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Using the algebraic Bethe ansatz, we derive a matrix product representationof the exact Bethe-ansatz states of the six-vertex Heisenberg chain (either XXXor XXZ and spin-$\frac{1}{2}$) with open boundary conditions. In thisrepresentation, the components of the Bethe eigenstates are expressed as tracesof products of matrices which act on a tensor product of auxiliary spaces. Ascompared to the matrix product states of the same Heisenberg chain but withperiodic boundary conditions, the dimension of the exact auxiliary matrices isenlarged as if the conserved number of spin-flips considered would have beendoubled. This result is generic for any non-nested integrable model, as isclear from our derivation and we further show by providing an additionalexample of the same matrix product state construction for a well known model ofa gas of interacting bosons. Counterintuitively, the matrices do not depend onthe spatial coordinate despite the open boundaries and thus suggest genericways of exploiting (emergent) translational invariance both for finite size andin the thermodynamic limit.

Derivation of matrix product states for the Heisenberg spin chain with open boundary conditions