A new Q-matrix in the Eight-Vertex Model

TL;DR

This paper introduces a novel Q-matrix construction for the eight-vertex model at roots of unity with odd L, addressing cases where previous methods failed, and explores its properties and limits.

Contribution

It presents a new Q-matrix for the eight-vertex model at roots of unity with odd L, expanding the understanding of integrable models in this regime.

Findings

New Q-matrix depends on spectral parameter and free parameter t.

For t=0, Q-matrix has standard properties.

Six-vertex limit of Q-matrix exists at a specific parameter value.

Abstract

We construct a $Q$-matrix for the eight-vertex model at roots of unity for crossing parameter $\eta=2mK/L$ with odd $L$, a case for which the existing constructions do not work. The new $Q$-matrix $\Q$ depends as usual on the spectral parameter and also on a free parameter $t$. For $t=0$ $\Q$ has the standard properties. For $t\neq 0$, however, it does not commute with the operator $S$ and not with itself for different values of the spectral parameter. We show that the six-vertex limit of $\Q(v,t=iK'/2)$ exists.