On factorized overlaps: Algebraic Bethe Ansatz, twists, and Separation of Variables

TL;DR

This paper derives exact overlap formulas for integrable states in spin chains using algebraic Bethe Ansatz and Separation of Variables, extending to twisted boundary conditions and higher spins.

Contribution

It introduces new algebraic relations and recursion formulas for overlaps, generalizes integrability conditions to twisted boundaries, and embeds states into the Separation of Variables framework.

Findings

Derived exact overlap formulas for the XXX Heisenberg chain.

Extended overlap formulas to twisted boundary conditions.

Embedded integrable states into the Separation of Variables framework.

Abstract

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.