On factorized overlaps: Algebraic Bethe Ansatz, twists, and separation of variables

We investigate the exact overlaps between eigenstates of integrable spin chains and a special class states called “integrable initial/final states”. These satisfy integrability constraint, they are closely related to boundary conditions. derive new algebraic relations for states, which lead set recursion overlaps. solve these thus we overlap formulas, valid in XXX Heisenberg chain its higher generalizations. Afterwards generalize condition twisted conditions, corresponding Finally, embed into “Separation Variables” framework, an alternative representation chain. Our derivations proofs rigorous, can form basis future investigations involving more complicated models such as nested or long-range deformed systems.