Analysis and Development of Stochastic Multigrid Methods in Lattice Field Theory

TL;DR

This paper investigates stochastic multigrid algorithms in lattice field theory, analyzing their critical behavior, proposing a new multigrid method for gauge theories, and demonstrating its effectiveness in reducing critical slowing down.

Contribution

It introduces a new multigrid method called time slice blocking for nonabelian lattice gauge theory and provides a theoretical and numerical analysis of its performance.

Findings

Critical slowing down can be eliminated in 2D SU(2) gauge fields using the new method.

The absence of relevant terms in the Hamiltonian expansion predicts the effectiveness of multigrid algorithms.

Time slice blocking significantly reduces autocorrelation times in 2D gauge simulations.

Abstract

We study the relation between the dynamical critical behavior and the kinematics of stochastic multigrid algorithms. The scale dependence of acceptance rates for nonlocal Metropolis updates is analyzed with the help of an approximation formula. A quantitative study of the kinematics of multigrid algorithms in several interacting models is performed. We find that for a critical model with Hamiltonian H(phi) absence of critical slowing down can only be expected if the expansion of <H(phi+psi)> in terms of the shift psi contains no relevant term (mass term). The predictions of this rule are verified in a multigrid Monte Carlo simulation of the Sine Gordon model in two dimensions. Our analysis can serve as a guideline for the development of new algorithms: We propose a new multigrid method for nonabelian lattice gauge theory, the time slice blocking. For SU(2) gauge fields in two dimensions, critical slowing down is almost completely eliminated by this method, in accordance with the theoretical prediction. The generalization of the time slice blocking to SU(2) in four dimensions is investigated analytically and by numerical simulations. Compared to two dimensions, the local disorder in the four dimensional gauge field leads to kinematical problems.