Parafermions in the tau-2 model II

TL;DR

This paper demonstrates how to express the eigenvectors of the $ au_2(t)$ model Hamiltonian in terms of parafermionic raising operators, proving a long-standing conjecture and unifying different approaches.

Contribution

It proves the last unproven conjecture in Baxter's earlier work by analyzing raising operators, thus enabling explicit eigenvector construction for the $ au_2(t)$ model.

Findings

Eigenvectors expressed in terms of parafermionic raising operators

Proof of the remaining unproven conjecture in Baxter's $ au_2(t)$ model

Unification of different notations and approaches for the Hamiltonian

Abstract

Many years ago Baxter introduced an inhomogeneous two-dimensional classical spin model, now called the $\tau_2(t)$ model with free boundary conditions, and he specialized the resulting quantum spin-chain Hamiltonian in a special limit to a simple clock Hamiltonian. Recently, Fendley showed that this clock Hamiltonian can be expressed in terms of free "parafermions." Baxter followed this up by showing that this construction generalizes to the more general $\tau_2(t)$ model, provided some conjectures hold. In this paper, we will compare the different notations and approaches enabling us to express the Hamiltonians in terms of projection operators as introduced by Fendley. By examining the properties of the raising operators, we are then able to prove the last unproven conjecture in Baxter's paper left in our previous paper. Thus the eigenvectors can all be written in terms of these raising operators.