Multigrid Methods for the Computation of Propagators in Gauge Fields

TL;DR

This paper develops covariant multigrid methods for computing propagators in gauge fields, avoiding gauge fixing and applicable across various dimensions and gauge groups, with demonstrated effectiveness in disordered SU(2) gauge fields.

Contribution

It introduces a ground-state projection-based restriction operator for multigrid algorithms that is gauge-invariant and versatile across different gauge theories and dimensions.

Findings

Multigrid methods work in highly disordered gauge fields.

Ground-state projection defines effective averaging kernels.

Numerical tests in SU(2) gauge fields confirm method effectiveness.

Abstract

NOTE: this is a shortened version of the abstract of the paper. Multigrid methods for propagators in gauge fields are investigated. Gauge fields are incorporated in algorithms in a covariant way. This avoids the necessity for gauge fixing in computations of propagators. The kernel $C$ of the restriction operator which averages from one grid to the next coarser grid is defined by projection on the ground- state of a local Hamiltonian. The idea behind this definition is that the appropriate notion of smoothness depends on the dynamics. The ground-state projection choice of $C$ is usable in arbitrary space-time dimension $d$ and for arbitrary gauge group. We discuss proper averaging operations for bosons and for staggered fermions. The averaging kernels $C$ can be used not only in deterministic multigrid computations, but also in multigrid Monte Carlo simulations, and for the definition of block spins and blocked gauge fields in Monte Carlo renormalization group studies of gauge theories. Actual numerical computations of kernels and propagators are performed in compact four-dimensional $SU(2)$ gauge fields. A central result of the present work is that {\em the multigrid method works in arbitrarily disordered gauge fields, in principle\/}.