Scaling solutions in the derivative expansion

TL;DR

This paper uses the functional renormalization group with spike plot techniques to analyze scalar field theories, revealing detailed phase structures and fixed points in two and three dimensions without relying on conformal invariance.

Contribution

It introduces the application of spike plot techniques to FRG equations at order ∂², providing new insights into the phase structure and fixed points of scalar theories.

Findings

Recovered phase structure consistent with conformal field theory in 2D.

Provided refined analysis of the Wilson-Fisher fixed point in 3D.

Highlighted the effectiveness of shooting techniques in non-perturbative studies.

Abstract

Scalar field theories with $\mathbb{Z}_{2}$-symmetry are the traditional playground of critical phenomena. In this work these models are studied using functional renormalization group (FRG) equations at order $\partial^2$ of the derivative expansion and, differently from previous approaches, the spike plot technique is employed to find the relative scaling solutions in two and three dimensions. The anomalous dimension of the first few universality classes in $d=2$ is given and the phase structure predicted by conformal field theory is recovered (without the imposition of conformal invariance), while in $d=3$ a refined view of the standard Wilson-Fisher fixed point is found. Our study enlightens the strength of shooting techniques in studying FRG equations, suggesting them as candidates to investigate strongly non-perturbative theories even in more complex cases.