Spectral properties of the overlap Dirac operator in QCD

1996

In this lecture we argue that the fluctuations of Dirac eigenvalues on the finest scale, i.e. on the scale of the average level spacing do not depend on the underlying dynamics and can be obtained from a chiral random matrix theory with the same low energy effective theory. We present three pieces of evidence supporting that such microscopic correlations of lattice QCD Dirac spectra are given by chiral random matrix theory. First, we find that the spectral correlations of eigenvalues in the bulk of the spectrum obey the Dyson-Mehta-Wigner statistics. Second, we show that the valence quark mass dependence for sufficiently small quark masses, as calculated by the Columbia group, can be obtained from the microscopic spectral density of chiral random matrix theory. Third, in the framework of chiral random matrix models, we present results showing that the microscopic spectral density is strongly universal, i.e. is insensitive to the details of the probability distribution.

Nuclear Physics B, 1999

We study the spectrum of the QCD Dirac operator by means of the valence quark mass dependence of the chiral condensate in partially quenched Chiral Perturbation Theory (pqChPT) in the supersymmetric formulation of Bernard and Golterman. We consider valence quark masses both in the ergodic domain (m v E c) and the diffusive domain (m v E c). These domains are separated by a mass scale E c ∼ F 2 /Σ 0 L 2 (with F the pion decay constant, Σ 0 the chiral condensate and L the size of the box). In the ergodic domain the effective super-Lagrangian reproduces the microscopic spectral density of chiral Random Matrix Theory (chRMT). We obtain a natural explanation of Damgaard's relation between the spectral density and the finite volume partition function with two additional flavors. We argue that in the ergodic domain the natural measure for the super-unitary integration in the pqChPT partition function is noncompact. We find that the tail of the two-point spectral correlation function derived from pqChPT agrees with the chRMT result in the ergodic domain. In the diffusive domain we extend the results for the slope of the Dirac spectrum first obtained by Smilga and Stern. We find that the spectral density diverges logarithmically for nonzero topological susceptibility. We study the transition between the ergodic and the diffusive domain and identify a range where chRMT and pqChPT coincide.

Continuous Advances in QCD 2008, 2008

In this lecture we discuss various aspects of QCD at nonzero chemical potential, including its phase diagram and the Dirac spectrum, and summarize what chiral random matrix theory has contributed to this subject. To illustrate the importance of the phase of the fermion determinant, we particularly highlight the differences between QCD and phase quenched QCD.

Nuclear Physics B, 2001

We analyze the statistical properties of the spectrum of the QCD Dirac operator at low energy in a finite box of volume L 4 by means of partially quenched Chiral Perturbation Theory (pqChPT), a low-energy effective field theory based on the symmetries of QCD. We derive the two-point spectral correlation function from the discontinuity of the chiral susceptibility. For eigenvalues much smaller than Ec = F 2 /ΣL 2 , where F is the pion decay constant and Σ is the absolute value of the quark condensate, our result for the two-point correlation function coincides with the result previously obtained from chiral Random Matrix Theory (chRMT). The departure from the chRMT result above that scale is described by the contribution of the nonzero momentum modes. In terms of the variance of the number of eigenvalues in an interval containing n eigenvalues on average, it results in a crossover from a log n-behavior to a n 2 log n-behavior.

Lattice Fermions and Structure of the Vacuum, 2000

1999

In the first part of these lectures we discuss the infrared limit of the spectrum of the QCD Dirac operator. We discuss the global symmetries of the QCD partition function and show that the Dirac spectrum near zero virtuality is determined by the pattern of spontaneous chiral symmetry breaking of a QCD-like partition function with additional bosonic valence quarks and their super-symmetric partners. We show the existence of an energy scale below which the fluctuations of the QCD Dirac spectrum are given by a chiral Random Matrix Theory (chRMT) with the global symmetries of the QCD partition function. Physically, for valence quark masses below this scale the partition function is dominated by the zero momentum modes. In the theory of disordered systems, this energy scale is known as the Thouless energy. In the second part of these lectures we discuss chRMT as a schematic model for the QCD partition function at nonzero temperature and chemical potential. We discuss novel features resulting from the non-Hermiticity of the Dirac operator. The analysis by Stephanov of the failure of the quenched approximation, the properties of Yang-Lee zeros, as well as the phase diagram of the chRMT partition function are discussed. We argue that a localization transition does not occur in the presence of light quarks. Several results will be derived in full detail. We mention the flavor symmetries of the QCD Dirac operator for two colors, the calculation of the valence quark mass dependence of the chiral condensate and the reduction of the chRMT partition function to the finite volume partition function.

Proceedings of The 30th International Symposium on Lattice Field Theory — PoS(Lattice 2012)

We introduce Random Matrix Models for the Hermitian Wilson-Dirac operator of QCD-like theories. We show that they are equivalent to the ε-limit of the chiral Lagrangian for Wilson chiral perturbation theory. Results are obtained for two-color QCD with quarks in the fundamental representation of the color group as well as any-color QCD with quarks in the adjoint representation. For N c = 2 we also have obtained the lattice spacing dependence of the quenched average spectral density for a fixed value of the index of the Dirac operator. Comparisons with direct numerical simulations of the random matrix ensemble are shown.

Physical Review D, 2013

Arxiv preprint hep-th/9310049, 1993

We study the spectrum of the QCD Dirac operator near zero virtuality. We argue that it can be described by a random matrix theory with the chiral structure of QCD. In the large N limit, this model reduces to the low energy limit of the QCD partition function put forward by Leutwyler and Smilga. We conjecture that the microscopic limit of its spectral density is universal and reproduces that of QCD. Using random matrix methods we obtain its exact analytical expression. This result is compared to numerically calculated spectra for a liquid of instantons, and we find a very satisfactory agreement.

Proceedings of XXIX International Symposium on Lattice Field Theory — PoS(Lattice 2011)

We study discretization effects of the Wilson and staggered Dirac operator with N c > 2 using chiral random matrix theory (chRMT). We obtain analytical results for the joint probability density of Wilson-chRMT in terms of a determinantal expression over complex pairs of eigenvalues, and real eigenvalues corresponding to eigenvectors of positive or negative chirality as well as for the eigenvalue densities. The explicit dependence on the lattice spacing can be readily read off from our results which are compared to numerical simulations of Wilson-chRMT. For the staggered Dirac operator we have studied random matrices modeling the transition from non-degenerate eigenvalues at non-zero lattice spacing to degenerate ones in the continuum limit.