Non-diagonal boundary conditions for gl(1|1) super spin chains

TL;DR

This paper extends the algebraic Bethe ansatz to a supersymmetric fermionic model with non-diagonal boundary conditions using super matrices, providing solutions for the eigenvalue problem in complex boundary scenarios.

Contribution

It introduces a method to incorporate non-diagonal boundary conditions into the graded Quantum Inverse Scattering Method for a supersymmetric fermionic model.

Findings

Eigenvalues obtained via algebraic Bethe ansatz for special boundary conditions.

Spectrum derived from functional relations for generic boundary conditions.

Framework applicable to models with supersymmetry and complex boundary interactions.

Abstract

We study a one-dimensional model of free fermions with $\mathfrak{gl}(1|1)$ supersymmetry and demonstrate how non-diagonal boundary conditions can be incorporated into the framework of the graded Quantum Inverse Scattering Method (gQISM) by means of \emph{super matrices} with entries from a superalgebra. For super hermitian twists and open boundary conditions subject to a certain constraint, we solve the eigenvalue problem for the super transfermatrix by means of the graded algebraic Bethe ansatz technique (gABA) starting from a fermionic coherent state. For generic boundary conditions the algebraic Bethe ansatz can not be applied. In this case the spectrum of the super transfer matrix is obtained from a functional relation.