Spectrum of the Dirac Operator and Multigrid Algorithm with Dynamical Staggered Fermions

TL;DR

This paper analyzes the spectra of the staggered Dirac operator in quenched and unquenched SU(2) gauge fields and investigates how these spectra influence the performance of multigrid and conjugate gradient algorithms.

Contribution

It provides a detailed spectral analysis of the Dirac operator and relates these spectra to the efficiency of multigrid and conjugate gradient algorithms in lattice QCD simulations.

Findings

Conjugate gradient convergence depends mainly on the condition number and lattice size.

Multigrid convergence is influenced by the spectrum in a more complex manner.

CG convergence in unquenched fields can be predicted from quenched simulations.

Abstract

Complete spectra of the staggered Dirac operator $\Dirac$ are determined in quenched four-dimensional $SU(2)$ gauge fields, and also in the presence of dynamical fermions. Periodic as well as antiperiodic boundary conditions are used. An attempt is made to relate the performance of multigrid (MG) and conjugate gradient (CG) algorithms for propagators with the distribution of the eigenvalues of~$\Dirac$. The convergence of the CG algorithm is determined only by the condition number~$\kappa$ and by the lattice size. Since~$\kappa$'s do not vary significantly when quarks become dynamic, CG convergence in unquenched fields can be predicted from quenched simulations. On the other hand, MG convergence is not affected by~$\kappa$ but depends on the spectrum in a more subtle way.