$U_q osp(2,2)$ Lattice Models

TL;DR

This paper constructs new lattice models with $U_q osp(2,2)$ symmetry, providing solutions to the graded Yang-Baxter equation, and offers Bethe ansatz solutions and conjectures for their ground states.

Contribution

It introduces novel trigonometric $R$-matrices with three continuous parameters and analyzes their Bethe ansatz solutions, expanding the understanding of $U_q osp(2,2)$ lattice models.

Findings

New solutions to the graded Yang-Baxter equation with three parameters.

Bethe ansatz solutions for certain $R$-matrices.

Conjectured solutions for models with complex representation theory.

Abstract

In this paper I construct lattice models with an underlying $U_q osp(2,2)$ superalgebra symmetry. I find new solutions to the graded Yang-Baxter equation. These {\it trigonometric} $R$-matrices depend on {\it three} continuous parameters, the spectral parameter, the deformation parameter $q$ and the $U(1)$ parameter, $b$, of the superalgebra. It must be emphasized that the parameter $q$ is generic and the parameter $b$ does not correspond to the `nilpotency' parameter of \cite{gs}. The rational limits are given; they also depend on the $U(1)$ parameter and this dependence cannot be rescaled away. I give the Bethe ansatz solution of the lattice models built from some of these $R$-matrices, while for other matrices, due to the particular nature of the representation theory of $osp(2,2)$, I conjecture the result. The parameter $b$ appears as a continuous generalized spin. Finally I briefly discuss the problem of finding the ground state of these models.