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The Bethe Wavefunction

Book Description

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.

Table of Contents

  1. Cover
  2. Half-title Page
  3. Epigraph
  4. Title Page
  5. Copyright Page
  6. Contents
  7. Foreword
  8. Translator’s note
  9. Introduction
  10. 1. The chain of spin-1/2 atoms
    1. 1.1 Model for a one-dimensional metal
    2. 1.2 Bethe’s method
    3. 1.3 Parameters and quantum numbers
    4. 1.4 Asymptotic positioning of complex momenta
    5. 1.5 State classification and counting
  11. 2. Thermodynamic limit of the Heisenberg–Ising chain
    1. 2.1 Results for the ground state and elementary excitations
    2. 2.2 Calculation method for the elementary excitations
    3. 2.3 Thermodynamics at nonzero temperature: Energy and entropy functionals (Δ ≥ 1)
    4. 2.4 Thermodynamics at nonzero temperature: Thermodynamic functions
    5. Appendix A
  12. 3. Thermodynamics of the spin- chain: Limiting cases
    1. 3.1 The Ising limit
    2. 3.2 The T = ±0 limits
    3. 3.3 T = ∞ limit
  13. 4. δ-Interacting bosons
    1. 4.1 The elementary symmetric wavefunctions
    2. 4.2 Normalization of states in the continuum
    3. 4.3 Periodic boundary conditions
    4. 4.4 Thermodynamic limit
    5. Appendix B
    6. Appendix C
    7. Appendix D
  14. 5. Bethe wavefunctions associated with a reflection group
    1. 5.1 Bosonic gas on a finite interval
    2. 5.2 The generalized kaleidoscope
    3. 5.3 The open chain
    4. Appendix E
  15. 6. Continuum limit of the spin chain
    1. 6.1 δ-Interacting bosons and the Heisenberg–Ising chain
    2. 6.2 Luttinger and Thirring models
    3. 6.3 Massive Thirring model
    4. 6.4 Diagonalization of
  16. 7. The six-vertex model
    1. 7.1 The ice model
    2. 7.2 The transfer matrix
    3. 7.3 Diagonalization
    4. 7.4 The free energy
    5. Appendix F
    6. Appendix G
  17. 8. The eight-vertex model
    1. 8.1 Definition and equivalences
    2. 8.2 The transfer matrix and the symmetries of the self-dual model
    3. 8.3 Relation of the XYZ Hamiltonian to the transfer matrix
    4. 8.4 One-parameter family of commuting transfer matrices
    5. 8.5 A representation of the symmetric group π[sub(N)]
    6. 8.6 Diagonalization of the transfer matrix
    7. 8.7 The coupled equations for the spectrum
    8. Appendix H
    9. Appendix I
  18. 9. The eight-vertex model: Eigenvectors and thermodynamics
    1. 9.1 Reduction to an Ising-type model
    2. 9.2 Equivalence to a six-vertex model
    3. 9.3 The thermodynamic limit
    4. 9.4 Various results on the critical exponents
  19. 10. Identical particles with δ-interactions
    1. 10.1 The Bethe hypothesis
    2. 10.2 Yang’s representation
    3. 10.3 Ternary relations algebra and integrability
    4. 10.4 On the models of Hubbard and Lai
  20. 11. Identical particles with δ-interactions: General solution for two internal states
    1. 11.1 The spin-1/2 fermion problem
    2. 11.2 The operatorial method
    3. 11.3 Sketch of the original solution of the fermion problem
    4. 11.4 On the thermodynamic limit of the fermion system in the vicinity of its ground state
    5. Appendix J
    6. Appendix K
    7. Appendix L
  21. 12. Identical particles with δ-interactions: General solution for n components and limiting cases
    1. 12.1 The transfer matrix Z(k) in a symmetry-adapted basis
    2. 12.2 Recursive diagonalization of matrix Z
    3. 12.3 Zero coupling limit
  22. 13. Various corollaries and extensions
    1. 13.1 A class of completely integrable spin Hamiltonians
    2. 13.2 Other examples of integrable systems
    3. 13.3 Ternary relation and star–triangle relation
    4. 13.4 Ternary relation with Z[sub(5)] symmetry
    5. 13.5 Ternary relations with Z[sup(g)sub(2)] symmetry
    6. 13.6 Notes on a system of distinguishable particles
    7. Appendix M
    8. Appendix N
  23. 14. On the Toda chain
    1. 14.1 Definition
    2. 14.2 Backlund transformation
    3. 14.3 The solitary wave
    4. 14.4 Complete integrability
    5. 14.5 The M-soliton solution for the infinite chain
    6. 14.6 The quantum chain
    7. 14.7 The integral equation for the eigenfunctions
    8. 14.8 Ternary relations and action–angle variables
  24. References
  25. Index