Spectrum of the Dirac Operator and Inversion Algorithms with Dynamical Staggered Fermions

Complete spectra of the staggered Dirac operator \Dirac are determined in four-dimensional $SU(2)$ gauge fields with and without dynamical fermions. An attempt is made to relate the performance of multigrid and conjugate gradient algorithms for propagators with the distribution of the eigenvalues of~\Dirac.Comment: 3 pages, 1 figure, uuencoded tar-compressed .ps-file, contribution to LATTICE'94, report# HUB-IEP-94/2

Universal correlations in spectra of the lattice QCD Dirac operator

Recently, Kalkreuter obtained complete Dirac spectra for $SU(2)$ lattice gauge theory both for staggered fermions and for Wilson fermions. The lattice size was as large as $12^4$. We performed a statistical analysis of these data and found that the eigenvalue correlations can be described by the Gaussian Symplectic Ensemble for staggered fermions and by the Gaussian Orthogonal Ensemble for Wilson fermions. In both cases long range spectral fluctuations are strongly suppressed: the variance of a sequence of levels containing $n$ eigenvalues on average is given by $\Sigma_2(n) \sim 2 (\log n)/\beta\pi^2$ ($\beta$ is equal to 4 and 1, respectively) instead of $\Sigma_2(n) = n$ for a random sequence of levels. Our findings are in agreement with the anti-unitary symmetry of the lattice Dirac operator for $N_c=2$ with staggered fermions which differs from Wilson fermions (with the continuum anti-unitary symmetry). For $N_c = 3$, we predict that the eigenvalue correlations are given by the Gaussian Unitary Ensemble.Comment: Talk present at LATTICE96(chirality in QCD), 3 pages, Late

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

Multigrid for propagators of staggered fermions in four-dimensional SU(2)SU(2) gauge fields

Multigrid (MG) methods for the computation of propagators of staggered fermions in non-Abelian gauge fields are discussed. MG could work in principle in arbitrarily disordered systems. The practical variational MG methods tested so far with a ``Laplacian choice'' for the restriction operator are not competitive with the conjugate gradient algorithm on lattices up to $18^4$. Numerical results are presented for propagators in $SU(2)$ gauge fields.Comment: 4 pages, 3 figures (one LaTeX-figure, two figures appended as encapsulated ps files); Contribution to LATTICE '92, requires espcrc2.st

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

Neural multigrid for gauge theories and other disordered systems

We present evidence that multigrid works for wave equations in disordered systems, e.g. in the presence of gauge fields, no matter how strong the disorder, but one needs to introduce a "neural computations" point of view into large scale simulations: First, the system must learn how to do the simulations efficiently, then do the simulation (fast). The method can also be used to provide smooth interpolation kernels which are needed in multigrid Monte Carlo updates.Comment: 9 pages [2 figures appended in PostScript format], preprint DESY 92-126, Sept. 199

Another Look at Neural Multigrid

We present a new multigrid method called neural multigrid which is based on joining multigrid ideas with concepts from neural nets. The main idea is to use the Greenbaum criterion as a cost functional for the neural net. The algorithm is able to learn efficient interpolation operators in the case of the ordered Laplace equation with only a very small critical slowing down and with a surprisingly small amount of work comparable to that of a Conjugate Gradient solver. In the case of the two-dimensional Laplace equation with SU(2) gauge fields at beta=0 the learning exhibits critical slowing down with an exponent of about z = 0.4. The algorithm is able to find quite good interpolation operators in this case as well. Thereby it is proven that a practical true multigrid algorithm exists even for a gauge theory. An improved algorithm using dynamical blocks that will hopefully overcome the critical slowing down completely is sketched.Comment: 13 pages, 3 ps figures, uses IJMPC styl

Approach to the Continuum Limit of the Quenched Hermitian Wilson-Dirac Operator

We investigate the approach to the continuum limit of the spectrum of the Hermitian Wilson-Dirac operator in the supercritical mass region for pure gauge SU(2) and SU(3) backgrounds. For this we study the spectral flow of the Hermitian Wilson-Dirac operator in the range $0\le m\le 2$. We find that the spectrum has a gap for $0 < m \le m_1$ and that the spectral density at zero, $\rho(0;m)$, is non-zero for $m_1\le m\le 2$. We find that $m_1\to 0$ and, for $m \ne 0, \rho(0;m)\to 0$ (exponential in the lattice spacing) as one goes to the continuum limit. We also compute the topological susceptibility and the size distribution of the zero modes. The topological susceptibility scales well in the lattice spacing for both SU(2) and SU(3). The size distribution of the zero modes does not appear to show a peak at a physical scale.Comment: 19 pages revtex with 9 postscript figures included by eps

Probing the Region of Massless Quarks in Quenched Lattice QCD using Wilson Fermions

We study the spectrum of $H(m)=\gamma_5 W(-m)$ with $W(m)$ being the Wilson-Dirac operator on the lattice with bare mass equal to $m$. The background gauge fields are generated using the SU(3) Wilson action at $\beta=5.7$ on an $8^3\times 16$ lattice. We find evidence that the spectrum of $H(m)$ is gapless for $1.02 < m < 2.0$, implying that the physical quark is massless in this whole region.Comment: 22 pages, LaTeX file, uses elsart.sty, includes 11 figures A typographical error in one reference has been fixe

A study of chiral symmetry in quenched QCD using the Overlap-Dirac operator

We compute fermionic observables relevant to the study of chiral symmetry in quenched QCD using the Overlap-Dirac operator for a wide range of the fermion mass. We use analytical results to disentangle the contribution from exact zero modes and simplify our numerical computations. Details concerning the numerical implementation of the Overlap-Dirac operator are presented.Comment: 24 pages revtex with 5 postscript figures included by eps