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A Pedestrian's Way to Baxter's Bethe Ansatz for the Periodic XYZ Chain

Xin Zhang, Andreas Klümper, Vladislav Popkov

Abstract

A chiral coordinate Bethe ansatz method is developed to study the periodic XYZ chain. We construct a set of chiral vectors with fixed number of kinks. All vectors are factorized and have simple structures. Under roots of unity conditions, the Hilbert space has an invariant subspace and our vectors form a basis of this subspace. We propose a Bethe ansatz solely based on the action of the Hamiltonian on the chiral vectors, avoiding the use of transfer matrix techniques. This allows to parameterize the expansion coefficients and derive the homogeneous Bethe ansatz equations whose solutions give the exact energies and eigenstates. Our analytic results agree with earlier approaches, notably by Baxter, and are supported by numerical calculations.

A Pedestrian's Way to Baxter's Bethe Ansatz for the Periodic XYZ Chain

Abstract

A chiral coordinate Bethe ansatz method is developed to study the periodic XYZ chain. We construct a set of chiral vectors with fixed number of kinks. All vectors are factorized and have simple structures. Under roots of unity conditions, the Hilbert space has an invariant subspace and our vectors form a basis of this subspace. We propose a Bethe ansatz solely based on the action of the Hamiltonian on the chiral vectors, avoiding the use of transfer matrix techniques. This allows to parameterize the expansion coefficients and derive the homogeneous Bethe ansatz equations whose solutions give the exact energies and eigenstates. Our analytic results agree with earlier approaches, notably by Baxter, and are supported by numerical calculations.
Paper Structure (16 sections, 119 equations, 1 figure, 7 tables)

This paper contains 16 sections, 119 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. XYZ Model and Chiral Subspace Conditions
  3. Chiral basis vectors
  4. $M=0$ case: perfect helix without kinks
  5. One-kink and two-kinks cases $M=1,2$
  6. Generalization of the chiral Bethe Ansatz to arbitrary $M$ case
  7. XXZ and XX limits of the XYZ model
  8. Elliptic theta functions
  9. Expressions of some functions
  10. Derivation of the XYZ eigenstates and BAE for the $M=2$ case
  11. Derivation of the $S$-matrix in (\ref{['S;matrix']})
  12. Derivation of BAE
  13. $M=1$ case
  14. $M=2$ case
  15. Degeneration of the inhomogeneous BAE
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Visualization of the vectors $\{\ket{{\mathit d};n_1,n_2}\}$ for $M=2,s=2$. Any state in (\ref{['Basis;M']}) corresponds to a directed path. Here we denote $\otimes_{n=1}^N\psi(\eta y_{{\mathit d},n})\equiv\ket{{\mathit d};n_1,n_2,\ldots,n_M}$.