Random field ϕ3 model and Parisi-Sourlas supersymmetry

Abstract

Abstract We use the RG framework set up in [1] to explore the ϕ3 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 − ϵ where the RG flow is perturbative. Allowed perturbations are singlets under the Sn 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 ≤ 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 ϕ3 model) that match the ones of the pure Lee-Yang fixed point according to dimensional reduction in all 2 ≤ d ≤ 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.