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.

Random Matrix Theory (RMT) is a powerful statistical tool to model spectral fluctuations. This approach has also found fruitful application in Quantum Chromodynamics (QCD). Importantly, RMT provides very efficient means to separate different scales in the spectral fluctuations. We try to identify the equivalent of a Thouless energy in complete spectra of the QCD Dirac operator for staggered fermions from SU(2) lattice gauge theory for different lattice size and gauge couplings. We focus on the bulk of the spectrum. In disordered systems, the Thouless energy sets the universal scale for which RMT applies. This relates to recent theoretical studies which suggest a strong analogy between QCD and disordered systems. The wealth of data allows us to analyze several statistical measures in the bulk of the spectrum with high quality. We find deviations which allows us to give an estimate for this universal scale. Other deviations than these are seen whose possible origin is discussed. Moreover, we work out higher order correlators as well, in particular three-point correlation functions.

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 2 π , in agreement with theoretical expectations.Recently, it has been shown by several authors that chiral random matrix theory (chRMT) is able to reproduce quantitatively spectral properties of the Dirac operator obtained from QCD lattice data. This statement is valid both for fluctuation properties in the bulk of the spectrum and for microscopic spectral properties near zero virtuality, see the reviews [1,2] and Refs. [3][4][5][6]. This result implies that the spectral fluctuation properties of the Dirac operator are universal, i.e., determined solely by the underlying symmetry of

We investigate the properties of sparse matrix ensembles with particular regard for the spectral ergodicity hypothesis, which claims the identity of ensemble and spectral averages of spectral correlators. An apparent violation of the spectral ergodicity is observed. This effect is studied with the aid of the normal modes of the random matrix spectrum, which describe fluctuations of the eigenvalues around their average positions. This analysis reveals that spectral ergodicity is not broken, but that different energy scales of the spectra are examined by the two averaging techniques. Normal modes are shown to provide a useful complement to traditional spectral analysis with possible applications to a wide range of physical systems.Comment: 22 pages, 15 figure

The partition function of QCD is analyzed for an arbitrary number of flavors, N f , and arbitrary quark masses including the contributions from all topological sectors in the Leutwyler-Smilga regime. For given N f and arbitrary vacuum angle, θ, the partition function can be reduced to N f − 2 angular integrations of single Bessel functions. For two and three flavors, the θ dependence of the QCD vacuum is studied in detail. For N f = 2 and 3, the chiral condensate decreases monotonically as θ increases from zero to π and the chiral condensate develops a cusp at θ = π for degenerate quark masses in the macroscopic limit. We find a discontinuity at θ = π in the first derivative of the energy density with respect to θ for degenerate quark masses.This corresponds to the first-order phase transition in which CP is spontaneously broken, known as Dashen's phenomena. *