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.

我们研究了\(\mathfrak {gl}_{1|1}\)超对称 XXX 自旋链。我们给出了作用于多项式评估的任何循环张量积的哈密顿量的代数的明确描述\(\mathfrak {gl}_{1|1}\)扬安模块。因此，在哈密顿量的代数的公共特征向量（直到成比例）和用 Drinfeld 多项式写成的显式多项式的单调因数之间存在双射。特别是我们的结果意味着哈密顿算子代数的每个公共特征空间都具有一维。我们还给出了广义特征空间的维度。我们表明，当张量积不可约时，所有特征向量都可以使用 Bethe ansatz 构造。我们根据第一个转移矩阵和杨根的中心来表达与向量表示的对称化和反对称化相关的转移矩阵。