On the solutions of the $Z_n$-Belavin model with arbitrary number of sites

TL;DR

This paper analyzes the periodic $Z_n$-Belavin model with any number of sites using the off-diagonal Bethe Ansatz, deriving eigenvalues through a unified inhomogeneous $T-Q$ relation, and recovers known results in special cases.

Contribution

It introduces a unified inhomogeneous $T-Q$ relation for the $Z_n$-Belavin model with arbitrary sites, extending previous homogeneous solutions.

Findings

Eigenvalues expressed via a unified inhomogeneous $T-Q$ relation

Recovery of homogeneous $T-Q$ relation for specific site numbers

Extension of Bethe Ansatz solutions to arbitrary site numbers

Abstract

The periodic $Z_n$-Belavin model on a lattice with an arbitrary number of sites $N$ is studied via the off-diagonal Bethe Ansatz method (ODBA). The eigenvalues of the corresponding transfer matrix are given in terms of an unified inhomogeneous $T-Q$ relation. In the special case of $N=nl$ with $l$ being also a positive integer, the resulting $T-Q$ relation recovers the homogeneous one previously obtained via algebraic Bethe Ansatz.