Global solutions of functional fixed point equations via pseudo-spectral methods

TL;DR

This paper introduces a pseudo-spectral method to accurately solve functional renormalisation group equations, enabling the analysis of global solutions and critical phenomena in various quantum field models.

Contribution

The paper presents a novel pseudo-spectral approach for solving functional fixed point equations, including new results for multi-critical models, the Gross-Neveu model, and scalar-tensor theories.

Findings

Accurate global fixed point potentials obtained.

Critical exponents and eigenfunctions computed.

New results for models in fractional and three dimensions.

Abstract

We apply pseudo-spectral methods to construct global solutions of functional renormalisation group equations in field space to high accuracy. For this, we introduce a basis to resolve both finite as well as asymptotic regions of effective potentials. Our approach is benchmarked using the critical behaviour of the scalar $O(1)$ model, providing results for the global fixed point potential as well as leading critical exponents and their respective global eigenfunctions. We provide new results for (1) multi-critical $O(1)$ models in fractional dimensions, (2) the three-dimensional Gross-Neveu model at both small and large $N$, and (3) the scalar-tensor model, also in three dimensions.