Solving functional flow equations with pseudo-spectral methods

TL;DR

This paper demonstrates the application of pseudo-spectral methods to solve functional flow equations with high precision, successfully analyzing models like the O(N) model and quantum potentials, including challenging cases.

Contribution

It introduces pseudo-spectral methods for high-accuracy integration of functional flow equations, extending previous work and tackling complex models with numerical difficulties.

Findings

Accurate flow analysis of O(N) models in various dimensions and N values.

Successful resolution of flows over many orders of magnitude.

Effective handling of complex quantum-mechanical potential flows.

Abstract

We apply pseudo-spectral methods to integrate functional flow equations with high accuracy, extending earlier work on functional fixed point equations \cite{Borchardt:2015rxa}. The advantages of our method are illustrated with the help of two classes of models: first, to make contact with literature, we investigate flows of the O$(N)$-model in 3 dimensions, for $N=1, 4$ and in the large $N$ limit. For the case of a fractal dimension, $d=2.4$, and $N=1$, we follow the flow along a separatrix from a multicritical fixed point to the Wilson-Fisher fixed point over almost 13 orders of magnitude. As a second example, we consider flows of bounded quantum-mechanical potentials, which can be considered as a toy model for Higgs inflation. Such flows pose substantial numerical difficulties, and represent a perfect test bed to exemplify the power of pseudo-spectral methods.