Derivations for the MPS overlap formulas of rational spin chains

A bstract We derive a universal formula for the overlaps between integrable matrix product states (MPS) and Bethe eigenstates in $$ {\mathfrak{gl}}_N $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>gl</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> symmetric spin chains. This formula expresses the normalized overlap as a product of a MPS-independent Gaudin-determinant ratio and a MPS-dependent scalar factor constructed from eigenvalues of commuting operators, defined via the K -matrix associated with the MPS. Our proof is fully representation-independent and relies solely on algebraic Bethe Ansatz techniques and the KT -relation. We also propose a generalization of the overlap formula to $$ {\mathfrak{so}}_N $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>so</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> and $$ {\mathfrak{sp}}_N $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>sp</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> spin chains, supported by algebra embeddings and low-rank isomorphisms. These results significantly broaden the class of integrable initial states for which exact overlap formulas are available, with implications for quantum quenches and defect CFTs.