Fusion approach for quantum integrable system associated with the $\mathfrak{gl}(1|1)$ Lie superalgebra

TL;DR

This paper provides an exact solution for a quantum integrable system linked to the $rak{gl}(1|1)$ Lie superalgebra, using fusion techniques to derive operator identities, spectra, and Bethe ansatz equations, especially for systems with broken $U(1)$ symmetry.

Contribution

It introduces a systematic fusion-based framework to solve and analyze spectra of $rak{gl}(1|1)$ related quantum integrable models with open boundaries and broken symmetries.

Findings

Derived the complete energy spectrum and Bethe ansatz equations.

Established operator identities among fused transfer matrices.

Provided a method to construct Bethe states from the spectrum.

Abstract

In this work we obtain the exact solution of quantum integrable system associated with the Lie superalgebra $\mathfrak{gl}(1|1)$, both for periodic and for generic open boundary conditions. By means of the fusion technique we derive a closed set of operator identities among the fused transfer matrices. These identities allow us to determine the complete energy spectrum and the corresponding Bethe ansatz equations of the model. Our approach furnishes a systematic framework for studying the spectra of quantum integrable models based on Lie superalgebras, in particular when the $U(1)$ symmetry is broken. The derivation of the Bethe states from the exact spectrum is also addressed.