36 research outputs found
Spectrum of the SU(3) Dirac operator on the lattice: Transition from random matrix theory to chiral perturbation theory
We calculate complete spectra of the Kogut-Susskind Dirac operator on the
lattice in quenched SU(3) gauge theory for various values of coupling constant
and lattice size. From these spectra we compute the connected and disconnected
scalar susceptibilities and find agreement with chiral random matrix theory up
to a certain energy scale, the Thouless energy. The dependence of this scale on
the lattice volume is analyzed. In the case of the connected susceptibility
this dependence is anomalous, and we explain the reason for this. We present a
model of chiral perturbation theory that is capable of describing the data
beyond the Thouless energy and that has a common range of applicability with
chiral random matrix theory.Comment: 8 pages, RevTeX, 15 .eps 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
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
, where 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
, 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
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
Small eigenvalues of the SU(3) Dirac operator on the lattice and in Random Matrix Theory
We have calculated complete spectra of the staggered Dirac operator on the
lattice in quenched SU(3) gauge theory for \beta = 5.4 and various lattice
sizes. The microscopic spectral density, the distribution of the smallest
eigenvalue, and the two-point spectral correlation function are analyzed. We
find the expected agreement of the lattice data with universal predictions of
the chiral unitary ensemble of random matrix theory up to a certain energy
scale, the Thouless energy. The deviations from the universal predictions are
determined using the disconnected scalar susceptibility. We find that the
Thouless energy scales with the lattice size as expected from theoretical
arguments making use of the Gell-Mann--Oakes--Renner relation.Comment: REVTeX, 5 pages, 4 figure
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 , in agreement with theoretical expectations.Comment: 11 pages, 7 figures, misprint in Eq. (13) corrected, minor
modifications, to appear in Phys. Lett.
Small eigenvalues of the staggered Dirac operator in the adjoint representation and Random Matrix Theory
The low-lying spectrum of the Dirac operator is predicted to be universal,
within three classes, depending on symmetry properties specified according to
random matrix theory. The three universal classes are the orthogonal, unitary
and symplectic ensemble. Lattice gauge theory with staggered fermions has
verified two of the cases so far, unitary and symplectic, with staggered
fermions in the fundamental representation of SU(3) and SU(2). We verify the
missing case here, namely orthogonal, with staggered fermions in the adjoint
representation of SU(N_c), N_c=2, 3.Comment: 3 pages, revtex, 2 postscript figure
Microscopic universality with dynamical fermions
It has recently been demonstrated in quenched lattice simulations that the
distribution of the low-lying eigenvalues of the QCD Dirac operator is
universal and described by random-matrix theory. We present first evidence that
this universality continues to hold in the presence of dynamical quarks. Data
from a lattice simulation with gauge group SU(2) and dynamical staggered
fermions are compared to the predictions of the chiral symplectic ensemble of
random-matrix theory with massive dynamical quarks. Good agreement is found in
this exploratory study. We also discuss implications of our results.Comment: 5 pages, 3 figures, minor modifications, to appear in Phys. Rev. D
(Rapid Commun.
The microscopic spectrum of the QCD Dirac operator with finite quark masses
We compute the microscopic spectrum of the QCD Dirac operator in the presence
of dynamical fermions in the framework of random-matrix theory for the chiral
Gaussian unitary ensemble. We obtain results for the microscopic spectral
correlators, the microscopic spectral density, and the distribution of the
smallest eigenvalue for an arbitrary number of flavors, arbitrary quark masses,
and arbitrary topological charge.Comment: 11 pages, RevTeX, 2 figures (included), minor typos corrected and
discussion extended, version to appear in Phys. Rev.