| Summary: | We consider integrable Matrix Product States (MPS) in integrable spin chains
and show that they correspond to "operator valued" solutions of the so-called
twisted Boundary Yang-Baxter (or reflection) equation. We argue that the
integrability condition is equivalent to a new linear intertwiner relation,
which we call the "square root relation", because it involves half of the steps
of the reflection equation. It is then shown that the square root relation
leads to the full Boundary Yang-Baxter equations. We provide explicit solutions
in a number of cases characterized by special symmetries. These correspond to
the "symmetric pairs" $(SU(N),SO(N))$ and $(SO(N),SO(D)\otimes SO(N-D))$, where
in each pair the first and second elements are the symmetry groups of the spin
chain and the integrable state, respectively. These solutions can be considered
as explicit representations of the corresponding twisted Yangians, that are new
in a number of cases. Examples include certain concrete MPS relevant for the
computation of one-point functions in defect AdS/CFT.
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