Equilibrium properties of disordered spin models with two scale interactions

TL;DR

This paper investigates the equilibrium properties of a composite disordered spin model combining sparse and dense interactions, revealing novel phase transitions and reentrant behaviors through analytical methods.

Contribution

It introduces a new composite model of disordered spins with mixed interactions and analyzes its phases using the replica method, highlighting complex transition behaviors.

Findings

Reentrant spin glass to ferromagnetic transitions at lower temperatures

Discontinuous paramagnetic to ferromagnetic transition with regular connectivity

Sensitivity of high temperature transitions to sparse sub-graph connectivity

Abstract

Methods for understanding classical disordered spin systems with interactions conforming to some idealized graphical structure are well developed. The equilibrium properties of the Sherrington-Kirkpatrick model, which has a densely connected structure, have become well understood. Many features generalize to sparse Erdos-Renyi graph structures above the percolation threshold, and to Bethe lattices when appropriate boundary conditions apply. In this paper we consider spin states subject to a combination of sparse strong interactions with weak dense interactions, which we term a composite model. The equilibrium properties are examined through the replica method, with exact analysis of the high temperature paramagnetic, spin glass and ferromagnetic phases by perturbative schemes. We present results of a replica symmetric variational approximations where perturbative approaches fail at lower temperature. Results demonstrate novel reentrant behaviors from spin glass to ferromagnetic phases as temperature is lowered, including transitions from replica symmetry broken to replica symmetric phases. The nature of high temperature transitions is found to be sensitive to the connectivity profile in the sparse sub-graph, with regular connectivity a discontinuous transition from the paramagnetic to ferromagnetic phases is apparent.