Handbook Article on Applications of Random Matrix Theory to QCD (original) (raw)

The Infrared Limit of the QCD Dirac Spectrum and Applications oc chiral Random Matrix Theory to QCD

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.

Lessons from Random Matrix Theory for QCD at Finite Density

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.

Random matrix theory and QCD at nonzero chemical potential

Nuclear Physics A, 1998

In this lecture we give a brief review of chiral Random Matrix Theory (chRMT) and its applications to QCD at nonzero chemical potential. We present both analytical arguments involving chiral perturbation theory and numerical evidence from lattice QCD simulations showing that correlations of the smallest Dirac eigenvalues are described by chRMT. We discuss the range of validity of chRMT and emphasize the importance of universality. For chRMT's at µ = 0 we identify universal properties of the Dirac eigenvalues and study the effect of quenching on the distribution of Yang-Lee zeros.

Chiral random matrix theory and the spectrum of the Dirac operator near zero virtuality

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.

Random matrix model approach to chiral symmetry

Nuclear Physics B - Proceedings Supplements, 1997

We review the application of random matrix theory (RMT) to chiral symmetry in QCD. Starting from the general philosophy of RMT we introduce a chiral random matrix model with the global symmetries of QCD. Exact results are obtained for universal properties of the Dirac spectrum: i) finite volume corrections to valence quark mass dependence of the chiral condensate, and ii) microscopic fluctuations of Dirac spectra. Comparisons with lattice QCD simulations are made. Most notably, the variance of the number of levels in an interval containing n levels on average is suppressed by a factor (log n)/π 2 n. An extension of the random matrix model model to nonzero temperatures and chemical potential provides us with a schematic model of the chiral phase transition. In particular, this elucidates the nature of the quenched approximation at nonzero chemical potential.

From chiral random matrix theory to chiral perturbation theory

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.

Random matrix theory and spectral sum rules for the Dirac operator in QCD

Nuclear Physics A, 1993

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 other words, the validity of the sum rules depends only on the symmetries of the theory but not on its details. To illustrate this point we show that the sum rules hold for an interacting liquid of instantons. The physical interpretation is that the way the thermodynamic limit of the spectral density near zero is approached is universal. However, its value, i.e. the chiral condensate, is not.

Random Matrix Models for the Hermitian Wilson-Dirac operator of QCD-like theories

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.

Chiral symmetry and the spectrum of the QCD Dirac operator

Nuclear Physics A, 2000

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.

Chiral random matrix models in QCD

Arxiv preprint hep-ph/9812376, 1998

Abstract: We review some motivation behind the introduction of chiral random matrix models in QCD, with particular emphasis on the importance of the Gell-Mann-Oakes (GOR) relation for these arguments. We show why the microscopic limit is universal in power counting, ...