Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Supersymmetric Bethe Ansatz and Baxter Equations from Discrete Hirota Dynamics

Vladimir Kazakov, Alexander Sorin, Anton Zabrodin

Abstract

We show that eigenvalues of the family of Baxter Q-operators for supersymmetric integrable spin chains constructed with the gl(K|M)-invariant $R$-matrix obey the Hirota bilinear difference equation. The nested Bethe ansatz for super spin chains, with any choice of simple root system, is then treated as a discrete dynamical system for zeros of polynomial solutions to the Hirota equation. Our basic tool is a chain of Backlund transformations for the Hirota equation connecting quantum transfer matrices. This approach also provides a systematic way to derive the complete set of generalized Baxter equations for super spin chains.

Supersymmetric Bethe Ansatz and Baxter Equations from Discrete Hirota Dynamics

Abstract

We show that eigenvalues of the family of Baxter Q-operators for supersymmetric integrable spin chains constructed with the gl(K|M)-invariant -matrix obey the Hirota bilinear difference equation. The nested Bethe ansatz for super spin chains, with any choice of simple root system, is then treated as a discrete dynamical system for zeros of polynomial solutions to the Hirota equation. Our basic tool is a chain of Backlund transformations for the Hirota equation connecting quantum transfer matrices. This approach also provides a systematic way to derive the complete set of generalized Baxter equations for super spin chains.

Paper Structure

This paper contains 43 sections, 286 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Motivation and background
  3. A sketch of the results
  4. Fusion relations for transfer matrices and Hirota equation
  5. Quantum transfer matrices
  6. Fusion procedure
  7. Functional relations for transfer matrices
  8. Simple examples of the functional equation
  9. Analogy with formulas for characters and identities for symmetric functions
  10. Normalization and boundary conditions
  11. Hierarchy of Hirota equations
  12. Hirota equation and boundary conditions for superalgebra $gl(K|M)$
  13. Auxiliary linear problems and Bäcklund transformations
  14. First Bäcklund transformation
  15. Second Bäcklund transformation
  16. ...and 28 more sections

Figures (22)

  • Figure 1: The domain ("fat hook") of non-vanishing transfer matrices $T(a,s,u)$ for the supersymmetric spin chain with the $gl(K|M)$ symmetry.
  • Figure 2: The $R$-matrix.
  • Figure 3: The Yang-Baxter equation.
  • Figure 4: The monodromy matrix.
  • Figure 5: The Young diagram decorated by spectral parameters.
  • ...and 17 more figures