Guerra interpolation for inverse freezing

In these short notes we adapt and systematically apply Guerra’s interpolation techniques on a class of disordered mean field spin glasses equipped with crystal fields and multi-value spin variables. These models undergo the phenomenon of inverse melting or inverse freezing. In particular, we focus on the Ghatak-Sherrington model, its extension provided by Katayama and Horiguchi and the disordered Blume-Emery-Griffiths-Capel model in the mean-field limit deepened by Crisanti and Leuzzi and by Schupper and Shnerb. Once shown how all these models can be retrieved as particular limits of a unique broader Hamiltonian, we study their free energies. We provide explicit expressions of their annealed and quenched expectations, inspecting the cases of replica symmetry and (first step) broken-replica-symmetry. We recover several results previously obtained via heuristic approaches (mainly replica trick) to prove full agreement with the existing Literature. As a sideline, we inspect also the onset of replica symmetry breaking by providing analytically the expression of the De-Almeida and Thouless instability line for a replica symmetric description: in this general setting, the latter is new also from a physical viewpoint.