127 research outputs found
Finite volume partition functions and Itzykson-Zuber integrals
We find the finite volume QCD partition function for arbitrary quark masses.
This is a generalization of a result obtained by Leutwyler and Smilga for equal
quark masses. Our result is derived in the sector of zero topological charge
using a generalization of the Itzykson-Zuber integral appropriate for arbitrary
complex matrices. We present a conjecture regarding the result for arbitrary
topological charge which reproduces the Leutwyler-Smilga result in the limit of
equal quark masses. We derive a formula of the Itzykson-Zuber type for
arbitrary {\em rectangular} complex matrices, extending the result of Guhr and
Wettig obtained for {\em square} matrices.Comment: 11 pages, LATEX. A minor typo in equation (12) has been corrected in
the revised versio
The Factorization Method for Monte Carlo Simulations of Systems With a Complex Action
We propose a method for Monte Carlo simulations of systems with a complex
action. The method has the advantages of being in principle applicable to any
such system and provides a solution to the overlap problem. In some cases, like
in the IKKT matrix model, a finite size scaling extrapolation can provide
results for systems whose size would make it prohibitive to simulate directly.Comment: Lattice2003(nonzero), 3 pages, 4 figures, Proceedings for Lattice
2003, July 2003, Tsukuba, Japa
Smallest Dirac Eigenvalue Distribution from Random Matrix Theory
We derive the hole probability and the distribution of the smallest
eigenvalue of chiral hermitian random matrices corresponding to Dirac operators
coupled to massive quarks in QCD. They are expressed in terms of the QCD
partition function in the mesoscopic regime. Their universality is explicitly
related to that of the microscopic massive Bessel kernel.Comment: 4 pages, 1 figure, REVTeX. Minor typos in subscripts corrected.
Version to appear in Phys. Rev.
Universal correlations in spectra of the lattice QCD Dirac operator
Recently, Kalkreuter obtained complete Dirac spectra for lattice
gauge theory both for staggered fermions and for Wilson fermions. The lattice
size was as large as . 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
eigenvalues on average is given by
( is equal to 4 and 1, respectively) instead of for a
random sequence of levels. Our findings are in agreement with the anti-unitary
symmetry of the lattice Dirac operator for with staggered fermions
which differs from Wilson fermions (with the continuum anti-unitary symmetry).
For , we predict that the eigenvalue correlations are given by the
Gaussian Unitary Ensemble.Comment: Talk present at LATTICE96(chirality in QCD), 3 pages, Late
Random matrix model approach to chiral symmetry
We review the application of random matrix theory (RMT) to chiral symmetry in
QCD. Starting from the general philosophy of RMT we introduce a chiral random
matrix model with the global symmetries of QCD. Exact results are obtained for
universal properties of the Dirac spectrum: i) finite volume corrections to
valence quark mass dependence of the chiral condensate, and ii) microscopic
fluctuations of Dirac spectra. Comparisons with lattice QCD simulations are
made. Most notably, the variance of the number of levels in an interval
containing levels on average is suppressed by a factor .
An extension of the random matrix model model to nonzero temperatures and
chemical potential provides us with a schematic model of the chiral phase
transition. In particular, this elucidates the nature of the quenched
approximation at nonzero chemical potential.Comment: Talk present at LATTICE96(chirality in QCD), plenary session, 7
pages, Late
Universality near zero virtuality
In this paper we study a random matrix model with the chiral and flavor
structure of the QCD Dirac operator and a temperature dependence given by the
lowest Matsubara frequency. Using the supersymmetric method for random matrix
theory, we obtain an exact, analytic expression for the average spectral
density. In the large-n limit, the spectral density can be obtained from the
solution to a cubic equation. This spectral density is non-zero in the vicinity
of eigenvalue zero only for temperatures below the critical temperature of this
model. Our main result is the demonstration that the microscopic limit of the
spectral density is independent of temperature up to the critical temperature.
This is due to a number of `miraculous' cancellations. This result provides
strong support for the conjecture that the microscopic spectral density is
universal. In our derivation, we emphasize the symmetries of the partition
function and show that this universal behavior is closely related to the
existence of an invariant saddle-point manifold.Comment: 23 pages, Late
The Microscopic Spectral Density of the QCD Dirac Operator
We derive the microscopic spectral density of the Dirac operator in
Yang-Mills theory coupled to fermions in the fundamental
representation. An essential technical ingredient is an exact rewriting of this
density in terms of integrations over the super Riemannian manifold
. The result agrees exactly with earlier calculations based on
Random Matrix Theory.Comment: 26 pages, Late
The Spectral Density of the QCD Dirac Operator and Patterns of Chiral Symmetry Breaking
We study the spectrum of the QCD Dirac operator for two colors with fermions
in the fundamental representation and for two or more colors with adjoint
fermions. For flavors, the chiral flavor symmetry of these theories is
spontaneously broken according to and , respectively, rather than the symmetry breaking pattern for QCD with three or more colors and fundamental
fermions. In this paper we study the Dirac spectrum for the first two symmetry
breaking patterns. Following previous work for the third case we find the Dirac
spectrum in the domain by means of partially
quenched chiral perturbation theory. In particular, this result allows us to
calculate the slope of the Dirac spectrum at . We also show that
for (with the linear size of the system)
the Dirac spectrum is given by a chiral Random Matrix Theory with the
symmetries of the Dirac operator.Comment: 27 pages Latex, corrected typo
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
Chiral Lagrangian and spectral sum rules for dense two-color QCD
We analytically study two-color QCD with an even number of flavors at high
baryon density. This theory is free from the fermion sign problem. Chiral
symmetry is broken spontaneously by the diquark condensate. Based on the
symmetry breaking pattern we construct the low-energy effective Lagrangian for
the Nambu-Goldstone bosons. We identify a new epsilon-regime at high baryon
density in which the quark mass dependence of the partition function can be
determined exactly. We also derive Leutwyler-Smilga-type spectral sum rules for
the complex eigenvalues of the Dirac operator in terms of the fermion gap. Our
results can in principle be tested in lattice QCD simulations.Comment: 24 pages, 1 table, no figur
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