Auxiliary matrices on both sides of the equator

TL;DR

This paper investigates auxiliary matrices in the six-vertex model at roots of unity, revealing differences based on chain length and roots of unity order, and proves a conjecture on degeneracies using functional equations.

Contribution

It provides a detailed analysis of auxiliary matrices at roots of unity, including differences for even and odd chain lengths, and proves a conjecture on degeneracies using new functional equations.

Findings

Eigenvalues differ for even and odd chain lengths at roots of unity.

For odd sites and even roots of unity, two solutions to Baxter's TQ-equation are found.

The conjecture on degeneracies of the six-vertex model is proven.

Abstract

The spectra of previously constructed auxiliary matrices for the six-vertex model at roots of unity are investigated for spin-chains of even and odd length. The two cases show remarkable differences. In particular, it is shown that for even roots of unity and an odd number of sites the eigenvalues contain two linear independent solutions to Baxter's TQ-equation corresponding to the Bethe ansatz equations above and below the equator. In contrast, one finds for even spin-chains only one linear independent solution and complete strings. The other main result is the proof of a previous conjecture on the degeneracies of the six-vertex model at roots of unity. The proof rests on the derivation of a functional equation for the auxiliary matrices which is closely related to a functional equation for the eight-vertex model conjectured by Fabricius and McCoy.