P. Claeys

Starting from integrable su(2) (quasi-)spin Richardson–Gaudin (RG) 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 RG models. This allows for the construction of the full set of conserved charges, the Bethe ansatz state, and the resulting RG 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 RG XXZ models.

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We propose an extension of the numerical approach for integrable Richardson-Gaudin models based on a new set of eigenvalue-based variables [A. Faribault et al., Phys. Rev. B 83, 235124 (2011); O. El Araby et al., Phys. Rev. B 85, 115130 (2012)]. Starting solely from the Gaudin algebra, the approach is generalized towards the full class of XXZ Richardson-Gaudin models. This allows for a fast and robust numerical determination of the spectral properties of these models, avoiding the singularities usually arising at the so-called singular points. We also provide different determinant expressions for the normalization of the Bethe ansatz states and form factors of local spin operators, opening up possibilities for the study of larger systems, both integrable and nonintegrable. These expressions can be written in terms of the new set of variables and generalize the results previously obtained for rational Richardson-Gaudin models [A. Faribault and D. Schuricht, J. Phys. A 45, 485202 (2012)] and Dicke-Jaynes-Cummings-Gaudin models [H. Tschirhart and A. Faribault,  J. Phys. A 47, 405204 (2014)]. Remarkably, these results are independent of the explicit parametrization of the Gaudin algebra, exposing a universality in the properties of Richardson-Gaudin integrable systems deeply linked to the underlying algebraic structure.