https://cds.cern.ch/record/362266/files/9808010.pdf

Locality properties of Neuberger's lattice Dirac operator

▪ Abstract Random matrix theory is a powerful way to describe universal correlations of eigenvalues of complex systems. It also may serve as a schematic model for disorder in quantum systems. In this review, we discuss both types of applications of chiral random matrix theory to the QCD partition function. We show that constraints imposed by chiral symmetry and its spontaneous breaking determine the structure of low-energy effective partition functions for the Dirac spectrum. We thus derive exact results for the low-lying eigenvalues of the QCD Dirac operator. We argue that the statistical properties of these eigenvalues are universal and can be described by a random matrix theory with the global symmetries of the QCD partition function. The total number of such eigenvalues increases with the square root of the Euclidean four-volume. The spectral density for larger eigenvalues (but still well below a typical hadronic mass scale) also follows from the same low-energy effective partition function. The valid...

Random matrix theory and chiral symmetry in qcd

Approximate Ginsparg-Wilson fermions: A First test

We show how to introduce a quark chemical potential in the overlap Dirac operator. The resulting operator satisfies a Ginsparg-Wilson relation and has exact zero modes. It is no longer gamma5 Hermitian, but its nonreal eigenvalues still occur in pairs. We compute the spectral density of the operator on the lattice and show that, for small eigenvalues, the data agree with analytical predictions of non-Hermitian chiral random matrix theory for both trivial and nontrivial topology. We also explain an observed change in the number of zero modes as a function of chemical potential.

/pdf/overlap-dirac-operator-at-nonzero-chemical-potential-and-e5fpq9wa7k.pdf

Overlap Dirac operator at nonzero chemical potential and random matrix theory

We discuss the eigenvalue distribution of the overlap Dirac operator in quenched QCD on lattices of size 84 , 104 and 124 at β = 5.85 and β = 6. We distinguish the topological sectors and study the distributions of the leading non-zero eigenvalues, which are stereographically mapped onto the imaginary axis. Thus they can be compared to the predictions of random matrix theory applied to the -expansion of chiral perturbation theory. We find a satisfactory agreement, if the physical volume exceeds about (1.2 fm)4 . For the unfolded level spacing distribution we find an accurate agreement with the random matrix conjecture on all volumes that we considered.

Spectral properties of the overlap Dirac operator in QCD

Eigenvalue spectrum of massless Dirac operators on the lattice

We reanalyze data from available finite-temperature QCD simulations near the chiral transition, with the help of chiral random matrix theory (chiral RMT) The statistical properties of the lowest-lying eigenvalues of the staggered Dirac operator for SU(3) lattice gauge theory with dynamical fermions are examined We consider temperatures below, near, and above the critical temperature ${T}_{c}$ for the chiral phase transition Below and above ${T}_{c}$ the statistics are in agreement with the exact analytical predictions in the microscopic scaling regime Above ${T}_{c}$ we observe a gap in the spectral density and a distribution compatible with the Airy distribution Near ${T}_{c}$ the eigenvalue correlations appear inconsistent with chiral RMT

/pdf/microscopic-universality-and-the-chiral-phase-transition-in-2avmi3p2mt.pdf

Microscopic Universality and the Chiral Phase Transition in two Flavor QCD

Chiral properties of the fixed point action of the Schwinger model

We perform a comparison between different lattice regularizations of the Dirac operator for massless fermions in the framework of the single and two flavor Schwinger model. We consider a) the Wilson-Dirac operator at the critical value of the hopping parameter; b) Neuberger's overlap operator; c) the fixed point operator. We test chiral properties of the spectrum, dispersion relations and rotational invariance of the mesonic bound state propagators.

Wilson, fixed point and Neuberger's lattice Dirac operator for the Schwinger model

We compare the lower edge spectral fluctuations of the staggered lattice Dirac operator for the Schwinger model with the predictions of chiral Random Matrix Theory (chRMT). We verify their range of applicability, checking in particular the role of non-trivial topological sectors and the flavor symmetry of the staggered fermions for finite lattice spacing. Approaching the continuum limit we indeed find clear signals for topological modes in the eigenvalue spectrum. These findings indicate problems in the verification of the chRMT predictions.

F. Farchioni

Papers

Eigenvalue spectrum of massless Dirac operators on the lattice

Microscopic Universality and the Chiral Phase Transition in two Flavor QCD

Chiral properties of the fixed point action of the Schwinger model

Wilson, fixed point and Neuberger's lattice Dirac operator for the Schwinger model

Effects of Topology in the Dirac Spectrum of Staggered Fermions