Eigenstates of triangularisable open XXX spin chains and closed-form solutions for the steady state of the open SSEP

Abstract

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.