Numerical analysis of the spectrum of the Dirac operator in four-dimensional SU(2) gauge fields

Two numerical algorithms for the computation of eigenvalues of Dirac operators in lattice gauge theories are described: one is an accelerated conjugate gradient method, the other one a standard Lanczos method. Results obtained by Cullum's and Willoughby's variant of the Lanczos method (whose convergence behaviour is closely linked with the local spectral density) are presented for euclidean Wilson fermions in quenched and unquenched SU(2) gauge fields. Complete spectra are determined on lattices up to $8^3 \cdot 12$, and we derive numerical values for fermionic determinants and results for spectral densities.Comment: 6 pages, uuencoded tar-compressed ps-file, contribution to the Proceedings of the International Symposium Ahrenshoop on the Theory of Elementary Particles, Buckow'95, talk also given at the DESY Workshop 199

Idealized Multigrid Algorithm for Staggered Fermions

An idealized multigrid algorithm for the computation of propagators of staggered fermions is investigated. Exemplified in four-dimensional $SU(2)$ gauge fields, it is shown that the idealized algorithm preserves criticality under coarsening. The same is not true when the coarse grid operator is defined by the Galerkin prescription. Relaxation times in computations of propagators are small, and critical slowing is strongly reduced (or eliminated) in the idealized algorithm. Unfortunately, this algorithm is not practical for production runs, but the investigations presented here answer important questions of principle.Comment: 11 pages, no figures, DESY 93-046; can be formatted with plain LaTeX article styl

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

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.Comment: 19 pages, 8 figures, HUB-IEP-94/12 and KL-TH 19/94; comes as a uuencoded tar-compressed .ps-fil

An Accelerated Conjugate Gradient Algorithm to Compute Low-Lying Eigenvalues --- a Study for the Dirac Operator in SU(2) Lattice QCD

The low-lying eigenvalues of a (sparse) hermitian matrix can be computed with controlled numerical errors by a conjugate gradient (CG) method. This CG algorithm is accelerated by alternating it with exact diagonalisations in the subspace spanned by the numerically computed eigenvectors. We study this combined algorithm in case of the Dirac operator with (dynamical) Wilson fermions in four-dimensional \SUtwo gauge fields. The algorithm is numerically very stable and can be parallelized in an efficient way. On lattices of sizes $4^4-16^4$ an acceleration of the pure CG method by a factor of~$4-8$ is found.Comment: 25 pages, uuencoded tar-compressed .ps-fil

Improving meson two-point functions in lattice QCD

We describe and test a method to compute Euclidean meson two-point functions in lattice QCD. The contribution from the low-lying eigenmodes of the Dirac operator is averaged over all positions of the quark sources. The contribution from the higher modes is estimated in the traditional way with one or a few source points per lattice. In some channels, we observe a significant improvement in the two-point functions for small quark masses.Comment: 10 pages, 7 figure

Short distance current correlators: Comparing lattice simulations to the instanton liquid

Point to point correlators of currents are computed in quenched QCD using a chiral lattice fermion action, the overlap action. I compare correlators made of exact quark propagators with correlators restricted to low (less than 500 MeV) eigenvalue eigenmodes of the Dirac operator. In many cases they show qualitative resemblence (typically at small values of the quark mass and distances larger than 0.4 fm) and they differ qualitatively at larger quark masses or at very short distance. Lattice results are in qualitative agreement (and in the difference of vector and axial vector channels, quantitative agreement) with the expectations of instanton liquid models. The scalar channel shows the effects of a quenched finite volume zero mode artifact, a negative correlator.Comment: 18 pages, Revtex, 11 postscript figures. Some changes. Version to appear in Phys. Rev.

Scaling tests of the improved Kogut-Susskind quark action

Improved lattice actions for Kogut-Susskind quarks have been shown to improve rotational symmetry and flavor symmetry. In this work we find improved scaling behavior of the rho and nucleon masses expressed in units of a length scale obtained from the static quark potential, and better behavior of the Dirac operator in instanton backgrounds.Comment: 4 pages, 4 figures, Revte