On the complete-spectrum characterization of quantum integrable spin chains via the inhomogeneous T-Q relation

TL;DR

This paper proves that functional relations from the off-diagonal Bethe Ansatz fully characterize the spectrum of XXZ spin chains and that each eigenvalue can be described by a minimal inhomogeneous T-Q relation, applicable with or without inhomogeneity.

Contribution

It establishes the necessary and sufficient conditions for spectrum characterization and introduces a minimal inhomogeneous T-Q relation for eigenvalues in quantum integrable spin chains.

Findings

Functional relations are necessary and sufficient for spectrum characterization.

Eigenvalues can be parameterized by a minimal inhomogeneous T-Q relation.

Results apply to models with and without inhomogeneity.

Abstract

With the XXZ spin chains as examples, we prove two theorems: (1) the functional relations derived from the off-diagonal Bethe Ansatz scheme are the sufficient and necessary conditions to characterize the complete spectrum of the corresponding transfer matrix; (2) each eigenvalue of the transfer matrix can be parameterized by a minimal inhomogeneous T-Q relation. These statements hold for both with and without inhomogeneity. The proof can be generalized to other finite-dimensional quantum integrable models.