Truncation Effects in the Functional Renormalization Group Study of Spontaneous Symmetry Breaking

TL;DR

This paper investigates how the local potential approximation within the functional renormalization group framework predicts spontaneous symmetry breaking in O(N) models, revealing limitations and consistency with theoretical expectations.

Contribution

It demonstrates that the local potential approximation can qualitatively capture SSB phenomena and highlights its limitations in predicting SSB where it should not occur.

Findings

LPA yields qualitatively correct SSB results for continuous symmetries.

SSB always appears in LPA solutions, even where it is theoretically forbidden.

Wilson-Fisher fixed points are identified as indicators of SSB in the analysis.

Abstract

We study the occurrence of spontaneous symmetry breaking (SSB) for O(N) models using functional renormalization group techniques. We show that even the local potential approximation (LPA) when treated exactly is sufficient to give qualitatively correct results for systems with continuous symmetry, in agreement with the Mermin-Wagner theorem and its extension to systems with fractional dimensions. For general N (including the Ising model N=1) we study the solutions of the LPA equations for various truncations around the zero field using a finite number of terms (and different regulators), showing that SSB always occurs even where it should not. The SSB is signalled by Wilson-Fisher fixed points which for any truncation are shown to stay on the line defined by vanishing mass beta functions.