Beautiful Models

70 Years Of Exactly Solved Quantum Many-body Problems

By: Bill Sutherland

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Paperback | 22 June 2004

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400 Pages

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This invaluable book provides a broad introduction to the fascinating and beautiful subject of many-body quantum systems that can be solved exactly. The subject began with Bethe's famous solution of the one-dimensional Heisenberg magnet more than 70 years ago, soon after the invention of quantum mechanics. Since then, the diversity and scope of such systems have been steadily growing.Beautiful Models is self-contained and unified in presentation. It may be used as an advanced textbook by graduate students and even ambitious undergraduates in physics. It is also suitable for the non-experts in physics who wish to have an overview of some of the classic and fundamental models in the subject. The explanations in the book are detailed enough to capture the interest of the curious reader, and complete enough to provide the necessary background material needed to go further into the subject and explore the research literature.

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?This book is self-contained and unified in presentation. It may be used as an advanced textbook by graduate students."

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Non-Fiction Science Physics Classical Mathematics Quantum Physics & Quantum Mechanics & Quantum Field Theory Materials & States of Matter Statistical Physics Booktopia Publisher Services World Scientific Publishing

Preface	p. vii
Overview	p. 1
Orientation	p. 1
An experiment--ballistic expansion	p. 6
One dimension versus two or three dimensions	p. 8
Quantum mechanics	p. 12
The essence of the Bethe ansatz	p. 12
A simple example--the 1/r[superscript 2] potential	p. 15
References and history	p. 18
Integrability and Nondiffraction	p. 27
What does it mean to say that a system is integrable?	p. 27
Scattering without diffraction	p. 30
Proof of integrability for the hyperbolic potential	p. 36
The potentials	p. 36
Proof of integrability	p. 38
The asymptotic momenta are conserved	p. 42
The [delta]-function potential	p. 44
Adding periodic boundary conditions	p. 45
This is not a low-density approximation	p. 49
Techniques	p. 53
Our fundamental equation	p. 53
The ground state	p. 54
Response of the ground state	p. 61
Excitations near the ground state	p. 64
Zero-temperature thermodynamics	p. 68
Finite temperature thermodynamics	p. 70
The inverse-square potential--an example	p. 72
The Classical Limit	p. 75
Classical diffraction--pictures from an exhibition	p. 75
The classical limit	p. 90
Classical system at zero temperature	p. 93
The inverse-square limit--example 1	p. 95
The Toda limit--example 2	p. 97
Solitons and constant profile solutions	p. 100
Groundstate Wavefunctions of Product Form	p. 102
An ansatz for the ground state wavefunction	p. 102
Motivation for the product form	p. 103
An inverse problem	p. 104
The trigonometric case	p. 106
Details	p. 108
Excited states for the trigonometric case	p. 109
Summary	p. 109
A transformation	p. 110
Change of basis	p. 111
Off-diagonal elements of H'	p. 113
Energy eigenvalues	p. 114
Validity of the asymptotic Bethe ansatz	p. 115
An example	p. 116
Ground state correlations for the trigonometric case	p. 116
The normalization constant	p. 116
Analogy with a classical plasma	p. 118
Important correlation functions	p. 119
Correlations for special values of [lambda]	p. 123
The Heisenberg-Ising Model	p. 131
The model and its symmetries	p. 131
The model	p. 131
Rotational symmetry	p. 132
Translational symmetry	p. 132
Ground state in the thermodynamic limit	p. 133
Other models	p. 134
Quantum lattice gas	p. 134
Statistics of the lattice gas	p. 135
Twisted lattice gas	p. 137
What we are going to find--a preview	p. 138
Bethe's ansatz	p. 143
The wavefunction	p. 143
Well-separated particles	p. 143
Near-neighbors	p. 144
Two-body phase shift [theta]	p. 145
Periodic boundary conditions	p. 145
Twisted boundary conditions and recapitulation	p. 146
The dilute limit M [double less-than sign] N	p. 146
M = 0	p. 146
M = 1	p. 146
M = 2	p. 148
A change of variables	p. 154
[Delta less than sign] -1, antiferromagnet	p. 154
1 [greater than sign Delta greater than sign] -1, paramagnet	p. 155
[Delta] = -1, Heisenberg antiferromagnet	p. 155
The ground state	p. 157
Preliminaries	p. 157
Half-filled lattice	p. 158
[Delta] = -cosh [lambda less than sign] -1, antiferromagnet	p. 159
1 [greater than sign Delta] = -cos [mu greater than sign] -1, paramagnet	p. 160
[Delta] = -1, Heisenberg antiferromagnet	p. 161
Perturbations and excitations of the ground state	p. 161
The general results	p. 162
Half-filling	p. 166
[Delta less than sign] -1, antiferromagnet	p. 167
1 [greater than sign Delta greater than sign] -1, paramagnet	p. 168
[Delta] = -1, Heisenberg antiferromagnet	p. 168
Zero-temperature thermodynamics	p. 170
Fundamental integral equation	p. 170
Near half-filling	p. 171
The function [kappa]	p. 172
Evaluation for the paramagnet	p. 174
Evaluation for the antiferromagnet	p. 177
What do our calculations really mean?	p. 178
The low-density limit and complex solutions	p. 180
M-strings	p. 180
Dispersion relations	p. 182
Boosting the bound states	p. 184
Further results	p. 186
Consistency	p. 189
How can we show integrability?	p. 189
The direct approach	p. 189
The constructive approach	p. 190
The indirect approach	p. 192
How can we show non-integrability?	p. 193
Consistency conditions	p. 197
Solution to consistency conditions	p. 205
Two components, conjugation symmetry	p. 205
Consistency conditions	p. 205
Necessary conditions for r and t	p. 208
Bound states	p. 210
Two components, different statistics	p. 213
Three or more components	p. 215
Periodic and twisted boundary conditions	p. 217
Eigenvectors of the transfer matrix, two-components	p. 220
The basic equations	p. 220
The winding matrix and the transfer operator	p. 222
Twisted boundary conditions	p. 224
Properties of the winding matrix and the transfer operator	p. 225
Eigenvalues	p. 228
Explicit expressions for the equations	p. 230
Solution to the general problem	p. 231
Eigenvectors of the transfer matrix: multicomponent system	p. 232
Basic equations	p. 233
The winding matrix and the transfer operator	p. 235
Properties of the winding matrix and the transfer operator	p. 236
Eigenvalues	p. 238
Summary of results	p. 242
The constructive approach	p. 242
Exchange Models	p. 244
Hyperbolic exchange models	p. 245
Integrability	p. 246
The two-body phase shifts	p. 248
The inverse-square exchange potential	p. 249
The [delta]-function potential	p. 253
The case F[superscript 2]--repulsive	p. 254
The case B[superscript 2]	p. 256
The case F[superscript 2]--attractive	p. 258
The case BF	p. 261
The Sinh-Cosh Model	p. 262
The sinh-cosh potential	p. 262
Integrability	p. 263
The two-body problem	p. 265
Phase shifts	p. 267
Spin waves	p. 269
Solution and results for zero temperature and zero spin/charge	p. 270
Exchange Lattice Systems by Freezing	p. 275
Freezing continuum models to give lattice models	p. 275
Review of the continuum exchange model	p. 277
The limit [lambda] to [infinity]	p. 279
The first order equation	p. 280
The densities [rho](x) and e(x)	p. 282
Results for the nearest-neighbor model	p. 284
Results for the hyperbolic model	p. 286
The inverse-square lattice at finite temperature	p. 289
The Hubbard Model	p. 295
The model and its symmetries	p. 295
The model	p. 295
Other statistics	p. 296
The phase diagram and symmetries	p. 298
Bethe's ansatz	p. 299
The wavefunction	p. 299
Well-separated particles	p. 300
Double occupancy	p. 300
Two-body scattering amplitudes	p. 302
A change of variables	p. 303
The repulsive case	p. 304
The fundamental equations	p. 304
The ground state	p. 305
The attractive case	p. 308
The bound state	p. 308
Consistency conditions and the scattering of pairs	p. 312
A fluid of pairs	p. 313
Ground state properties at the symmetric point	p. 314
Excitations at the symmetric point	p. 314
Ground state response	p. 314
Holes and particles in the fluid of pairs	p. 315
Unbound particles	p. 316
Some Two-Body Problems	p. 319
The two-body problem	p. 319
The potential v(r) = c[delta](r)	p. 324
The potential v(r) = [lambda]([lambda] - 1)/r[superscript 2]	p. 326
The potential v(r) = [lambda]([lambda] - 1)/sinh[superscript 2]r	p. 327
The potential v(r) = -[lambda]([lambda] - 1)/cosh[superscript 2]r	p. 331
Combination potentials	p. 336
Representations	p. 338
The permutation group	p. 338
The twisted permutation group	p. 342
Two ways to write the Bethe ansatz	p. 345
Two-body scattering	p. 347
The consistency conditions	p. 349
Representations	p. 350
Twisted boundary conditions	p. 353
Realizations	p. 360
Lattice gases	p. 360
Other realizations	p. 363
Nearest-neighbor hopping	p. 365
Periodic boundary conditions	p. 368
Bibliography	p. 371
Index	p. 377
Table of Contents provided by Rittenhouse. All Rights Reserved.

Beautiful Models

70 Years Of Exactly Solved Quantum Many-body Problems

At a Glance

Paperback

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