Thermodynamics of the L\'evy spin glass

TL;DR

This paper analyzes the thermodynamics of the Levy spin glass, a mean-field model with power-law couplings, revealing the absence of a replica-symmetric phase and detailing the effects of replica symmetry breaking.

Contribution

It introduces a method to study the Levy spin glass by combining small and strong couplings, and calculates thermodynamic functions and the de Almeida-Thouless line.

Findings

No replica-symmetric spin glass phase exists.

Leading corrections to ground-state energy are determined.

Effects of replica symmetry breaking are minimal.

Abstract

We investigate the L\'evy glass, a mean-field spin glass model with power-law distributed couplings characterized by a divergent second moment. By combining extensively many small couplings with a spare random backbone of strong bonds the model is intermediate between the Sherrington-Kirkpatrick and the Viana-Bray model. A truncated version where couplings smaller than some threshold $\eps$ are neglected can be studied within the cavity method developed for spin glasses on locally tree-like random graphs. By performing the limit $\eps\to 0$ in a well-defined way we calculate the thermodynamic functions within replica symmetry and determine the de Almeida-Thouless line in the presence of an external magnetic field. Contrary to previous findings we show that there is no replica-symmetric spin glass phase. Moreover we determine the leading corrections to the ground-state energy within one-step replica symmetry breaking. The effects due to the breaking of replica symmetry appear to be small in accordance with the intuitive picture that a few strong bonds per spin reduce the degree of frustration in the system.