......Page 107
3 Multi-component Sutherland model......Page 112
3.1 Triangular form of Hamiltonian......Page 113
3.2.1 Eigenstates of identical particles......Page 118
3.2.2 Wave function of ground state......Page 121
3.2.3 Eigenstates with bosonic Fock condition......Page 123
3.3 Energy spectrum with most general internal symmetry......Page 125
3.4.1 Quasi-particles......Page 128
3.4.2 Quasi-holes......Page 129
3.5.1 Multi-component bosons and fermions......Page 134
3.5.2 Explicit results for U(2) anyons......Page 137
3.5.3 Generalization to U(K) symmetry......Page 141
3.6.1 Non-symmetric Jack polynomials......Page 143
3.6.2 Jack polynomials with U(2) symmetry......Page 147
3.7 Dynamics of U(2) Sutherland model......Page 149
3.7.1 Hole propagator…......Page 150
3.7.2 Unified description of correlation functions......Page 152
3.8.1 Hole propagator......Page 156
3.8.2 Density correlation function......Page 160
4 Spin chain with 1/r2 interactions......Page 164
4.1 Mapping to hard-core bosons......Page 165
4.2.1 Hole representation of lattice fermions......Page 166
4.2.2 Gutzwiller wave function in Jastrow form......Page 169
4.3 Projection to the Sutherland model......Page 170
4.4 Static structure factors......Page 171
4.5 *Derivation of static correlation functions......Page 177
4.6 Spectrum of magnons......Page 185
4.7.1 Localized spinons......Page 187
4.7.2 Spectrum of spinons......Page 189
4.7.3 Polarized ground state......Page 192
4.8.1 Degeneracy beyond SU(2) symmetry......Page 194
4.8.2 Local current operators......Page 197
4.8.3 Freezing trick......Page 199
4.9.1 Removal of phonons......Page 202
4.9.2 Completeness of spinon basis......Page 204
4.9.3 Semionic statistics of spinons......Page 207
4.9.4 Variants of Young diagrams......Page 208
4.10.1 Energy functional of spinons......Page 210
4.10.2 Thermodynamic potential of spinons......Page 214
4.10.3 Susceptibility and specific heat......Page 217
4.10.4 *Thermodynamics by freezing trick......Page 219
4.11.1 Brief survey on dynamical theory......Page 222
4.11.2 Exact analytic results......Page 225
4.11.3 Dynamics in magnetic field......Page 229
4.11.3 Dynamics in magnetic field…......Page 230
Antispinon creation in S…......Page 231
4.11.4 Comments on experimental results......Page 233
5 SU(K) spin chain......Page 234
5.1 Coordinate representation of ground state......Page 235
5.2 Spectrum and motif......Page 237
5.3 Statistical parameters via freezing trick......Page 243
5.4 Dynamical structure factor......Page 245
6 Supersymmetric t–J model with 1/r2 interaction......Page 247
6.1 Global supersymmetry in ¡ model......Page 248
6.2 Mapping to U(1,1) Sutherland model......Page 249
6.3 Static structure factors......Page 253
6.4.1 Energy of polynomial wave functions......Page 259
6.4.2 Spinons and antispinons......Page 264
6.4.3 Holons and antiholons......Page 267
6.5.1 Yangian generators......Page 270
6.5.2 Ribbon diagrams and supermultiplets......Page 273
6.5.3 Motif as representation of supermultiplets......Page 275
6.6.1 Parameters for exclusion statistics......Page 276
6.6.2 Energy and thermodynamic potential......Page 279
6.6.3 Fully polarized limit......Page 281
6.6.4 Distribution functions at low temperature......Page 282
6.6.5 Magnetic susceptibility......Page 284
6.6.6 Charge susceptibility......Page 286
6.6.7 Entropy and specific heat......Page 288
6.7.1 Coupling of external fields to quasi-particles......Page 292
6.7.2 Dynamical spin structure factor......Page 294
6.7.3 Dynamical structure factor in magnetic fields......Page 301
6.7.4 Dynamical charge structure factor......Page 304
6.7.5 Electron addition spectrum......Page 307
6.7.6 Electron removal spectrum......Page 309
6.7.7 Momentum distribution......Page 314
6.8 *Derivation of dynamics for finite-sized t–j model......Page 316
6.8.1 Electron addition spectrum......Page 317
6.8.2 Dynamical spin structure factor......Page 320
Part II Mathematics related to 1/r2 systems......Page 323
7 Jack polynomials......Page 325
7.1.1 Composition......Page 326
7.1.2 Cherednik–Dunkl operators......Page 328
Integral norm......Page 333
Combinatorial inner product......Page 334
7.1.5 Generating operators......Page 338
7.1.6 Arms and legs of compositions......Page 343
7.1.7 Evaluation formula......Page 347
7.2.1 Antisymmetric Jack polynomials......Page 348
7.2.2 Integral norm......Page 352
7.2.3 Binomial formula......Page 355
7.2.4 Combinatorial norm......Page 356
7.3.1 Relation to non-symmetric Jack polynomials......Page 360
7.3.2 Evaluation formula......Page 365
7.3.3 Symmetry-changing operator......Page 366
7.3.4 Bosonic of partitionsdescription......Page 369
7.3.5 Integral norm......Page 373
7.3.6 Combinatorial norm......Page 374
7.3.8 Power-sum decomposition......Page 378
7.3.9 Duality......Page 379
7.3.10 Skew Jack functions and Pieri formula......Page 382
7.4.1 Relation to non-symmetric Jack polynomials......Page 385
7.4.2 Integral norm......Page 386
7.4.4 Jack polynomials......Page 387
7.4.5 Evaluation formula......Page 389
7.4.6 Binomial formula......Page 391
7.4.7 Power-sum decomposition......Page 395
7.5.1 Relation to non-symmetric Jack polynomials......Page 396
7.5.2 Evaluation formula......Page 398
7.5.3 Bosonization for separated states......Page 399
7.5.4 Factorization for separated states......Page 400
7.5.5 Binomial formula for separated states......Page 402
7.5.6 Integral norm......Page 403
8 Yang–Baxter relations and orthogonal eigenbasis......Page 405
8.1 Fock condition and R-matrix......Page 406
8.2 R-matrix and monodromy matrix......Page 411
8.3 Yangian......Page 415
8.4 Relation to U(2) Sutherland model......Page 417
8.5.1 Examples for small systems......Page 420
8.5.2 Orthogonal eigenbasis for N-particle systems......Page 430
8.6 Norm of Yangian Gelfand–Zetlin basis......Page 433
9 SU(K) and supersymmetric Yangians......Page 436
9.1 Construction of monodromy matrix......Page 437
9.2 Quantum determinant vs. ordinary determinant......Page 440
9.3 Capelli determinant......Page 441
9.4 Quantum determinant of SU(K) Yangian......Page 444
9.5 Alternative construction of monodromy matrix......Page 445
9.6 Drinfeld polynomials......Page 449
9.7 Extension to supersymmetry......Page 452
10.1 Macdonald symmetric polynomials......Page 455
10.2 Uglov polynomials......Page 458
10.3 Reduction to single-component bosons......Page 459
10.4 From Yangian Gelfand–Zetlin basis to Uglov polynomials......Page 463
10.5 Dynamical correlation functions......Page 464
Afterword......Page 469
References......Page 472
Index of symbols......Page 478
Index......Page 485
