Nested Algebraic Bethe Ansatz for Open Spin Chains with Even Twisted Yangian Symmetry

We present a nested algebraic Bethe ansatz for a one-dimensional open spin chain whose boundary quantum spaces are irreducible \(\mathfrak {so}_{2n}\)- or \(\mathfrak {sp}_{2n}\)-representations, and the monodromy matrix satisfies the defining relations of the Olshanskii twisted Yangian \(Y^\pm (\mathfrak {gl}_{2n})\). We use a generalization of the Bethe ansatz introduced by De Vega and Karowski which allows us to relate the spectral problem of a \(\mathfrak {so}_{2n}\)- or \(\mathfrak {sp}_{2n}\)-symmetric open spin chain to that of a \(\mathfrak {gl}_{n}\)-symmetric periodic spin chain. We explicitly derive the structure of the Bethe vectors and the nested Bethe equations.