Random Field $\phi^3$ Model and Parisi-Sourlas Supersymmetry

TL;DR

This paper investigates the random field $\,\phi^3$ model using RG methods, providing evidence for a supersymmetric fixed point related to the Lee-Yang CFT and confirming the dimensional reduction conjecture in dimensions 2 to 8.

Contribution

It demonstrates the existence of a supersymmetric fixed point in the random field $\,\phi^3$ model and classifies operators affecting the RG flow, supporting the Parisi-Sourlas conjecture.

Findings

Confirmed supersymmetric fixed point in $d\le 8$

Classified operators into susy-writable, susy-null, non-susy-writable

Matched critical exponents with Lee-Yang universality class

Abstract

We use the RG framework set up in arXiv:2009.10087 to explore the $\phi^3$ theory with a random field interaction. According to the Parisi-Sourlas conjecture this theory admits a fixed point with emergent supersymmetry which is related to the pure Lee-Yang CFT in two less dimensions. We study the model using replica trick and Cardy variables in $d=8-\epsilon$ where the RG flow is perturbative. Allowed perturbations are singlets under the $S_n$ symmetry that permutes the $n$ replicas. These are decomposed into operators with different scaling dimensions: the lowest dimensional part, `leader', controls the RG flow in the IR; the other operators, `followers', can be neglected. The leaders are classified into: susy-writable, susy-null and non-susy-writable according to their mixing properties. We construct low lying leaders and compute the anomalous dimensions of a number of them. We argue that there is no operator that can destabilize the SUSY RG flow in $d\le 8$. This agrees with the well known numerical result for critical exponents of Branched Polymers (which are in the same universality class as the random field $\phi^3$ model) that match the ones of the pure Lee-Yang fixed point according to dimensional reduction in all $2\le d\le 8$. Hence this is a second strong check of the RG framework that was previously shown to correctly predict loss of dimensional reduction in random field Ising model.