Universal and non-universal behavior in Dirac spectra

We have computed ensembles of complete spectra of the staggered Dirac operator using four-dimensional SU(2) gauge fields, both in the quenched approximation and with dynamical fermions. To identify universal features in the Dirac spectrum, we compare the lattice data with predictions from chiral random matrix theory for the distribution of the low-lying eigenvalues. Good agreement is found up to some limiting energy, the so-called Thouless energy, above which random matrix theory no longer applies. We determine the dependence of the Thouless energy on the simulation parameters using the scalar susceptibility and the number variance.Comment: LATTICE98(confine), 9 pages, 11 figure

Kramers Equation Algorithm with Kogut-Susskind Fermions on Lattice

We compare the performance of the Kramers Equation Monte Carlo (KMC) Algorithm with that of the Hybrid Monte Carlo (HMC) algorithm for numerical simulations with dynamical Kogut-Susskind fermions. Using the lattice Gross-Neveu model in 2 space-time dimensions, we calculate the integrated autocorrelation time of different observables at a number of couplings in the scaling region on 16^2 and 32^2 lattices while varying the parameters of the algorithms for optimal performance. In our investigation the performance of KMC is always significantly below than that of HMC for the observables used. We also stress the importance of having a large number of configurations for the accurate estimation of the integrated autocorrelation time.Comment: revised version to appear in Phys. Lett. B, 9 pages, 3 ps figure

Can we do better than Hybrid Monte Carlo in Lattice QCD?

The Hybrid Monte Carlo algorithm for the simulation of QCD with dynamical staggered fermions is compared with Kramers equation algorithm. We find substantially different autocorrelation times for local and nonlocal observables. The calculations have been performed on the parallel computer CRAY T3D.Comment: Talk presented at LATTICE96(algorithms), LaTeX 3 pages, uses espcrc2, epsf, 2 postscript figure

Equivalent of a Thouless energy in lattice QCD Dirac spectra

Random matrix theory (RMT) is a powerful statistical tool to model spectral fluctuations. In addition, RMT provides efficient means to separate different scales in spectra. Recently RMT has found application in quantum chromodynamics (QCD). In mesoscopic physics, the Thouless energy sets the universal scale for which RMT applies. We try to identify the equivalent of a Thouless energy in complete spectra of the QCD Dirac operator with staggered fermions and $SU_c(2)$ lattice gauge fields. Comparing lattice data with RMT predictions we find deviations which allow us to give an estimate for this scale.Comment: LATTICE99 (theor. devel.), 3 pages, 4 figure

Random Matrix Theory, Chiral Perturbation Theory, and Lattice Data

Recently, the chiral logarithms predicted by quenched chiral perturbation theory have been extracted from lattice calculations of hadron masses. We argue that the deviations of lattice results from random matrix theory starting around the so-called Thouless energy can be understood in terms of chiral perturbation theory as well. Comparison of lattice data with chiral perturbation theory formulae allows us to compute the pion decay constant. We present results from a calculation for quenched SU(2) with Kogut-Susskind fermions at \beta=2.0 and 2.2.Comment: LaTeX, 12 pages, 7 .eps figure

Random Matrix Theory and Chiral Logarithms

Recently, the contributions of chiral logarithms predicted by quenched chiral perturbation theory have been extracted from lattice calculations of hadron masses. We argue that a detailed comparison of random matrix theory and lattice calculations allows for a precise determination of such corrections. We estimate the relative size of the m*log(m), m, and m^2 corrections to the chiral condensate for quenched SU(2).Comment: LaTeX (elsart.cls), 9 pages, 6 .eps figures, added reference, altered discussion of Eq.(9

Beyond the Thouless energy

The distribution and the correlations of the small eigenvalues of the Dirac operator are described by random matrix theory (RMT) up to the Thouless energy $E_c\propto 1/\sqrt{V}$, where $V$ is the physical volume. For somewhat larger energies, the same quantities can be described by chiral perturbation theory (chPT). For most quantities there is an intermediate energy regime, roughly $1/V<E<1/\sqrt{V}$, where the results of RMT and chPT agree with each other. We test these predictions by constructing the connected and disconnected scalar susceptibilities from Dirac spectra obtained in quenched SU(2) and SU(3) simulations with staggered fermions for a variety of lattice sizes and coupling constants. In deriving the predictions of chPT, it is important to take into account only those symmetries which are exactly realized on the lattice.Comment: LATTICE99(Theoretical Developments), 3 pages, 3 figures, typo in Ref. [10] correcte

Microscopic universality in the spectrum of the lattice Dirac operator

Large ensembles of complete spectra of the Euclidean Dirac operator for staggered fermions are calculated for SU(2) lattice gauge theory. The accumulation of eigenvalues near zero is analyzed as a signal of chiral symmetry breaking and compared with parameter-free predictions from chiral random matrix theory. Excellent agreement for the distribution of the smallest eigenvalue and the microscopic spectral density is found. This provides direct evidence for the conjecture that these quantities are universal functions.Comment: 4 pages, 3 figures (included), REVTeX 3.1; updated version to appear in Phys. Rev. Let

Crossover to Non-universal Microscopic Spectral Fluctuations in Lattice Gauge Theory

The spectrum of the Dirac operator near zero virtuality obtained in lattice gauge simulations is known to be universally described by chiral random matrix theory. We address the question of the maximum energy for which this universality persists. For this purpose, we analyze large ensembles of complete spectra of the Euclidean Dirac operator for staggered fermions. We calculate the disconnected scalar susceptibility and the microscopic number variance for the chiral symplectic ensemble of random matrices and compare the results with lattice Dirac spectra for quenched SU(2). The crossover to a non-universal regime is clearly identified and found to scale with the square of the linear lattice size and with $f_{\pi}^2$, in agreement with theoretical expectations.Comment: 11 pages, 7 figures, misprint in Eq. (13) corrected, minor modifications, to appear in Phys. Lett.