On the determinant representations of Gaudin models' scalar products and form factors

TL;DR

This paper introduces new determinant formulas for scalar products and form factors in rational Gaudin models that do not require Bethe solutions, simplifying numerical analysis and potentially aiding non-equilibrium studies.

Contribution

It presents alternative determinant representations that avoid Bethe equations, using partition functions with domain wall boundary conditions, and expresses form factors solely in terms of eigenvalues of conserved charges.

Findings

New determinant formulas for scalar products and form factors.

Form factors depend only on eigenvalues of conserved charges.

Simplifies numerical procedures for studying decoherence in Gaudin models.

Abstract

We propose alternative determinant representations of certain form factors and scalar products of states in rational Gaudin models realized in terms of compact spins. We use alternative pseudo-vacuums to write overlaps in terms of partition functions with domain wall boundary conditions. Contrarily to Slavnovs determinant formulas, this construction does not require that any of the involved states be solutions to the Bethe equations; a fact that could prove useful in certain non-equilibrium problems. Moreover, by using an atypical determinant representation of the partition functions, we propose expressions for the local spin raising and lowering operators form factors which only depend on the eigenvalues of the conserved charges. These eigenvalues define eigenstates via solutions of a system of quadratic equations instead of the usual Bethe equations. Consequently, the current work allows important simplifications to numerical procedures addressing decoherence in Gaudin models.