Duality and Symmetry in Chiral Potts Model

TL;DR

This paper uncovers a duality in the N-state chiral Potts model that relates low and high temperature regimes, connecting its spectral properties to symmetries of related models and extending the understanding of integrable systems.

Contribution

It introduces an Ising-type duality in the general N-state chiral Potts model, linking spectral and symmetry properties across different temperature regimes.

Findings

Established the duality relation between low and high temperature chiral Potts models.

Connected the Onsager-algebra symmetry with the $sl_2$-loop-algebra symmetry.

Derived exact relationships between eigenstates of related models.

Abstract

We discover an Ising-type duality in the general $N$-state chiral Potts model, which is the Kramers-Wannier duality of planar Ising model when N=2. This duality relates the spectrum and eigenvectors of one chiral Potts model at a low temperature (of small $k'$) to those of another chiral Potts model at a high temperature (of $k'^{-1}$). The $\tau^{(2)}$-model and chiral Potts model on the dual lattice are established alongside the dual chiral Potts models. With the aid of this duality relation, we exact a precise relationship between the Onsager-algebra symmetry of a homogeneous superintegrable chiral Potts model and the $sl_2$-loop-algebra symmetry of its associated spin-$\frac{N-1}{2}$ XXZ chain through the identification of their eigenstates.