Analytic Bethe ansatz and functional equations associated with any simple root systems of the Lie superalgebra sl(r+1|s+1)

TL;DR

This paper develops an analytic Bethe ansatz framework for all simple root systems of the Lie superalgebra sl(r+1|s+1), providing explicit eigenvalue formulas, determinant expressions, and functional relations, enhancing understanding of its integrable models.

Contribution

It introduces a comprehensive analytic Bethe ansatz approach applicable to any simple root system of sl(r+1|s+1), including new determinant formulas and functional relations.

Findings

Eigenvalue formulas expressed via Young supertableaux

Determinant expressions as quantum analogues of classical formulas

Functional relations derived from Hirota bilinear difference equation

Abstract

The Lie superalgebra sl(r+1|s+1) admits several inequivalent choices of simple root systems. We have carried out analytic Bethe ansatz for any simple root systems of sl(r+1|s+1). We present transfer matrix eigenvalue formulae in dressed vacuum form, which are expressed as the Young supertableaux with some semistandard-like conditions. These formulae have determinant expressions, which can be viewed as quantum analogue of Jacobi-Trudi and Giambelli formulae for sl(r+1|s+1). We also propose a class of transfer matrix functional relations, which is specialization of Hirota bilinear difference equation. Using the particle-hole transformation, relations among the Bethe ansatz equations for various kinds of simple root systems are discussed.