Quantum Loop Subalgebra and Eigenvectors of the Superintegrable Chiral Potts Transfer Matrices

TL;DR

This paper explores the structure of eigenspaces of the superintegrable chiral Potts transfer matrix, revealing degeneracies and algebraic decompositions, and constructs eigenvectors assuming certain algebraic relations hold.

Contribution

It generalizes the quantum loop algebra framework to the Q≠0 case and constructs eigenvectors under conjectured algebraic relations.

Findings

Eigenspaces are highly degenerate for Q≠0.

Eigenspaces split into two parts with 2^{r-1} eigenvectors each.

Eigenvectors are constructed assuming the validity of Serre relations.

Abstract

It has been shown in earlier works that for Q=0 and L a multiple of N, the ground state sector eigenspace of the superintegrable tau_2(t_q) model is highly degenerate and is generated by a quantum loop algebra L(sl_2). Furthermore, this loop algebra can be decomposed into r=(N-1)L/N simple sl_2 algebras. For Q not equal 0, we shall show here that the corresponding eigenspace of tau_2(t_q) is still highly degenerate, but splits into two spaces, each containing 2^{r-1} independent eigenvectors. The generators for the sl_2 subalgebras, and also for the quantum loop subalgebra, are given generalizing those in the Q=0 case. However, the Serre relations for the generators of the loop subalgebra are only proven for some states, tested on small systems and conjectured otherwise. Assuming their validity we construct the eigenvectors of the Q not equal 0 ground state sectors for the transfer matrix of the superintegrable chiral Potts model.