Michel Gaudin's book La fonction d'onde de Bethe is a uniquely influential masterpiece on exactly solvable models of quantum mechanics and statistical physics. Available in English for the first time, this translation brings his classic work to a new generation of graduate students and researchers in physics. It presents a mixture of mathematics interspersed with powerful physical intuition, retaining the author's unmistakably honest tone. The book begins with the Heisenberg spin chain, starting from the coordinate Bethe Ansatz and culminating in a discussion of its thermodynamic properties. Delta-interacting bosons (the Lieb-Liniger model) are then explored, and extended to exactly solvable models associated to a reflection group. After discussing the continuum limit of spin chains, the book covers six- and eight-vertex models in extensive detail, from their lattice definition to their thermodynamics. Later chapters examine advanced topics such as multi-component delta-interacting systems, Gaudin magnets and the Toda chain.

Foreword page ix

Translator’s note x

Introduction xii

1 The chain of spin-1/2 atoms 1

1.1 Model for a one-dimensional metal 1

1.2 Bethe’s method 3

1.3 Parameters and quantum numbers 8

1.4 Asymptotic positioning of complex momenta 15

1.5 State classification and counting 19

2 Thermodynamic limit of the Heisenberg–Ising chain 27

2.1 Results for the ground state and elementary excitations 27

2.2 Calculation method for the elementary excitations 30

2.3 Thermodynamics at nonzero temperature: Energy and entropy functionals ( ≥ 1) 33

2.4 Thermodynamics at nonzero temperature: Thermodynamic functions 37

Appendix A 42

3 Thermodynamics of the spin-1/2 chain: Limiting cases 44

3.1 The Ising limit 44

3.2 The T = ±0 limits 46

3.3 T = ∞ limit 52

4 δ-Interacting bosons 54

4.1 The elementary symmetric wavefunctions 54

4.2 Normalization of states in the continuum 56

4.3 Periodic boundary conditions 63

4.4 Thermodynamic limit 67

Appendix B 70

Appendix C 75

Appendix D 77

5 Bethe wavefunctions associated with a reflection group 79

5.1 Bosonic gas on a finite interval 79

5.2 The generalized kaleidoscope 83

5.3 The open chain 89

Appendix E 91

6 Continuum limit of the spin chain 94

6.1 δ-Interacting bosons and the Heisenberg–Ising chain 94

6.2 Luttinger and Thirring models 99

6.3 Massive Thirring model 105

6.4 Diagonalization of HT 107

7 The six-vertex model 112

7.1 The ice model 112

7.2 The transfer matrix 114

7.3 Diagonalization 119

7.4 The free energy 124

Appendix F 137

Appendix G 140

8 The eight-vertex model 142

8.1 Definition and equivalences 142

8.2 The transfer matrix and the symmetries of the self-dual model 146

8.3 Relation of the XYZ Hamiltonian to the transfer matrix 150

8.4 One-parameter family of commuting transfer matrices 153

8.5 A representation of the symmetric group πN 158

8.6 Diagonalization of the transfer matrix 162

8.7 The coupled equations for the spectrum 167

Appendix H 173

Appendix I 175

9 The eight-vertex model: Eigenvectors and thermodynamics 179

9.1 Reduction to an Ising-type model 179

9.2 Equivalence to a six-vertex model 185

9.3 The thermodynamic limit 194

9.4 Various results on the critical exponents 199

10 Identical particles with δ-interactions 203

10.1 The Bethe hypothesis 203

10.2 Yang’s representation 207

10.3 Ternary relations algebra and integrability 211

10.4 On the models of Hubbard and Lai 220

11 Identical particles with δ-interactions: General solution for two internal states 223

11.1 The spin-1/2 fermion problem 223

11.2 The operatorial method 231

11.3 Sketch of the original solution of the fermion problem 233

11.4 On the thermodynamic limit of the fermion system in the vicinity of its ground state 236

Appendix J 244

Appendix K 249

Appendix L 250

12 Identical particles with δ-interactions: General solution for components and limiting cases 253

12.1 The transfer matrix Z(k) in a symmetry-adapted basis 253

12.2 Recursive diagonalization of matrix Z 257

12.3 Zero coupling limit 264

13 Various corollaries and extensions 268

13.1 A class of completely integrable spin Hamiltonians 268

13.2 Other examples of integrable systems 275

13.3 Ternary relation and star–triangle relation 279

13.4 Ternary relation with Z5 symmetry 282

13.5 Ternary relations with Zg 2 symmetry 287

13.6 Notes on a system of distinguishable particles 292

Appendix M 294

Appendix N 298

14 On the Toda chain 301

14.1 Definition 301

14.2 Bäcklund transformation 301

14.3 The solitary wave 303

14.4 Complete integrability 304

14.5 The M-soliton solution for the infinite chain 307

14.6 The quantum chain 308

14.7 The integral equation for the eigenfunctions 310

14.8 Ternary relations and action–angle variables 311

References 314

Index 323

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