Scheme of the replica symmetry breaking for short- ranged Ising spin glass within the Bethe- Peierls method

TL;DR

This paper applies the Bethe-Peierls method to short-range Ising spin glasses, deriving an equation for the order parameter near the transition, revealing dimension-dependent behavior and connecting to the Parisi solution in infinite dimensions.

Contribution

It extends the Bethe-Peierls approach to short-range spin glasses, deriving a dimension-dependent order parameter equation near the transition.

Findings

Equation for the spin glass parameter function was derived.

The form of the equation is similar to the SK model but depends on dimension.

In the limit of infinite dimension, the Parisi solution is recovered.

Abstract

Within the Bethe- Peierls method the for short- ranged Ising spin glass, recently formulated by Serva and Paladin, the equation for the spin glass parameter function near the transition to the paramagnetic phase has been carried out. The form of this equation is qualitatively similar to that for Sherrington- Kirpatrick model, but quantitatively the order parametr function depends of the dimension d of the system. In the case d tends to infinity one obtains well known Parisi solution.