In this article we study the relation between the eigenstates of open rational spin $\frac{1}{2}$ Heisenberg chains with different boundary conditions. The focus lies on the relation between the spin chain with diagonal boundary conditions and the spin chain with triangular boundary conditions as well as the class of spin chains that can be brought to such form by certain similarity transformations in the physical space. The boundary driven Symmetric Simple Exclusion Process (open SSEP) belongs to the latter. We derive a transformation that maps the eigenvectors of the diagonal spin chain to the eigenvectors of the triangular chain. This transformation yields an essential simplification for determining the states beyond half-filling. It allows to first determine the eigenstates of the diagonal chain through the Bethe ansatz on the fully excited reference state and subsequently map them to the triangular chain for which only the vacuum serves as a reference state. In particular the transformed reference state, i.e. the fully excited eigenstate of the triangular chain, is presented at any length of the chain. It can be mapped to the steady state of the open SSEP. This results in a concise closed-form expression for the probabilities of particle distributions and correlation functions in the steady state. Further, the complete set of eigenstates of the Markov generator is expressed in terms of the eigenstates of the diagonal open chain.

中文翻译：

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可三角化的开放 XXX 自旋链的本征态和开放 SSEP 稳态的封闭形式解

在本文中，我们研究了不同边界条件下开有理自旋 $\frac{1}{2}$ 海森堡链的本征态之间的关系。重点在于具有对角边界条件的自旋链和具有三角形边界条件的自旋链之间的关系以及通过物理空间中的某些相似变换可以得到这种形式的自旋链的类别。边界驱动的对称简单排除过程（open SSEP）属于后者。我们推导出一种变换，将对角自旋链的特征向量映射到三角链的特征向量。这种转换为确定半填充以外的状态提供了基本的简化。它允许首先在完全激发的参考态上通过 Bethe ansatz 确定对角链的本征态，然后将它们映射到三角形链，其中只有真空作为参考态。特别地，变换后的参考态，即三角形链的完全激发的本征态，在链的任何长度处呈现。它可以映射到开放 SSEP 的稳定状态。这导致稳定状态下粒子分布和相关函数的概率的简明封闭形式表达式。此外，马尔可夫生成器的完整本征态集用对角开链的本征态表示。特别地，变换后的参考态，即三角形链的完全激发的本征态，在链的任何长度处呈现。它可以映射到开放 SSEP 的稳定状态。这导致稳定状态下粒子分布和相关函数的概率的简明封闭形式表达式。此外，马尔可夫生成器的完整本征态集用对角开链的本征态表示。特别地，变换后的参考态，即三角形链的完全激发的本征态，在链的任何长度处呈现。它可以映射到开放 SSEP 的稳定状态。这导致稳定状态下粒子分布和相关函数的概率的简明封闭形式表达式。此外，马尔可夫生成器的完整本征态集用对角开链的本征态表示。