Loop expansion around the Bethe approximation through the $M$-layer construction

TL;DR

This paper introduces a loop expansion method around the Bethe approximation using the $M$-layer construction, aiming to analyze phase transitions and critical behavior beyond mean-field theory.

Contribution

It develops a diagrammatic loop expansion for the $1/M$ perturbative series around the Bethe approximation, applicable to generic models on graphs, and connects it to field theory techniques.

Findings

The $1/M$ expansion can be expressed as a sum of Feynman diagrams with field theory-like prefactors.

The diagrams are evaluated by considering the original lattice structure with chains and Bethe trees.

Near critical points, the expansion diverges, providing a basis for non-mean-field critical exponent analysis.

Abstract

For every physical model defined on a generic graph or factor graph, the Bethe $M$-layer construction allows building a different model for which the Bethe approximation is exact in the large $M$ limit and it coincides with the original model for $M=1$. The $1/M$ perturbative series is then expressed by a diagrammatic loop expansion in terms of so-called fat-diagrams. Our motivation is to study some important second-order phase transitions that do exist on the Bethe lattice but are either qualitatively different or absent in the corresponding fully connected case. In this case the standard approach based on a perturbative expansion around the naive mean field theory (essentially a fully connected model) fails. On physical grounds, we expect that when the construction is applied to a lattice in finite dimension there is a small region of the external parameters close to the Bethe critical point where strong deviations from mean-field behavior will be observed. In this region, the $1/M$ expansion for the corrections diverges and it can be the starting point for determining the correct non-mean-field critical exponents using renormalization group arguments. In the end, we will show that the critical series for the generic observable can be expressed as a sum of Feynman diagrams with the same numerical prefactors of field theories. However, the contribution of a given diagram is not evaluated associating Gaussian propagators to its lines as in field theories: one has to consider the graph as a portion of the original lattice, replacing the internal lines with appropriate one-dimensional chains, and attaching to the internal points the appropriate number of infinite-size Bethe trees to restore the correct local connectivity of the original model.