Bethe equations for the critical three-state Potts spin chain with toroidal boundary conditions

TL;DR

This paper develops Bethe ansatz equations for the critical three-state Potts quantum chain with twisted boundary conditions, connecting spectral analysis with conformal field theory predictions.

Contribution

It introduces a new parameterization of the spectrum using Bethe equations for the Potts chain with toroidal boundary conditions, including analysis of eigenvalues and eigenstates.

Findings

Bethe equations accurately describe the spectrum for small lattices.

Low-lying excitations exhibit fractional spins consistent with conformal field theory.

Framework allows construction of integrable Hamiltonians with mixed boundary conditions.

Abstract

In this paper, we consider the parameterization of the spectra of the three-state critical Potts quantum chain with integrable twisted boundary conditions in terms of Bethe ansatz type equations. The Bethe equations are found by investigating the structure of the eigenvalues of the respective twisted transfer matrices, and with the help of certain identities satisfied by the product of transfer matrix operators. We have studied the completeness of the spectrum in terms of the Bethe roots for small lattice sizes and have computed the eigenstate momenta. We found that the spins of the low-lying excitations can have fractional values in accordance with predictions of the underlying conformal field theory. We argue that our framework can be used to build integrable Hamiltonians whose spectra are determined by mixing different toroidal boundary conditions.