Off-diagonal Bethe Ansatz solution of the $\tau_2$-model

TL;DR

This paper applies the off-diagonal Bethe Ansatz method to solve the quantum $ au_2$-model with inhomogeneity, deriving eigenvalues and Bethe equations, and confirming completeness through numerical analysis.

Contribution

It introduces an off-diagonal Bethe Ansatz solution for the inhomogeneous $ au_2$-model, providing explicit eigenvalues and Bethe equations.

Findings

Eigenvalues expressed via inhomogeneous T-Q relation.

Bethe Ansatz equations derived for the model.

Numerical checks confirm the completeness of the spectrum.

Abstract

The generic quantum $\tau_2$-model (also known as Baxter-Bazhanov-Stroganov (BBS) model) with periodic boundary condition is studied via the off-diagonal Bethe Ansatz method. The eigenvalues of the corresponding transfer matrix (solutions of the recursive functional relations in $\tau_j$-hierarchy) with generic site-dependent inhomogeneity parameters are given in terms of an inhomogeneous T-Q relation with polynomial Q-functions. The associated Bethe Ansatz equations are obtained. Numerical solutions of the Bethe Ansatz equations for small number of sites indicate that the inhomogeneous T-Q relation does indeed give the complete spectrum.