Auxiliary Field Loop Expansion of the Effective Action for Stochastic Partial Differential Equations

TL;DR

This paper introduces an auxiliary field loop expansion method for stochastic PDEs, deriving effective actions for various models, and compares non-perturbative results with existing theories, revealing new insights into symmetry and renormalization.

Contribution

It develops a non-perturbative auxiliary field loop expansion approach for stochastic PDEs, providing new effective actions and insights into symmetry and renormalization.

Findings

The approach yields the effective potential for several models including KPZ and Ginzburg-Landau.

For KPZ, the method does not show fluctuation-induced symmetry breaking, contradicting earlier studies.

Some renormalization group flows are obtained directly from the effective potential.

Abstract

We present an alternative to the perturbative diagrammatic approach for studying stochastic dynamics. Our approach is based on an auxiliary field loop expansion for the path integral representation for the generating functional of the noise induced correlation functions. We derive two different effective actions, one based on the Onsager-Machlup (OM) approach, and the other on the Martin-Siggia-Rose (MSR) response function approach. In particular we determine the leading order approximation for the effective action and effective potential for arbitrary spatial dimensions for several simple systems. These include the Kardar-Parisi-Zhang (KPZ) equation, the chemical reaction annihilation and diffusion process $A+A \rightarrow 0$, and the Ginzburg-Landau (GL) model for spin relaxation. We show how to obtain the effective potential of the OM approach from the effective potential in the MSR approach. For the KPZ equation we find that our approximation, which is non-perturbative and obeys broken symmetry Ward identities, does not lead to the appearance of a fluctuation induced symmetry breakdown. This contradicts the results of earlier studies. We also obtain some of the renormalization group flows directly from the effective potential and compare our results with exact and perturbative results.