The Renormalization Group and Two Dimensional Multicritical Effective Scalar Field Theory

TL;DR

This paper investigates the existence and properties of multicritical fixed points in two-dimensional scalar field theories using a derivative expansion of the exact renormalization group equations, achieving results consistent with conformal field theory.

Contribution

The authors develop a derivative expansion approach to identify and analyze multicritical fixed points in 2D scalar theories, providing new non-perturbative insights and critical exponents.

Findings

Identified the first ten fixed points at second order in derivatives.

Achieved agreement with conformal field theory predictions within 0.2% to 33%.

Computed critical exponents and operator dimensions for these fixed points.

Abstract

Direct verification of the existence of an infinite set of multicritical non-perturbative FPs (Fixed Points) for a single scalar field in two dimensions, is in practice well outside the capabilities of the present standard approximate non-perturbative methods. We apply a derivative expansion of the exact RG (Renormalization Group) equations in a form which allows the corresponding FP equations to appear as non-linear eigenvalue equations for the anomalous scaling dimension $\eta$. At zeroth order, only continuum limits based on critical sine-Gordon models, are accessible. At second order in derivatives, we perform a general search over all $\eta\ge.02$, finding the expected first ten FPs, and {\sl only} these. For each of these we verify the correct relevant qualitative behaviour, and compute critical exponents, and the dimensions of up to the first ten lowest dimension operators. Depending on the quantity, our lowest order approximate description agrees with CFT (Conformal Field Theory) with an accuracy between 0.2\% and 33\%; this requires however that certain irrelevant operators that are total derivatives in the CFT are associated with ones that are not total derivatives in the scalar field theory.