Derivation of Matrix Product States for the Heisenberg Spin Chain with Open Boundary Conditions

Using the algebraic Bethe Ansatz, we derive an exact matrix product representation of the Bethe-Ansatz states of the XXZ spin-1/2 Heisenberg chain with open boundary conditions. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices that act on a tensor product of auxiliary spaces. As compared to the matrix product states of the same Heisenberg chain but with periodic boundary conditions, the dimension of the exact auxiliary matrices is enlarged as if the conserved number of spin-flips considered would have been doubled. Our method is generic for any non-nested integrable model, and we show this by deriving a matrix product representation of the Bethe eigenstates of the Lieb-Liniger model. Counterintuitively, the matrices do not depend on the spatial coordinate despite the open boundaries [1], and thus they suggest generic ways of exploiting translational invariance both for finite size and in the thermodynamic limit. // [1] Zhongtao Mei and C. J. Bolech, Phys. Rev. E 95, 032127 (2017).