The Q-operator and Functional Relations of the Eight-vertex Model at Root-of-unity $\eta = \frac{2m K}{N}$ for odd N

TL;DR

This paper constructs a new Q-operator for the root-of-unity eight-vertex model at specific parameters, enabling the verification of functional relations and connecting it with the chiral Potts model.

Contribution

It introduces a novel Q-operator for the root-of-unity eight-vertex model where previous operators did not exist, and verifies key functional relations.

Findings

Constructed a new Q-operator for odd N at root-of-unity parameters.

Verified the functional relations of the eight-vertex model using the new Q-operator.

Connected the eight-vertex model with the superintegrable N-state chiral Potts model.

Abstract

Following Baxter's method of producing Q_{72}-operator, we construct the Q-operator of the root-of-unity eight-vertex model for the crossing parameter $\eta = \frac{2m K}{N}$ with odd $N$ where Q_{72} does not exist. We use this new Q-operator to study the functional relations in the Fabricius-McCoy comparison between the root-of-unity eight-vertex model and the superintegrable N-state chiral Potts model. By the compatibility of the constructed Q-operator with the structure of Baxter's eight-vertex (solid-on-solid) SOS model, we verify the set of functional relations of the root-of-unity eight-vertex model using the explicit form of the Q-operator and fusion weights of SOS model.