Eigenvalue-based determinants for scalar products and form factors in Richardson-Gaudin integrable models coupled to a bosonic mode

TL;DR

This paper develops eigenvalue-based determinant formulas for scalar products and form factors in Richardson-Gaudin models coupled to bosonic modes, extending previous results and simplifying the analysis of these integrable systems.

Contribution

It introduces a pseudo-deformation approach to derive conserved charges and Bethe Ansatz states for bosonic-coupled Richardson-Gaudin models, extending eigenvalue-based determinant expressions.

Findings

Derived simplified quadratic equations for eigenvalues

Extended determinant formulas to bosonic-coupled models

Linked results to previously known Richardson-Gaudin models

Abstract

Starting from integrable $su(2)$ (quasi-)spin Richardson-Gaudin XXZ models we derive several properties of integrable spin models coupled to a bosonic mode. We focus on the Dicke-Jaynes-Cummings-Gaudin models and the two-channel $(p+ip)$-wave pairing Hamiltonian. The pseudo-deformation of the underlying $su(2)$ algebra is here introduced as a way to obtain these models in the contraction limit of different Richardson-Gaudin models. This allows for the construction of the full set of conserved charges, the Bethe Ansatz state, and the resulting Richardson-Gaudin equations. For these models an alternative and simpler set of quadratic equations can be found in terms of the eigenvalues of the conserved charges. Furthermore, the recently proposed eigenvalue-based determinant expressions for the overlaps and form factors of local operators are extended to these models, linking the results previously presented for the Dicke-Jaynes-Cummings-Gaudin models with the general results for Richardson-Gaudin XXZ models.