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

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.

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• Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields
• A variational method for integrability-breaking Richardson-Gaudin models
• Exact solution for the inhomogeneous Dicke model in the canonical ensemble: thermodynamical limit and finite-size corrections
• An exact solution of spherical mean-field plus orbit-dependent non-separable pairing model with two non-degenerate j-orbits
• Determinant representation of the domain-wall boundary condition partition function of a Richardson-Gaudin model containing one arbitrary spin
• On the exact solvability of the anisotropic central spin model: An operator approach
• Steady-states of out-of-equlibrium inhomogeneous Richardson-Gaudin quantum integrable models in quantum optics