Conformal Invariance in (2+1)-Dimensional Stochastic Systems

TL;DR

This paper explores conformal invariance in (2+1)-dimensional stochastic systems, proposing a method to derive the energy-momentum tensor and applying it to the KPZ surface growth model, confirming its conformal fixed point.

Contribution

It introduces a general solution to translation Ward identities in stochastic systems using the Martin-Siggia-Rose formalism, incorporating replicated fields to address dimensional reduction issues.

Findings

Derived a conformal energy-momentum tensor for (2+1)D stochastic systems.

Applied the method to the KPZ model, confirming its $c=1$ conformal fixed point.

Validated the approach through perturbative analysis of the KPZ ultraviolet behavior.

Abstract

Stochastic partial differential equations can be used to model second order thermodynamical phase transitions, as well as a number of critical out-of-equilibrium phenomena. In (2+1) dimensions, many of these systems are conjectured (and some are indeed proved) to be described by conformal field theories. We advance, in the framework of the Martin-Siggia-Rose field theoretical formalism of stochastic dynamics, a general solution of the translation Ward identities, which yields a putative conformal energy-momentum tensor. Even though the computation of energy-momentum correlators is obstructed, in principle, by dimensional reduction issues, these are bypassed by the addition of replicated fields to the original (2+1)-dimensional model. The method is illustrated with an application to the Kardar-Parisi-Zhang (KPZ) model of surface growth. The consistency of the approach is checked by means of a straightforward perturbative analysis of the KPZ ultraviolet region, leading, as expected, to its $c=1$ conformal fixed point.