Guerra interpolation for inverse freezing

TL;DR

This paper applies Guerra's interpolation techniques to analyze inverse freezing phenomena in mean-field spin glasses, providing explicit free energy expressions and insights into replica symmetry breaking and stability.

Contribution

It systematically extends Guerra's interpolation to a broad class of disordered spin models exhibiting inverse freezing, deriving explicit free energy formulas and stability conditions.

Findings

Explicit free energy expressions for models with inverse freezing.

Analytical derivation of the de Almeida-Thouless instability line.

Confirmation of previous heuristic results with rigorous methods.

Abstract

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 the replica trick) to prove full agreement with the existing literature. As a sideline, we also inspect the onset of replica symmetry breaking by providing analytically the expression of the de Almeida-Thouless instability line for a replica symmetric description: in this general setting, the latter is new also from a physical viewpoint.