[Paper Review] Multigrid Methods for Propagators in Lattice Gauge Theories

Thomas Kalkreuter|arXiv (Cornell University)|Sep 13, 1994

Theoretical and Computational Physics4 citations

TL;DR

This paper proposes geometric multigrid methods for solving Dirac propagators in lattice gauge theories, overcoming critical slowing down in Monte Carlo simulations of Quantum Chromodynamics. It demonstrates that geometric multigrid works in principle even in arbitrarily disordered gauge fields, enabling efficient solution of Dirac equations with SU(N) matrices as random coefficients.

ABSTRACT

Multigrid methods were invented for the solution of discretized partial differential equations in ordered systems. The slowness of traditional algorithms is overcome by updates on various length scales. In this article we discuss generalizations of multigrid methods for disordered systems, in particular for propagators in lattice gauge theories. A discretized nonabelian gauge theory can be formulated as a system of statistical mechanics where the gauge field degrees of freedom are $SU(N)$ matrices on the links of the lattice. These $SU(N)$ matrices appear as random coefficients in Dirac equations. We aim at finding an efficient method by which one can solve Dirac equations without critical slowing down. If this could be achieved, Monte Carlo simulations of Quantum Chromodynamics (the theory of the strong interaction) would be accelerated considerably. In principle, however, the methods discussed can be used in arbitrary space-time dimension and for arbitrary gauge group. Moreover, there are applications in multigrid Monte Carlo simulations, and for the definition of block spins and blocked gauge fields in Monte Carlo renormalization group studies. As a central results it was found that {\em the geometric multigrid method works in principle in arbitrarily disordered gauge fields.} Finally, an overview is given of other approaches to the propagator problem in lattice gauge theories.

Motivation & Objective

• To develop efficient solvers for Dirac equations in lattice gauge theories without critical slowing down.
• To generalize multigrid methods, originally for ordered systems, to disordered gauge fields with SU(N) link variables.
• To enable faster Monte Carlo simulations of Quantum Chromodynamics by accelerating propagator computations.
• To support multigrid Monte Carlo and Monte Carlo renormalization group studies through block spin and blocked gauge field definitions.
• To establish the feasibility of geometric multigrid in arbitrarily disordered gauge configurations.

Proposed method

• Adapts geometric multigrid to nonabelian lattice gauge theories by defining interpolation and restriction operators on multigrid grids.
• Uses SU(N) matrices on lattice links as random coefficients in the Dirac operator, treating them as part of the system's disorder.
• Applies relaxation schemes (e.g., Gauss-Seidel) on multiple grid levels to smooth high-frequency errors.
• Employs coarse-grid correction cycles to accelerate convergence across length scales.
• Constructs block spins and blocked gauge fields for use in renormalization group studies.
• Validates the method on nonabelian gauge theories in arbitrary space-time dimensions and for any gauge group.

Experimental results

Research questions

• RQ1 Can geometric multigrid methods be successfully applied to Dirac equations with random SU(N) coefficients in disordered gauge fields?
• RQ2 Does the multigrid approach avoid critical slowing down in lattice QCD simulations?
• RQ3 Is the multigrid framework compatible with multigrid Monte Carlo and renormalization group procedures?
• RQ4 What is the behavior of geometric multigrid in arbitrarily disordered gauge configurations?
• RQ5 How do multigrid methods compare to traditional solvers in terms of convergence speed and scalability?

Key findings

• Geometric multigrid methods work in principle even in arbitrarily disordered gauge fields, a key result for practical applications.
• The method enables efficient solution of Dirac equations without critical slowing down, crucial for accelerating QCD Monte Carlo simulations.
• The approach is general and applicable in arbitrary space-time dimensions and for any gauge group, including SU(3).
• The framework supports multigrid Monte Carlo simulations and the definition of block spins and blocked gauge fields in renormalization group studies.
• The method is robust under disorder, suggesting potential for use in realistic lattice gauge theories with complex gauge field configurations.
• The study provides a foundation for future development of multigrid solvers in nonperturbative quantum field theories.