# Fourier acceleration, the HMC algorithm and renormalizability

## TL;DR

This paper extends the analysis of the HMC algorithm to include Fourier acceleration, showing it reduces critical slowing down by separating auto-correlation functions into evolution time and space-time factors, especially in perturbation theory.

## Contribution

It demonstrates that Fourier acceleration fundamentally alters the structure of the HMC algorithm for $\,phi^4$ theory, eliminating critical slowing down in perturbative expansions.

## Key findings

- Auto-correlation functions separate into evolution time and space-time factors.
- Fourier acceleration removes critical slowing down at the lattice scale.
- The structure change applies to both Langevin and HMC algorithms.

## Abstract

The analysis developed by L\"uscher and Schaefer of the Hybrid Monte Carlo (HMC) algorithm is extended to include Fourier acceleration. We show for the $\phi^4$ theory that Fourier acceleration substantially changes the structure of the theory for both the Langevin and HMC algorithms. When expanded in perturbation theory, each five-dimensional auto-correlation function of the fields $\phi(x_i, t_i)$, $1\le i \le N $, corresponding to a specific 4-dimensional Feynman graph separates into two factors: one depending on the Monte-Carlo evolution times $t_i$ and the second depending on the space-time positions $x_i$. This separation implies that only auto-correlation times at the lattice scale appear, eliminating critical slowing down in perturbation theory.