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

我们研究可积自旋链的本征态与称为“可积初始/终态”的一类特殊状态之间的确切重叠。这些状态满足特殊的可积性约束，并且与可积边界条件密切相关。我们为可积状态导出了新的代数关系，这导致了一组精确重叠的递归关系。我们解决了这些递归关系，从而得出了新的重叠公式，这些公式在XXX Heisenberg链及其可积分的更高自旋推广中有效。然后，我们将可积性条件推广到扭曲的边界条件，并得出相应的精确重叠。最后，我们将可积状态嵌入“变量分离”框架中，并为XXX链的精确重叠得出另一种表示形式。