Replica theory for Levy spin glasses

TL;DR

This paper develops a replica theoretical framework for Levy spin glasses, analyzing their phase transition behavior and the role of rare strong bonds, with results supported by numerical simulations.

Contribution

It introduces a replica approach at imaginary temperature to analyze Levy spin glasses, deriving self-consistent equations and exploring the transition mechanisms.

Findings

Transition temperature determined via replica-symmetric equations

Percolation of rare strong bonds influences the transition

Numerical simulations confirm theoretical transition temperatures

Abstract

Infinite-range spin-glass models with Levy-distributed interactions show a spin-glass transition with similarities to both the Sherrington-Kirkpatrick model and to disordered spin systems on finite connectivity random graphs. Despite the diverging moments of the coupling distribution the transition can be analyzed within the replica approach by working at imaginary temperature. Within the replica-symmetric approximation a self-consistent equation for the distribution of local fields is derived and from the instability of the paramagnetic solution to this equation the glass-transition temperature is determined. The role of the percolation of rare strong bonds for the transition is elucidated. The results partly agree and partly disagree with those obtained within the cavity approach. Numerical simulations using parallel tempering are in agreement with the transition temperatures found.