Expansion of the effective action around non-Gaussian theories

TL;DR

This paper develops a diagrammatic perturbation framework for the effective action around non-Gaussian theories, extending known methods beyond Gaussian approximations and applying it to models like the Ising model.

Contribution

It introduces a generalized class of irreducible diagrams for non-Gaussian expansions, unifying and extending existing perturbation techniques.

Findings

Derived Feynman rules for non-Gaussian effective actions

Applied method to the Ising model, recovering TAP mean-field theory

Showed that Plefka and high-temperature expansions are special cases

Abstract

This paper derives the Feynman rules for the diagrammatic perturbation expansion of the effective action around an arbitrary solvable problem. The perturbation expansion around a Gaussian theory is well known and composed of one-line irreducible diagrams only. For the expansions around an arbitrary, non-Gaussian problem, we show that a more general class of irreducible diagrams remains in addition to a second set of diagrams that has no analogue in the Gaussian case. The effective action is central to field theory, in particular to the study of phase transitions, symmetry breaking, effective equations of motion, and renormalization. We exemplify the method on the Ising model, where the effective action amounts to the Gibbs free energy, recovering the Thouless-Anderson-Palmer mean-field theory in a fully diagrammatic derivation. Higher order corrections follow with only minimal effort compared to existing techniques. Our results show further that the Plefka expansion and the high-temperature expansion are special cases of the general formalism presented here.