Properties of the String Operator in the Eight-Vertex Model

TL;DR

This paper proves a conjecture about the behavior of string operators in the eight-vertex model, showing their relation to symmetry operators and their role in generating eigenstates at elliptic roots of unity.

Contribution

It provides a proof that the 'naive' string operator vanishes and clarifies its relation to symmetry operators for chains of odd length.

Findings

The 'naive' string operator vanishes as conjectured.

For odd-length chains, the string operator is proportional to the symmetry operator S or vanishes.

The results depend on the specific crossing parameter at elliptic roots of unity.

Abstract

The construction of creation operators of exact strings in eigenvectors of the eight vertex model at elliptic roots of unity of the crossing parameter which allow the generation of the complete set of degenerate eigenstates is based on the conjecture that the 'naive' string operator vanishes. In this note we present a proof of this conjecture. Furthermore we show that for chains of odd length the string operator is either proportional to the symmetry operator $S$ or vanishes depending on the precise form of the crossing parameter.