Eigenvectors in the Superintegrable Model II: Ground State Sector

TL;DR

This paper analyzes the eigenvectors of the transfer matrix in the ground state sector of the superintegrable chiral Potts model, revealing algebraic structures and explicit eigenvector constructions.

Contribution

It extends previous work by explicitly constructing eigenvectors and expressing the transfer matrix in terms of $sl_2$ algebra generators for the ground state sector.

Findings

Eigenvectors are expressed in terms of rotated $H_m$ eigenvectors.

Transfer matrix is represented using $sl_2$ algebra generators.

Degenerate eigenspaces are generated by $sl_2$ algebra elements.

Abstract

In 1993, Baxter gave $2^{m_Q}$ eigenvalues of the transfer matrix of the $N$-state superintegrable chiral Potts model with spin-translation quantum number $Q$, where $m_Q=\lfloor(NL-L-Q)/N\rfloor$. In our previous paper we studied the Q=0 ground state sector, when the size $L$ of the transfer matrix is chosen to be a multiple of $N$. It was shown that the corresponding $\tau_2$ matrix has a degenerate eigenspace generated by the generators of $r=m_0$ simple $sl_2$ algebras. These results enable us to express the transfer matrix in the subspace in terms of these generators $E_m^{\pm}$ and $H_m$ for $m=1,...,r$. Moreover, the corresponding $2^r$ eigenvectors of the transfer matrix are expressed in terms of rotated eigenvectors of $H_m$.