In quantum chromodynamics (QCD) at nonzero chemical potential, the eigenvalues of the Dirac operator are scattered in the complex plane. Can the fluctuation properties of the Dirac spectrum be described by universal predictions of non-Hermitian random matrix theory? We introduce an unfolding procedure for complex eigenvalues and apply it to data from lattice QCD at finite chemical potential m to construct the nearest-neighbor spacing distribution of adjacent eigenvalues in the complex plane. For intermediate values of m, we find agreement with predictions of the Ginibre ensemble of random matrix theory, both in the confinement and in the deconfinement phase. PACS numbers: 12.38.Gc, 05.45.Pq Physical systems which are described by non-Hermitian operators have recently attracted a lot of attention. They are of interest, e.g., in dissipative quantum chaos [1], disordered systems [2,3], neural networks [4], and quantum chromodynamics (QCD) at finite chemical potential [5].Often they display unusual and unexpected behavior, such as a delocalization transition in one dimension [2]. Consequently, analytical efforts have been made to develop mathematical methods to deal with non-Hermitian matrices (see, e.g., Refs. [6-10]).Of particular interest in the analysis of complicated quantum systems are the properties of the eigenvalues of the Hamilton operator. In the Hermitian case, it has been shown for many different systems that the spectral fluctuations on the scale of the mean level spacing are given by universal predictions of random matrix theory (RMT) [11]. In QCD, one studies the eigenvalues of the Dirac operator, and it was demonstrated in lattice simulations that at zero chemical potential, the local spectral fluctuation properties in the bulk of the spectrum are reproduced by Hermitian RMT both in the confinement and in the deconfinement phase [12].If one considers QCD at nonzero chemical potential, the Dirac operator loses its Hermiticity properties so that its eigenvalues become complex. The aim of the present paper is to investigate whether non-Hermitian RMT is able to describe the fluctuation properties of the complex eigenvalues of the QCD Dirac operator. The eigenvalues are generated on the lattice for various values of m. We define a two-dimensional unfolding procedure to separate the average eigenvalue density from the fluctuations and construct the nearest-neighbor spacing distribution P͑s͒ of adjacent eigenvalues in the complex plane. The data are then compared to analytical predictions of non-Hermitian RMT.We start with a few definitions. At m fi 0, the QCD Dirac operator on the lattice in the staggered formulation is given by [13]

A continuum expression for the trace of the massive dressed-quark propagator is used to explicate a connection between the infrared limit of the QCD Dirac operator's spectrum and the quark condensate appearing in the operator product expansion, and the connection is verified via comparison with a lattice-QCD simulation. The pseudoscalar vacuum polarisation provides a good approximation to the condensate over a larger range of current-quark masses.

We investigate the eigenvalue spectrum of the staggered Dirac matrix in SU(3) gauge theory and in full QCD on a 6 3 × 4 lattice. As a measure of the fluctuation properties of the eigenvalues, we study the nearest-neighbor spacing distribution P (s) for various values of β both in the confinement and in the deconfinement phase. In both phases except far into the deconfinement region, the lattice data agree with the Wigner surmise of random-matrix theory which is indicative of quantum chaos. We do not find signs of a transition to Poisson regularity at the deconfinement phase transition.

We compute eigenmodes of the lattice Dirac operator for quenched SU(3) gauge configurations on the 4-torus with topological charge ±1. We find a strong dependence of the zero modes on the boundary conditions which we impose for the Dirac operator. The lumps seen by the eigenmodes often change their position when changing the boundary conditions, while the local chirality of the lumps remains the same. Our results show that the zero mode of a charge ±1 configuration can couple to more than one object. We address the question whether these objects could be fractionally charged lumps.

Complete eigenvalue spectra of the staggered Dirac operator in quenched 4D compact QED are studied on 8 3 ϫ4 and 8 3 ϫ6 lattices. We investigate the behavior of the nearest-neighbor spacing distribution P(s) as a measure of the fluctuation properties of the eigenvalues in the strong coupling and the Coulomb phase. In both phases we find agreement with the Wigner surmise of the unitary ensemble of random-matrix theory indicating quantum chaos. Combining this with previous results on QCD, we conjecture that quite generally the nonlinear couplings of quantum field theories lead to a chaotic behavior of the eigenvalues of the Dirac operator.