Transfer matrix functional relations for the generalized tau_2(t_q) model

TL;DR

This paper generalizes the $ au_2(t_q)$ model to inhomogeneous cases, derives its functional relations, and re-establishes the eigenvalue structure, facilitating analysis of fixed-spin boundary conditions in the chiral Potts model.

Contribution

It introduces a column-inhomogeneous $ au_2(t_q)$ model and derives its functional relations without relying on traditional chiral Potts relations, enabling new boundary condition analyses.

Findings

Derived functional relations for the inhomogeneous $ au_2(t_q)$ model.

Re-established the eigenvalue structure of the transfer matrix.

Enabled analysis of fixed-spin boundary conditions in the model.

Abstract

The $N$-state chiral Potts model in lattice statistical mechanics can be obtained as a ``descendant'' of the six-vertex model, via an intermediate ``$Q$'' or ``$\tau_2 (t_q)$'' model. Here we generalize this to obtain a column-inhomogeneous $\tau_2 (t_q)$ model, and derive the functional relations satisfied by its row-to-row transfer matrix. We do {\em not} need the usual chiral Potts relations between the $N$th powers of the rapidity parameters $a_p, b_p, c_p, d_p$ of each column. This enables us to readily consider the case of fixed-spin boundary conditions on the left and right-most columns. We thereby re-derive the simple direct product structure of the transfer matrix eigenvalues of this model, which is closely related to the superintegrable chiral Potts model with fixed-spin boundary conditions.