An elliptic current operator for the 8 vertex model

TL;DR

This paper introduces an elliptic current operator for the 8 vertex model at roots of unity, revealing a non-nilpotent structure that allows arbitrary string additions to eigenstates, expanding the parameter space.

Contribution

It defines a new elliptic current operator for the 8 vertex model at roots of unity and shows it is not nilpotent, unlike in the six-vertex model, enabling more flexible eigenstate construction.

Findings

The current operator is not nilpotent, unlike in the six-vertex model.

Arbitrary strings can be added to eigenstates, enlarging parameter space.

Eigenstates can include arbitrary string centers, increasing degeneracy.

Abstract

We compute the operator which creates the missing degenerate states in the algebraic Bethe ansatz of the 8 vertex model at roots of unity and relate it to the concept of an elliptic current operator. We find that in sharp contrast with the corresponding formalism in the six-vertex model at roots of unity the current operator is not nilpotent with the consequence that in the construction of degenerate eigenstates of the transfer matrix an arbitrary number of exact strings can be added to the set of regular Bethe roots. Thus the original set of free parameters {s,t} of an eigenvector of T is enlarged to become {s,t,\lambda_{c,1}, ..., \lambda_{c,n}\} with arbitrary string centers \lambda_{c,j} and arbitrary n.