On the supersymmetric XXX spin chains associated to $\mathfrak{gl}_{1|1}$

TL;DR

This paper analyzes the $rak{gl}_{1|1}$ supersymmetric XXX spin chains, providing an explicit algebraic description, eigenvector classification, and Bethe ansatz construction, revealing the structure of eigenvalues and eigenspaces.

Contribution

It offers a detailed algebraic framework for the $rak{gl}_{1|1}$ supersymmetric XXX spin chains, including eigenvector classification and transfer matrix relations.

Findings

Eigenvectors correspond to divisors of an explicit polynomial.

Each eigenspace of the Hamiltonian algebra is one-dimensional.

Bethe ansatz constructs all eigenvectors in irreducible cases.

Abstract

We study the $\mathfrak{gl}_{1|1}$ supersymmetric XXX spin chains. We give an explicit description of the algebra of Hamiltonians acting on any cyclic tensor products of polynomial evaluation $\mathfrak{gl}_{1|1}$ Yangian modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the Drinfeld polynomials. In particular our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also give dimensions of the generalized eigenspaces. We show that when the tensor product is irreducible, then all eigenvectors can be constructed using Bethe ansatz. We express the transfer matrices associated to symmetrizers and anti-symmetrizers of vector representations in terms of the first transfer matrix and the center of the Yangian.