Chiral symmetry and the spectrum of the QCD Dirac operator

Abstract

According to the Banks-Casher formula the chiral order parameter is directly related to the spectrum of the Dirac operator. In this lecture, we will argue that some properties of the Dirac spectrum are universal and can be obtained from a random matrix theory with the global symmetries of the QCD partition function. In particular, this is true for the spectrum near zero on the scale of a typical level spacing. Alternatively, the chiral order parameter can be characterized by the zeros of the partition function. We will analyze such zeros for a random matrix model at nonzero chemical potential. IntroductionMany phenomena in nuclear physics, as for example the lightness of the pion mass and the absence of parity doublets, can be explained by the assumption that chiral symmetry is spontaneously broken. This assumption has been confirmed by numerous lattice QCD simulations (for a review see [1,2]). However, these studies also show that chiral symmetry [3] is restored at a critical temperature of T c ≈ 140 MeV . In spite of steady progress [4], the situation at nonzero baryon number density is much less clear [5]. It seems that the quenched approximation does not work [5,6], and the phase of the fermion determinant makes unquenched simulations virtually impossible.The order parameter of the chiral phase transition is the chiral condensate. It is directly related to the spectral density of the Euclidean Dirac operator [7]. One of the questions we wish to address is to what extent the Dirac spectrum shows universal features which can be obtained from a Random Matrix Theory (RMT) with the global symmetries of the QCD partition function (chiral Random Matrix Theory (chRMT)). As is well-known from the study of complex systems [8], only correlations on a scale set by the eigenvalue spacing are given by RMT. Such correlations may be important in mesoscopic systems. Typical examples are a finite nucleus [9], small metallic particles [10], quantum dots and disordered wires (see [11] for a review). In particular, universal conductance fluctuations have been understood in the framework of RMT [12]. In