Non-trivial fixed point of a $\psi^4_d$ fermionic theory, II. Anomalous exponent and scaling operators

TL;DR

This paper analyzes a fermionic $ ext{psi}^4_d$ model at a fixed point across dimensions 1 to 3, revealing an anomalous critical exponent for density responses and constructing scale-invariant operators using advanced RG techniques.

Contribution

It provides a rigorous construction of the fixed point, response functions, and scaling operators for a fermionic model with fractional kinetic terms, highlighting the anomalous exponent for density responses.

Findings

Critical exponent for field response is naive.

Density response exponent is anomalous and analytic.

Constructed scale-invariant operators with stretched exponential decay.

Abstract

We consider the Renormalization Group (RG) fixed-point theory associated with a fermionic $\psi^4_d$ model in $d=1,2,3$ with fractional kinetic term, whose scaling dimension is fixed so that the quartic interaction is weakly relevant in the RG sense. The model is defined in terms of a Grassmann functional integral with interaction $V^*$, solving a fixed-point RG equation in the presence of external fields, and a fixed ultraviolet cutoff. We define and construct the field and density scale-invariant response functions, and prove that the critical exponent of the former is the naive one, while that of the latter is anomalous and analytic. We construct the corresponding (almost-)scaling operators, whose two point correlations are scale-invariant up to a remainder term, which decays like a stretched exponential at distances larger than the inverse of the ultraviolet cutoff. Our proof is based on constructive RG methods and, specifically, on a convergent tree expansion for the generating function of correlations, which generalizes the approach developed by three of the authors in a previous publication [A. Giuliani, V. Mastropietro, S. Rychkov, JHEP 01 (2021) 026].