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

TL;DR

The paper addresses the exact spectral problem for quantum integrable systems based on the Lie superalgebra gl(1|1) under periodic and generic open boundaries. It introduces a two-branch fusion framework that yields a closed set of operator identities among fused transfer matrices, enabling a complete TQ description and Bethe equations for the spectrum. The authors derive explicit TQ relations for both periodic and open cases, revealing how Grassmann boundary parameters affect eigenstates while leaving energies governed by the same structural equations. The methodology extends naturally to q-deformations and higher-rank superalgebras, offering a versatile route to solving Lie superalgebra–based integrable models without relying on a reference state.

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.

Figures (2)

• Figure 1: The fusion procedure of $R$-matrix.
• Figure 2: Schematic diagram of the transfer matrix fusion procedure. The blue and red lines represent the fist and second fusion branches respectively. The spectral parameter $u_j$ must be set to a specific value at each step, as shown in Eq. \ref{['p']}.