The TQ equation of the 8 vertex model for complex elliptic roots of unity

TL;DR

This paper extends Baxter's TQ equation for the 8 vertex model to complex elliptic roots of unity, exploring various cases based on the parity of parameters, thereby broadening the understanding of the model's solutions.

Contribution

It introduces a generalization of the TQ equation for complex elliptic roots of unity, considering different parameter parity cases, which was not previously addressed.

Findings

Multiple cases depending on the parity of parameters are identified.

The generalized TQ equation is formulated for complex elliptic roots of unity.

The study broadens the theoretical understanding of the 8 vertex model solutions.

Abstract

We extend our studies of the TQ equation introduced by Baxter in his 1972 solution of the 8 vertex model with parameter $\eta$ given by $2L\eta=2m_1K+im_2K'$ from $m_2=0$ to the more general case of complex $\eta.$ We find that there are several different cases depending on the parity of $m_1$ and $m_2$.