Algebraic Bethe ansatz for the gl(1$|$2) generalized model II: the three gradings

TL;DR

This paper develops an algebraic Bethe ansatz method for all models with a specific 9x9 rational R-matrix and three grading options, including models like the supersymmetric t-J and U models, broadening solvable models.

Contribution

It provides a unified Bethe ansatz solution for all models with the given R-matrix and gradings, extending applicability to various supersymmetric and impurity models.

Findings

Bethe ansatz applicable to all three gradings

Explicit eigenvalue and Bethe equations derived

Solution extends to quantum transfer matrix spectrum

Abstract

The algebraic Bethe ansatz can be performed rather abstractly for whole classes of models sharing the same $R$-matrix, the only prerequisite being the existence of an appropriate pseudo vacuum state. Here we perform the algebraic Bethe ansatz for all models with $9 \times 9$, rational, gl(1$|$2)-invariant $R$-matrix and all three possibilities of choosing the grading. Our Bethe ansatz solution applies, for instance, to the supersymmetric t-J model, the supersymmetric $U$ model and a number of interesting impurity models. It may be extended to obtain the quantum transfer matrix spectrum for this class of models. The properties of a specific model enter the Bethe ansatz solution (i.e. the expression for the transfer matrix eigenvalue and the Bethe ansatz equations) through the three pseudo vacuum eigenvalues of the diagonal elements of the monodromy matrix which in this context are called the parameters of the model.