Non-Hermitian Random Matrix Theory and Lattice QCD with Chemical Potential

Abstract

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]

Microscopic Correlation Functions for the QCD Dirac Operator with Chemical Potential

“…For large values of µ the eigenvalue density on the lattice is no longer constant and develops a hole in the middle (see e.g. [19,25]). We will also see such a hole develop in the microscopic correlations in the limit of strong non-Hermiticity, as shown in Fig.…”

“…The phase diagram of QCD in the temperature density plane has also been predicted with such a model [18]. Recently complex Dirac eigenvalues calculated on the lattice have been confronted to a complex matrix model on the microscopic scale given by the inverse volume in the bulk of the spectrum [19]. The nearest neighbor distribution along a given direction in the complex plane was considered and a transition from the Unitary to the Ginibre ensemble was observed at increasing µ, ending in a Poisson distribution.…”

“…Although recent progress has been made [16] along the phase transition line the general phase diagram remains unexplored for µ = 0. It is in this context that analytical knowledge of microscopic correlation functions from complex chiral RMM could be very useful, in view of its success in predicting real eigenvalue correlations.Chiral RMM including a chemical potential have been already studied in several works [4,17,18,19]. In [4] the nature of the quenched limit has been analyzed in such a model, and the global density of complex eigenvalues together with its boundary have been calculated as a function of µ.…”

“…Chiral RMM including a chemical potential have been already studied in several works [4,17,18,19]. In [4] the nature of the quenched limit has been analyzed in such a model, and the global density of complex eigenvalues together with its boundary have been calculated as a function of µ.…”

Unquenched QCD Dirac operator spectra at nonzero baryon chemical potential

“…Spectra of the non-Hermitian Dirac operator at nonzero baryon density have been obtained from lattice QCD in the quenched case [1,41,42,43] and for QCD with two colors [44,45] and have been compared successfully to random matrix theory [41,42,43,45]. The microscopic spectral density of quenched lattice QCD at nonzero baryon density was first analyzed in [43] where quantitative agreement with analytical predictions [26,37] was found in an asymptotic domain where the results derived from the chiral Lagrangian [26] agree with the expression obtained in [37].…”