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An Elliptic Current Operator for the Eight Vertex Model

K. Fabricius

Published Aug 8, 2007• 5 citations
Publisher

Abstract

Whereas the tools to determine the eigenvalues of the eight-vertex transfer matrix T are well known there has been until recently incomplete knowledge about the eigenvectors of T. We describe the construction of eigenvectors of T corresponding to degenerate eigenvalues and discuss the related hidden elliptic symmetry.
DOI
10.22323/1.038.0003

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We compute the operator which creates the missing degenerate states in the algebraic Bethe ansatz of the eight-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 [12] 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 the transfer matrix T is enlarged to become { s , t , λ c ,1 , ..., λ c , n } with arbitrary string centres λ c , j and arbitrary n .
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 ‘naïve’ string operator vanishes. In this paper, 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.
Baxter's method of solving the eight-vertex model in lattice statistical mechanics is examined from an elementary point of view. It is shown that the algebraic operations in the method can be carried out without recourse to elliptic functions. These include: construction of certain subspaces invariant uti.der the action of the transfer matrix; reduction of the transfer matrix eigenvalue problem to an equivalent ice-type problem and construction of certain matrices which commute with the transfer matrix and satisfy a functional matrix equation.