Random matrix theory for the Hermitian Wilson Dirac operator and the chGUE-GUE transition

This article focuses on chiral random matrix theories with the global symmetries of quantum chromodynamics (QCD). In particular, it explains how random matrix theory (RMT) can be applied to the spectra of the Dirac operator both at zero chemical potential, when the Dirac operator is Hermitian, and at non-zero chemical potential, when the Dirac operator is non-Hermitian. Before discussing the spectra of these Dirac operators at non-zero chemical potential, the article considers spontaneous symmetry breaking in RMT and the QCD partition function. It then examines the global symmetries of QCD, taking into account the Dirac operator for a finite chiral basis, as well as the global symmetry breaking pattern and the Goldstone manifold in chiral random matrix theory (chRMT). It also describes the generating function for the Dirac spectrum and applications of chRMT to QCD to gauge degrees of freedom.

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Overlap Dirac Operator at Nonzero Chemical Potential and Random Matrix Theory

Physical Review Letters ◽

10.1103/physrevlett.97.012003 ◽

2006 ◽

Vol 97 (1) ◽

Cited By ~ 42

Author(s):

Jacques Bloch ◽

Tilo Wettig

Keyword(s):

Dirac Operator ◽

Random Matrix Theory ◽

Random Matrix ◽

Chemical Potential ◽

Matrix Theory

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Spectrum of the QCD Dirac operator and chiral random matrix theory

Physical Review Letters ◽

10.1103/physrevlett.72.2531 ◽

1994 ◽

Vol 72 (16) ◽

pp. 2531-2533 ◽

Cited By ~ 335

Author(s):

Jacobus Verbaarschot

Keyword(s):

Dirac Operator ◽

Random Matrix Theory ◽

Random Matrix ◽

Matrix Theory

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Distribution of the Dirac modes in QCD

EPJ Web of Conferences ◽

10.1051/epjconf/201817504005 ◽

2018 ◽

Vol 175 ◽

pp. 04005

Author(s):

M. Catillo ◽

L. Ya. Glozman

Keyword(s):

Dirac Operator ◽

Random Matrix Theory ◽

Random Matrix ◽

General Property ◽

Matrix Theory ◽

Spontaneous Breaking ◽

Gaussian Unitary Ensemble ◽

Zero Modes ◽

Dynamical Simulations ◽

Unitary Ensemble

It was established that distribution of the near-zero modes of the Dirac operator is consistent with the Chiral Random Matrix Theory (CRMT) and can be considered as a consequence of spontaneous breaking of chiral symmetry (SBCS) in QCD. The higherlying modes of the Dirac operator carry information about confinement physics and are not affected by SBCS. We study distributions of the near-zero and higher-lying modes of the overlap Dirac operator within NF = 2 dynamical simulations. We find that distributions of both near-zero and higher-lying modes are the same and follow the Gaussian Unitary Ensemble of Random Matrix Theory. This means that randomness, while consistent with SBCS, is not a consequence of SBCS and is related to some more general property of QCD in confinement regime.

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Distribution law of the Dirac eigenmodes in QCD

International Journal of Modern Physics A ◽

10.1142/s0217751x18500549 ◽

2018 ◽

Vol 33 (10) ◽

pp. 1850054

Author(s):

Marco Catillo ◽

Leonid Ya. Glozman

Keyword(s):

Dirac Operator ◽

Random Matrix Theory ◽

Random Matrix ◽

Matrix Theory ◽

Spontaneous Breaking ◽

Distribution Law ◽

Gaussian Unitary Ensemble ◽

Zero Modes ◽

Dynamical Simulations ◽

Unitary Ensemble

The near-zero modes of the Dirac operator are connected to spontaneous breaking of chiral symmetry in QCD (SBCS) via the Banks–Casher relation. At the same time, the distribution of the near-zero modes is well described by the Random Matrix Theory (RMT) with the Gaussian Unitary Ensemble (GUE). Then, it has become a standard lore that a randomness, as observed through distributions of the near-zero modes of the Dirac operator, is a consequence of SBCS. The higher-lying modes of the Dirac operator are not affected by SBCS and are sensitive to confinement physics and related [Formula: see text] and [Formula: see text] symmetries. We study the distribution of the near-zero and higher-lying eigenmodes of the overlap Dirac operator within [Formula: see text] dynamical simulations. We find that both the distributions of the near-zero and higher-lying modes are perfectly described by GUE of RMT. This means that randomness, while consistent with SBCS, is not a consequence of SBCS and is linked to the confining chromo-electric field.

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Spectrum of the SU(3) Dirac operator on the lattice: Transition from random matrix theory to chiral perturbation theory

Physical Review D ◽

10.1103/physrevd.65.034503 ◽

2002 ◽

Vol 65 (3) ◽

Author(s):

M. Göckeler ◽

H. Hehl ◽

P. E. L. Rakow ◽

A. Schäfer ◽

T. Wettig

Keyword(s):

Perturbation Theory ◽

Dirac Operator ◽

Random Matrix Theory ◽

Random Matrix ◽

Chiral Perturbation Theory ◽

Matrix Theory ◽

Chiral Perturbation

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Overlap Dirac Operator, Eigenvalues and Random Matrix Theory

Nuclear Physics B - Proceedings Supplements ◽

10.1016/s0920-5632(00)00352-2 ◽

2000 ◽

Vol 83-84 (1-3) ◽

pp. 446-448

Author(s):

R Edwards

Keyword(s):

Dirac Operator ◽

Random Matrix Theory ◽

Random Matrix ◽

Matrix Theory ◽

Operator Eigenvalues

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Random Matrix Theory and Chiral Symmetry in QCD

Annual Review of Nuclear and Particle Science ◽

10.1146/annurev.nucl.50.1.343 ◽

2000 ◽

Vol 50 (1) ◽

pp. 343-410 ◽

Cited By ~ 211

Author(s):

J.J.M. Verbaarschot ◽

T. Wettig

Keyword(s):

Partition Function ◽

Dirac Operator ◽

Chiral Symmetry ◽

Random Matrix Theory ◽

Random Matrix ◽

Matrix Theory ◽

Low Energy ◽

Spontaneous Breaking ◽

Schematic Model ◽

Qcd Phase Diagram

▪ Abstract Random matrix theory is a powerful way to describe universal correlations of eigenvalues of complex systems. It also may serve as a schematic model for disorder in quantum systems. In this review, we discuss both types of applications of chiral random matrix theory to the QCD partition function. We show that constraints imposed by chiral symmetry and its spontaneous breaking determine the structure of low-energy effective partition functions for the Dirac spectrum. We thus derive exact results for the low-lying eigenvalues of the QCD Dirac operator. We argue that the statistical properties of these eigenvalues are universal and can be described by a random matrix theory with the global symmetries of the QCD partition function. The total number of such eigenvalues increases with the square root of the Euclidean four-volume. The spectral density for larger eigenvalues (but still well below a typical hadronic mass scale) also follows from the same low-energy effective partition function. The validity of the random matrix approach has been confirmed by many lattice QCD simulations in a wide parameter range. Stimulated by the success of the chiral random matrix theory in the description of universal properties of the Dirac eigenvalues, the random matrix model is extended to nonzero temperature and chemical potential. In this way we obtain qualitative results for the QCD phase diagram and the spectrum of the QCD Dirac operator. We discuss the nature of the quenched approximation and analyze quenched Dirac spectra at nonzero baryon density in terms of an effective partition function. Relations with other fields are also discussed.

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