New Q matrices and their functional equations for the eight vertex model at elliptic roots of unity

TL;DR

This paper introduces a new Q matrix for the eight vertex model that exists for all roots of unity and explores the relations and functional equations among various known Q matrices, extending the understanding of the model's solutions.

Contribution

It demonstrates the existence of a new Q matrix applicable to all roots of unity and analyzes the relations and conjectured functional equations among different Q matrices.

Findings

New Q matrix exists for all roots of unity

Relations between known Q matrices are established

Conjectured functional equations for Q matrices

Abstract

The Q matrix invented by Baxter in 1972 to solve the eight vertex model at roots of unity exists for all values of N, the number of sites in the chain, but only for a subset of roots of unity. We show in this paper that a new Q matrix, which has recently been introduced and is non zero only for N even, exists for all roots of unity. In addition we consider the relations between all of the known Q matrices of the eight vertex model and conjecture functional equations for them.