Boundary Overlaps from Functional Separation of Variables

TL;DR

This paper introduces a novel application of the Functional Separation of Variables method to compute overlaps with integrable boundary states in an su(3) spin chain, providing determinant formulas for a wide class of states and operators.

Contribution

The paper develops a general FSoV-based framework for calculating overlaps with boundary states, including off-shell Bethe states and multiple operator insertions, in integrable systems.

Findings

Determinant formulas for overlaps with boundary states and off-shell Bethe states.

Explicit representations for Principal Operator insertions.

Construction of a complete basis of integrable boundary states.

Abstract

In this paper we show how the Functional Separation of Variables (FSoV) method can be applied to the problem of computing overlaps with integrable boundary states in integrable systems. We demonstrate our general method on the example of a particular boundary state, a singlet of the symmetry group, in an su(3) rational spin chain in an alternating fundamental--anti-fundamental representation. The FSoV formalism allows us to compute in determinant form not only the overlaps of the boundary state with the eigenstates of the transfer matrix, but in fact with any factorisable state. This includes off-shell Bethe states, whose overlaps with the boundary state have been out of reach with other methods. Furthermore, we also found determinant representations for insertions of so-called Principal Operators (forming a complete algebra of all observables) between the boundary and the factorisable state as well as certain types of multiple insertions of Principal Operators. Concise formulas for the matrix elements of the boundary state in the SoV basis and su(N) generalisations are presented. Finally, we managed to construct a complete basis of integrable boundary states by repeated action of conserved charges on the singlet state. As a result, we are also able to compute the overlaps of all of these states with integral of motion eigenstates.