Thermodynamic properties of a full-replica-symmetry-breaking Ising spin glass on lattice gas: The random Blume-Emery-Griffiths-Capel model

Abstract

The study of the mean-field static solution of the Random Blume-Emery-Griffiths-Capel model, an Ising-spin lattice gas with quenched random magnetic interaction, is performed. The model exhibits a paramagnetic phase, described by a stable Replica Symmetric solution. When the temperature is decreased or the density increases, the system undergoes a phase transition to a Full Replica Symmetry Breaking spin-glass phase. The nature of the transition can be either of the second order (like in the Sherrington-Kirkpatrick model) or, at temperature below a given critical value, of the first order in the Ehrenfest sense, with a discontinuous jump of the order parameter and accompanied by a latent heat. In this last case coexistence of phases takes place. The thermodynamics is worked out in the Full Replica Symmetry Breaking scheme, and the relative Parisi equations are solved using a pseudo-spectral method down to zero temperature.Since its discovery, the spin glass (SG) phase has played and still plays a fundamental role in the investigation and understanding of many basic properties of disordered and complex systems. The analysis of the mean-field approximation of theoretical models displaying such a phase has revealed different possible scenarios, including different kinds of transition from the paramagnetic phase to the SG phase, as well as different kinds of SG phases. Most of the work, however, has been concentrated on just two scenarios.In order of appearance in literature the first scenario is described by a Full Replica Symmetry Breaking (FRSB) solution characterized by a continuous order parameter function, 1 which continuously grows from zero by crossing the transition. The prototype model is the Sherrington-Kirkpatrick (SK) model, 2 a fully connected Ising-spin model with quenched random magnetic interactions.The second scenario, initially introduced by Derrida by means of the Random Energy Model (REM), 3 provides a transition with a jump in the order parameter to a stable low temperature phase in which the replica symmetry is spontaneously broken only once. The order parameter is a step function taking two values q min and the so-called Edwards-Anderson order parameter, 4 q EA , (else said self-overlap), with q min < q EA . In the paramagnetic phase they are both equal to zero. At the transition, q EA grows to a value larger than q min . No discontinuity appear, however, in the thermodynamic functions. Actually, at the transition to the one step Replica Symmetry Breaking (1RSB) SG phase, the Edwards-Anderson order parameter q EA can either grow continuously from zero or jump discontinuously to a finite value. The first case of this second scenario includes Potts-glasses with three or four states, 5 the spherical p-spin spin-glass model in strong magnetic field 6 and some inhomogeneous spherical p-spin model with a mixture of p = 2 and p > 3 interactions.7,8 The latter case includes, instead, Potts-glasses with more than four states 5 , quadrupolar glass models, 5,9 p-spin interaction spin-glass models with p...

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