Thermodynamics of One-Dimensional Solvable Models

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ISBN: 9780521551434 | 0521551439
Cover: Hardcover
Copyright: 3/28/1999

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Exactly solvable models are very important in physics. They are important not just from a theoretical point of view but also from the experimentalist's perspective because in such cases theoretical results and experimental results can be compared without ambiguity. This is a book about an important class of exactly solvable models in physics. The subject area is the Bethe-ansatz approach for a number of one-dimensional models, and the setting up of equations within this approach to determine the thermodynamics of these systems. It is a topic that crosses the boundaries between condensed matter physics, mathematics and field theory. The derivation and application of thermodynamic Bethe-ansatz equations for one-dimensional models are explained in detail. This technique is indispensable for physicists studying the low-temperature properties of one-dimensional substances. This book, written by one of the top physicists in this field, and the originator of much of the work in the subject, will be of great interest to theoretical condensed matter physicists.

Preface

Part one: Thermodynamics of non-interacting systems and ground states of interacting systems

(108)

1 Free energy and correlation functions of the XY model

(9)

1.1 The isotropic XY model

(6)

1.1.1 Introduction and historical overview

(2)

1.1.2 Energy eigenvalues of the Hamiltonian and the partition function

(1)

1.1.3 Correlation functions

(3)

1.2 The anisotropic XY model

(3)

1.2.1 The subspace alpha = 1

(1)

1.2.2 The subspace alpha = -1

(1)

1.2.3 The free energy

(1)

2 Systems with a delta-function potential

(24)

2.1 The boson problem

(10)

2.1.1 The c = 0 case

(1)

2.1.2 The c = (infinity) case

(1)

2.1.3 Scattering states of Bose particles with finite interaction

(2)

2.1.4 Periodic boundary conditions

(1)

2.1.5 Linear integral equation for the distribution of quasi-momenta

(1)

2.1.6 Bound states in the case c is less than 0

(2)

2.1.7 Elementary excitations for repulsive bosons

(2)

2.2 The fermion problem

(12)

2.2.1 The c = 0 case

(1)

2.2.2 The two-body problem

(1)

2.2.3 The three-body problem

(3)

2.2.4 The M = 1 and arbitrary N case

(1)

2.2.5 The arbitrary M and N case

(1)

2.2.6 Periodic boundary conditions

(2)

2.2.7 The ground state for c is greater than 0

(1)

2.2.8 The ground state for c is less than 0

(2)

2.2.9 Expansion from small n XXX

(1)

2.2.10 A unified form of the integral equations

(1)

2.3 Bosons and fermions with arbitrary spin

(2)

3 The isotropic Heisenberg model

(12)

3.1 The ferromagnetic case

(3)

3.2 The string solution of an infinite system

(1)

3.3 The Hulthen solution for an antiferromagnet

(2)

3.4 The des Cloizeaux-Pearson mode of an antiferromagnet

(2)

3.5 The magnetic susceptibility and magnetization curve for J is less than 0

(3)

3.6 Wiener-Hopf type integral equation

(2)

4 The XXZ model

(21)

4.1 Symmetry of the Hamiltonian

(1)

4.2 The Bethe-ansatz wave function

(1)

4.3 The string solution for Delta is greater than 1

(1)

4.4 The lowest energy state for Delta is less than -- 1

(2)

4.5 The magnetization curve for a field in the z-direction at Delta is less than -- 1

(1)

4.6 The lowest energy state for fixed M and -1 less than Delta less than 1

(1)

4.7 The magnetization curve for a field in the z-direction at -1 is less than equal to Delta is less than 1

(1)

4.8 Susceptibility for -1 is less than Delta is less than 1

(2)

4.9 The long range order of the XXZ model

(3)

4.10 Excitations from the ground state

(8)

4.10.1 Delta is less than -1 and the energy gap

(2)

4.10.2 Excitations for -1 is less than Delta is less than 1

(1)

4.10.3 Spin-wave like excitations for 0 is less than Delta is less than 1

(2)

4.10.4 The spin-wave bound state

(3)

5 XYZ and eight-vertex models

(18)

5.1 Transfer matrix of the eight-vertex model

(6)

5.1.1 Relation between the six-vertex model and the XXZ model

(3)

5.1.2 The eight-vertex model and the XYZ model

(3)

5.2 The symmetry of the XYZ model

(1)

5.3 Modulus l and modulus k

(2)

5.4 The case J(x) = 0

(1)

5.5 The ground state for J(z) is greater than 0

(2)

5.6 Long range Order

(1)

5.7 Elementary excitations

(5)

5.7.1 The nearly degenerate ground state

(1)

5.7.2 Spinon excitations

(1)

5.7.3 Spin-wave excitations

(2)

5.7.4 Spin-wave bound states

(2)

6 The Hubbard model

(24)

6.1 Symmetry of the Hamiltonian

(3)

6.1.1 Particle-hole symmetry

(2)

6.1.2 SU(2) symmetry

(1)

6.2 The Bethe-ansatz equation for the one-dimensional Hubbard model

(13)

6.2.1 The wave function for a finite system

(2)

6.2.2 Periodic boundary condition

(1)

6.2.3 Fredholm type integral equations for the ground state

(2)

6.2.4 Analytic solution for the ground state in the half-filled case

(1)

6.2.5 Spinon excitation in the half-filled case

(3)

6.2.6 Energy gap of the charge excitation

(2)

6.2.7 Susceptibility and magnetization curve of the half-filled case

(4)

6.3 1/U' expansion

101

(1)

6.4 Perturbation expansion in the half-filled case

102

(2)

6.5 Asymptotic expansion from U = 0

104

(5)

Part two: Finite temperature integral equations for un-nested systems

109

(50)

7 Repulsive delta-function bosons

109

(8)

7.1 Uniqueness of the solution

109

(1)

7.2 Holes of quasi-momenta and their distribution function

110

(1)

7.3 Thermodynamic equilibrium

111

(2)

7.4 Elementary excitations

113

(2)

7.5 Some special limits

115

(2)

7.5.1 The c = (infinity) limit

115

(1)

7.5.2 The c = 0 + limit

116

(1)

7.5.3 The T = 0 + limit

116

(1)

8 Thermodynamics of the XXX chain

117

(13)

8.1 String solution of an infinite XXX chain

117

(1)

8.2 String hypothesis for a long XXX chain

118

(2)

8.3 Thermodynamic Bethe-ansatz equations for the XXX chain

120

(3)

8.4 Some special cases and expansions

123

(7)

8.4.1 The J/T --> 0 case

123

(1)

8.4.2 High-temperature expansion or small J expansion

124

(2)

8.4.3 The low-temperature limit

126

(1)

8.4.4 The fugacity expansion

127

(3)

9 Thermodynamics of the XXZ model

130

(15)

9.1 Thermodynamic equations for the XXZ model for Delta is greater than 1

130

(3)

9.2 Theory for the |Delta| is less than 1 XXZ model

133

(10)

9.2.1 String solution of the infinite XXZ model with |Delta| is less than 1

133

(5)

9.2.2 Scattering phase shift among strings

138

(1)

9.2.3 Bethe-ansatz equation for the XXZ model with |Delta| is less than 1

139

(4)

9.3 Some special limits

143

(2)

9.3.1 The T --> (infinity) or J --> 0 limit

143

(1)

9.3.2 The case J is greater than 0, -1 is less than Delta is less than 0 and T --> 0

144

(1)

9.3.3 The case J is greater than 0, 0 is less than Delta is less than 1 and T --> 0 limit

144

(1)

9.3.4 The Delta = 0 case

144

(1)

10 Thermodynamics of the XYZ model

145

(7)

10.1 Bethe-ansatz equation for the XYZ model

145

(4)

10.2 Some special limits

149

(3)

10.2.1 The T --> (infinity) or J --> 0 limit

149

(1)

10.2.2 The J(z) is greater than 0, J(x) is less than 0 and T --> 0 limit

150

(1)

10.2.3 The J(z) is greater than 0, and T --> 0 limit

150

(1)

10.2.4 The J(x) = 0 case

151

(1)

11 Low-temperature thermodynamics

152

(7)

11.1 The XXZ model

152

(1)

11.1.1 The XXZ model at 2h is greater than J (1-Delta), (Delta) is less than 1

152

(1)

11.1.2 The case J(1-(Delta)/2 is greater than h is greater than 0

152

(1)

11.2 Roger's dilogarithm and specific heat at h = 0

153

(2)

11.2.1 Specific heat of the XXX antiferromagnet

153

(2)

11.3 The ferromagnetic chain and modified spin-wave theory

155

(1)

11.3.1 Numerical analysis of the thermodynamic Bethe-ansatz equation

155

(1)

11.3.2 Spin-wave calculation of the 1D ferromagnetic chain

155

(1)

11.4 The antiferromagnetic XXX model

156

(1)

11.5 The XYZ model at |J(X)| is less than is equal to J(y) is less than J(z)

157

(2)

Part three: Finite temperature integral equations for nested systems

159

(26)

12 S = 1/2 fermions with repulsive potential in the continuum

159

(8)

12.1 Derivation of the thermodynamic equations

159

(5)

12.2 Some special limits

164

(3)

12.2.1 c --> 0+

164

(2)

12.2.2 c --> (infinity)

166

(1)

13 S = 1/2 fermions with an attractive potential

167

(7)

13.1 Derivation of the thermodynamic equations

167

(4)

13.2 Some special limits

171

(3)

13.2.1 c --> 0 -

171

(1)

13.2.2 T --> 0+

172

(2)

14 Thermodynamics of the Hubbard model

174

(11)

14.1 Strings of the Hubbard model

174

(3)

14.2 Thermodynamic Bethe-ansatz equation for the Hubbard model

177

(2)

14.3 Some special limits

179

(6)

14.3.1 The limit U --> (infinity)

179

(1)

14.3.2 The limit U --> 0

180

(2)

14.3.3 The limit T --> 0

182

(1)

14.3.4 The limit t --> 0

183

(2)

Part four: The quantum transfer matrix and recent developments

185

(41)

15 The transfer matrix and correlation length

185

(4)

15.1 The transfer matrix for the Ising chain

185

(1)

15.2 The transfer matrix for the classical Heisenberg model

186

(3)

16 The spin 1/2 XXZ model in a magnetic field

189

(21)

16.1 The diagonal-to-diagonal transfer matrix

189

(9)

16.2 The limit of an infinite Trotter number

198

(3)

16.3 Analytical solutions for special cases

201

(4)

16.3.1 The Ising limit

201

(1)

16.3.2 The XY limit

202

(1)

16.3.3 The T = h = 0 case

203

(2)

16.4 Numerical calculations of the XXZ model

205

(5)

17 The XYZ model with no magnetic field

210

(13)

17.1 The transfer matrix for the XYZ model

210

(8)

17.1.1 Baxter's theory for the inhomogeneous eight-vertex model

211

(1)

17.1.2 Transcendental equations

212

(3)

17.1.3 The limit of M --> (infinity)

215

(3)

17.2 Special cases and numerical methods

218

(3)

17.2.1 The T --> 0 limit

218

(2)

17.2.2 The J(x) = 0 case (anisotropic XY chain)

220

(1)

17.3 Numerical calculations

221

(2)

18 Recent developments and related topics

223

(3)

18.1 Numerical analysis of the S = 1 chain

223

(3)

Appendix A The Young tableau and the theorem of Lieb and Mattis

226

(2)

Appendix B The number of string solutions

228

(3)

Appendix C The commuting transfer matrix and spectral parameter

231

(4)

Appendix D The matrix Q(v)

235

(4)

Appendix E Special functions

239

(7)

Bibliography

246

(5)

Index

251

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Thermodynamics of One-Dimensional Solvable Models

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