On the Exact Solution of Models based on Non-Standard Representations

TL;DR

This paper develops a systematic method to exactly solve models based on non-standard, higher-dimensional representations of algebraic structures, extending the algebraic Bethe ansatz beyond traditional fusion-type models.

Contribution

It introduces a new systematic approach for diagonalizing transfer-matrices associated with non-fusion, higher-dimensional representations of quantum algebras.

Findings

Successfully diagonalized a transfer-matrix from two different four-dimensional representations.

Extended algebraic Bethe ansatz applicability to non-fusion models.

Demonstrated the method on a model based on $U_{q}(\ ext{hat}\{gl'}(2,1;C))$.

Abstract

The algebraic Bethe ansatz is a powerful method to diagonalize transfer-matrices of statistical models derived from solutions of (graded) Yang Baxter equations, connected to fundamental representations of Lie (super-)algebras and their quantum deformations respectively. It is, however, very difficult to apply it to models based on higher dimensional representations of these algebras in auxiliary space, which are not of fusion type. A systematic approach to this problem is presented here. It is illustrated by the diagonalization of a transfer-matrix of a model based on the product of two different four-dimensional representations of $U_{q}(\hat{gl}'(2,1;C))$.