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.
Matrix product states for critical spin chains: Finite-size versus finite-entanglement scaling