On exact overlaps for $\mathfrak{gl}(N)$ symmetric spin chains

TL;DR

This paper derives exact, factorized formulas for overlaps between integrable two-site states and wave-functions in $ ext{gl}(N)$ symmetric spin chains, resolving a longstanding problem in nested algebraic Bethe ansatz.

Contribution

It provides a general derivation of overlap formulas for a broad class of integrable states in $ ext{gl}(N)$ spin chains, confirming previous conjectures.

Findings

Overlaps have factorized forms involving one-particle overlaps and Gaudin-like determinants.

Recurrence relations uniquely determine the off-shell overlaps.

Previous overlap formulas are validated by the new derivation.

Abstract

We study the integrable two-site states of the quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(N)$-invariant R-matrix. We investigate the overlaps between the integrable two-site states and the wave-functions. To find exact derivations for the factorized overlap formulas for the nested integrable systems is a longstanding unsolved problem. In this paper we give a derivation for a large class of the integrable states of the $\mathfrak{gl}(N)$ symmetric spin chain. The first part of the derivation is to calculate recurrence relations for the off-shell overlap that uniquely fix it. Using these recursions we prove that the normalized overlaps of the multi-particle states have factorized forms which contain the products of the one-particle overlaps and the ratio of the Gaudin-like determinants. We also show that the previously proposed overlap formulas agree with our general formula.