Universal spectral correlators and massive Dirac operators

Damgaard, Poul H.; Nishigaki, Shinsuke M.

doi:10.1016/s0550-3213(98)00123-0

Cited by 99 publications

(154 citation statements)

References 18 publications

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“…The form of the Gaussian weight w G (Φ) respects the flavor symmetries, but these symmetries do not exclude traces of higher powers of Φ † Φ in the exponent as well as other non-invariant terms. At zero chemical potential it was shown [50,51,52,53,54,55,56,57] that partition functions and eigenvalue correlations in the microscopic limit are independent of the form of this weight, which, in random matrix theory, is known as universality. The only condition for the probability density is that the spectral density near the origin is nonvanishing in the thermodynamic limit.…”

Section: Random Matrix Model For Qcd At Nonzero Chemical Potentialmentioning

confidence: 99%

Unquenched QCD Dirac operator spectra at nonzero baryon chemical potential

et al. 2005

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The microscopic spectral density of the QCD Dirac operator at nonzero baryon chemical potential for an arbitrary number of quark flavors was derived recently from a random matrix model with the global symmetries of QCD. In this paper we show that these results and extensions thereof can be obtained from the replica limit of a Toda lattice equation. This naturally leads to a factorized form into bosonic and fermionic QCD-like partition functions. In the microscopic limit these partition functions are given by the static limit of a chiral Lagrangian that follows from the symmetry breaking pattern. In particular, we elucidate the role of the singularity of the bosonic partition function in the orthogonal polynomials approach. A detailed discussion of the spectral density for one and two flavors is given.

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Section: Random Matrix Model For Qcd At Nonzero Chemical Potentialmentioning

confidence: 99%

Unquenched QCD Dirac operator spectra at nonzero baryon chemical potential

et al. 2005

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“…This can be done by showing that the microscopic spectral density is insensitive to deformations of the probability distribution. The interested reader is referred to the extensive literature on this subject [85,86,87,88,89,90,91,92].…”

Section: The Dirac Spectrum In Qcdmentioning

confidence: 99%

The supersymmetric method in random matrix theory and applications to QCD

Verbaarschot¹

2004

AIP Conference Proceedings

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Abstract.The supersymmetric method is a powerful method for the nonperturbative evaluation of quenched averages in disordered systems. Among others, this method has been applied to the statistical theory of S-matrix fluctuations, the theory of universal conductance fluctuations and the microscopic spectral density of the QCD Dirac operator.We start this series of lectures with a general review of Random Matrix Theory and the statistical theory of spectra. An elementary introduction of the supersymmetric method in Random Matrix Theory is given in the second and third lecture. We will show that a Random Matrix Theory can be rewritten as an integral over a supermanifold. This integral will be worked out in detail for the Gaussian Unitary Ensemble that describes level correlations in systems with broken time reversal invariance. We especially emphasize the role of symmetries.As a second example of the application of the supersymemtric method we discuss the calculation of the microscopic spectral density of the QCD Dirac operator. This is the eigenvalue density near zero on the scale of the average level spacing which is known to be given by chiral Random Matrix Theory. Also in this case we use symmetry considerations to rewrite the generating function for the resolvent as an integral over a supermanifold.The main topic of the second last lecture is the recent developments on the relation between the supersymmetric partition function and integrable hierarchies (in our case the Toda lattice hierarchy). We will show that this relation is an efficient way to calculate superintegrals. Several examples that were given in previous lectures will be worked out by means of this new method. Finally, we will discuss the quenched QCD Dirac spectrum at nonzero chemical potential. Because of the nonhermiticity of the Dirac operator the usual supersymmetric method has not been successful in this case. However, we will show that the supersymmetric partition function can be evaluated by means of the replica limit of the Toda lattice equation.

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“…This has been shown in great detail for the chiral Unitary Ensemble (chUE), which is the ensemble of Hermitian chiral random matrices with no anti-unitary symmetries [18,19,20,21,5,22,23,24,25,26,27,28] [ 29,12,30,31,32,33]. The chiral ensembles with real or quaternion real matrix elements (known as the chiral Orthogonal Ensemble (chOE) and the chiral Symplectic Ensemble (chSE), respectively) are mathematically much more complicated.…”

Section: Introductionmentioning

confidence: 99%

Spectral universality of real chiral random matrix ensembles

Klein

Verbaarschot

2000

Nuclear Physics B

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We investigate the universality of microscopic eigenvalue correlations for Random Matrix Theories with the global symmetries of the QCD partition function. In this article we analyze the case of real valued chiral Random Matrix Theories (β = 1) by relating the kernel of the correlations functions for β = 1 to the kernel of chiral Random Matrix Theories with complex matrix elements (β = 2), which is already known to be universal. Our proof is based on a novel asymptotic property of the skew-orthogonal polynomials: an integral over the corresponding wavefunctions oscillates about half its asymptotic value in the region of the bulk of the zeros. This results solves the puzzle that microscopic universality persists in spite of contributions to the microscopic correlators from the region near the largest zero of the skew-orthogonal polynomials. Our analytical results are illustrated by the numerical construction of the skew-orthogonal polynomials for an x 4 probability potential.

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Universal spectral correlators and massive Dirac operators

Cited by 99 publications

References 18 publications

Unquenched QCD Dirac operator spectra at nonzero baryon chemical potential

Unquenched QCD Dirac operator spectra at nonzero baryon chemical potential

The supersymmetric method in random matrix theory and applications to QCD

Spectral universality of real chiral random matrix ensembles

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