Diffusion of topological charge in lattice QCD simulations

TL;DR

This paper investigates how topological charge observables in lattice QCD autocorrelate over simulation time, modeling their behavior as a diffusion process and analyzing the impact of boundary conditions on autocorrelation times.

Contribution

It demonstrates that autocorrelation functions follow a diffusion equation, measures the diffusion coefficient, and relates boundary conditions to autocorrelation reduction in lattice QCD simulations.

Findings

Autocorrelation functions obey a diffusion equation.

Diffusion coefficient scales with the square of the lattice spacing.

Open boundary conditions reduce autocorrelations at a characteristic lattice spacing.

Abstract

We study the autocorrelations of observables constructed from the topological charge density, such as the topological charge on a time slice or in a subvolume, using a series of hybrid Monte Carlo simulations of pure SU(3) gauge theory with both periodic and open boundary conditions. We show that the autocorrelation functions of these observables obey a simple diffusion equation and we measure the diffusion coefficient, finding that it scales like the square of the lattice spacing. We use this result and measurements of the rate of tunneling between topological charge sectors to calculate the scaling behavior of the autocorrelation times of these observables on periodic and open lattices. There is a characteristic lattice spacing at which open boundary conditions become worthwhile for reducing autocorrelations and we show how this lattice spacing is related to the diffusion coefficient, the tunneling rate, and the lattice Euclidean time extent.