One-loop topological expansion for spin glasses in the large connectivity limit

TL;DR

This paper introduces a novel one-loop topological expansion method around the Bethe solution for high connectivity spin glasses, offering advantages over traditional approaches by directly using the original Hamiltonian and applicable at zero temperature.

Contribution

The paper presents a new expansion technique around the Bethe solution that simplifies calculations and can be extended to zero temperature cases, unlike traditional methods.

Findings

Method reproduces known results from standard field theory.

Advantage of direct use of the original Hamiltonian.

Applicable at zero temperature where traditional models fail.

Abstract

We apply for the first time a new one-loop topological expansion around the Bethe solution to the spin-glass model with field in the high connectivity limit, following the methodological scheme proposed in a recent work. The results are completely equivalent to the well known ones, found by standard field theoretical expansion around the fully connected model (Bray and Roberts 1980, and following works). However this method has the advantage that the starting point is the original Hamiltonian of the model, with no need to define an associated field theory, nor to know the initial values of the couplings, and the computations have a clear and simple physical meaning. Moreover this new method can also be applied in the case of zero temperature, when the Bethe model has a transition in field, contrary to the fully connected model that is always in the spin glass phase. Sharing with finite dimensional model the finite connectivity properties, the Bethe lattice is clearly a better starting point for an expansion with respect to the fully connected model. The present work is a first step towards the generalization of this new expansion to more difficult and interesting cases as the zero-temperature limit, where the expansion could lead to different results with respect to the standard one.