We discuss the spectrum of the QCD Dirac operator both at zero and at nonzero baryon chemical potential. We show that, in the ergodic domain of QCD, the Dirac spectrum can be obtained from the replica limit of a Toda lattice equation. At zero chemical potential this method explains the factorization of known results into compact and noncompact integrals, and at nonzero chemical potential it all...

We present two pieces of evidence in support of the conjecture that the microscopic spectral density of the Dirac operator is a universal quantity. First, we compare lattice data to predictions from random matrix theory. Second, we show that the functional form of the microscopic spectral correlations remains unchanged in randommatrix models which take account of finite temperature. Furthermore...

For an ideal D-dimensional Fermi gas under generic external confinement we derive the correcting coefficient (D − 2)/3D of the von Weizsacker term in the kinetic energy density. To obtain this coefficient we use the Kirzhnits semiclassical expansion of the number operator up to the second order in the Planck constant h̄. Within this simple and direct approach we determine the differential equati...

We present two pieces of evidence in support of the conjecture that the microscopic spectral density of the Dirac operator is a universal quantity. First, we compare lattice data to predictions from random matrix theory. Second, we show that the functional form of the microscopic spectral correlations remains unchanged in random matrix models which take account of finite temperature. Furthermor...

We compute by Monte Carlo methods the individual distributions of the kth smallest Dirac operator eigenvalues in QCD, and compare them with recent analytical predictions. We do this for both massless and massive quarks in an SU(3) gauge theory with staggered fermions. Very precise agreement is found in all cases. As a simple by-product we also extract the microscopic spectral density of the Dir...

The radial Dirac operator with a potential tending to infinity at infinity and satisfying a mild regularity condition is known to have a purely absolutely continuous spectrum covering the whole real line. Although having two singular end-points in the limit-point case, the operator has a simple spectrum and a generalised Fourier expansion in terms of a single solution. In the present paper, a s...

We construct a random matrix model that, in the large N limit, reduces to the low energy limit of the QCD partition function put forward by Leutwyler and Smilga. This equivalence holds for an arbitrary number of flavors and any value of the QCD vacuum angle. In this model, moments of the inverse squares of the eigenvalues of the Dirac operator obey sum rules, which we conjecture to be universal...

In this talk we present the results published recently in Ref. [1], where we showed 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 γ5-Hermitian, but its nonreal eigenvalues still occur in pairs. We compute the spectral density of the operator on the lattice and show ...

The spectral density ρ(λ) of the Dirac operator in QCD contains interesting informations as it is for example directly proportional to the chiral condensate Σ at the origin through the Banks-Casher relation . While the full spectrum is only accessible numerically in QCD on the lattice many analytic results for parts of the spectrum have been obtained during the past years using random matrix th...

Two-dimensional decaying turbulence is simulated using the lattice Boltzmann method with the Bhatnagar-Gross-Krook collision operator. The simulation results are analyzed in the spectral space and the auto power spectral density functions of the one-particle distribution functions are presented. It is shown that both the equilibrium and the non-equilibrium parts of the distribution functions ha...