An algebraic derivation of the eigenspaces associated with an Ising-like spectrum of the superintegrable chiral Potts model

TL;DR

This paper uses algebraic methods to derive the eigenspaces of an Ising-like spectrum in the superintegrable chiral Potts model, revealing the structure of Bethe eigenstates and their algebraic properties.

Contribution

It provides an algebraic derivation of invariant subspaces linked to the Ising-like spectrum using the $rak{sl}_2$ loop algebra and Bethe-ansatz, connecting Bethe states to eigenspaces.

Findings

Bethe eigenstates lead to an Ising-like spectrum.

Every regular Bethe state is an eigenvector of the transfer matrix.

The $ au_2$-model commutes with the $rak{sl}_2$ loop algebra.

Abstract

In terms of the $\mathfrak{sl}_{2}$ loop algebra and the algebraic Bethe-ansatz method, we derive the invariant subspace associated with a given Ising-like spectrum consisting of $2^{r}$ eigenvalues of the diagonal-to-diagonal transfer matrix of the superintegrable chiral Potts (SCP) model with arbitrary inhomogeneous parameters. We show that every regular Bethe eigenstate of the $\tau_2$-model leads to an Ising-like spectrum and is an eigenvector of the SCP transfer matrix which is given by the product of two diagonal-to-diagonal transfer matrices with a constraint on the spectral parameters. We also show in a sector that the $\tau_2$-model commutes with the $\mathfrak{sl}_{2}$ loop algebra, $L(\mathfrak{sl}_{2})$, and every regular Bethe state of the $\tau_2$-model is of highest weight. Thus, from physical assumptions such as the completeness of the Bethe ansatz, it follows in the sector that every regular Bethe state of the $\tau_2$-model generates an $L(\mathfrak{sl}_{2})$-degenerate eigenspace and it gives the invariant subspace, i.e. the direct sum of the eigenspaces associated with the Ising-like spectrum.