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.
Highlights
One dimensional quantum integrable models are very special systems, where the exact wave functions can be computed using analytic methods, even in the presence of interactions
We show that the integrable initial states can be represented in the Separation of Variables (SoV) framework, and we derive new overlap formulas using this approach
We introduce a new relation, which is very useful for the derivation of overlap formulas: it leads to a new recursion relation for the off-shell overlaps
Summary
One dimensional quantum integrable models are very special systems, where the exact wave functions can be computed using analytic methods, even in the presence of interactions. A new approach was initiated in [31], where the exact formulas were proven regarding all particle numbers, based on the analytic properties of the coordinate Bethe Ansatz expressions This method is based on the ideas of Korepin, developed for the derivation of the norm of the Bethe wave function [32]. It was shown in [31] that the method is applicable to the Heisenberg chain, and to spin chains with non-compact local spaces such as the so-called sl(2, R) chain This method was already used in [33] to give an alternative proof for the overlap formulas between the Lieb-Liniger Bethe states and the Bose-Einstein condensate state, originally derived in [5] and first proven in [34]. We show that the integrable initial states can be represented in the SoV framework, and we derive new overlap formulas using this approach
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