Overrelaxation algorithms for lattice field theories.

Abstract

We study overrelaxation algorithms for the thermalization of lattice field theories with multiquadratic and more general actions. Overrelaxation algorithms are one-parameter generalizations of the heat-bath algorithm which satisfy the detailed-balance condition; the parameter is the relaxation parameter \ensuremath{\omega}, 0<\ensuremath{\omega}<2, with \ensuremath{\omega}=1 corresponding to the heat bath. First, we show that the \ensuremath{\omega}\ensuremath{\rightarrow}0 (extreme underrelaxation) limit of the overrelaxation algorithm is equivalent to the Langevin equation approach. We analyze the thermalization of a free-field action, and show that for \ensuremath{\omega}\ensuremath{\sim}2 an overrelaxed Gauss-Seidel algorithm yields a critical slowing down which is independent of wavelength, and has a correlation time which is a factor N smaller than that for an unaccelerated Jacobi iteration, with N the linear dimension of the lattice in lattice units. For a general nonmultiquadratic action, we give a generalized overrelaxation algorithm which satisfies detailed balance with respect to an effective action which is explicitly computable in terms of the original action. In the case of SU(n) lattice gauge theory we use this construction to formulate an overrelaxed algorithm which has exact lattice gauge invariance, and which satisfies detailed balance with respect to an effective action differing from the Wilson action only by terms of relative order ${a}^{2}$ in the continuum limit, with a the lattice spacing.