261 research outputs found
Random matrix theory and
We suggest that the spectral properties near zero virtuality of three
dimensional QCD, follow from a Hermitean random matrix model. The exact
spectral density is derived for this family of random matrix models both for
even and odd number of fermions. New sum rules for the inverse powers of the
eigenvalues of the Dirac operator are obtained. The issue of anomalies in
random matrix theories is discussed.Comment: 10p., SUNY-NTG-94/1
Three-dimensional QCD in the adjoint representation and random matrix theory
In this paper we complete the derivations of finite volume partition
functions for QCD using random matrix theories by calculating the effective
low-energy partition function for three-dimensional QCD in the adjoint
representation from a random matrix theory with the same global symmetries. As
expected, this case corresponds to Dyson index , that is, the Dirac
operator can be written in terms of real quaternions. After discussing the
issue of defining Majorana fermions in Euclidean space, the actual matrix model
calculation turns out to be simple. We find that the symmetry breaking pattern
is , as expected from the correspondence
between symmetric (super)spaces and random matrix universality classes found by
Zirnbauer. We also derive the first Leutwyler--Smilga sum rule.Comment: LaTeX, 19 pages. Minor corrections, added comments, to appear on
Phys. Rev.
Universal Massive Spectral Correlators and QCD_3
Based on random matrix theory in the unitary ensemble, we derive the
double-microscopic massive spectral correlators corresponding to the Dirac
operator of QCD_3 with an even number of fermions N_f. We prove that these
spectral correlators are universal, and demonstrate that they satisfy exact
massive spectral sum rules of QCD_3 in a phase where flavor symmetries are
spontaneously broken according to U(N_f) -> U(N_f/2) x U(N_f/2).Comment: 5 pages, REVTeX. Misprint correcte
Microscopic universality in the spectrum of the lattice Dirac operator
Large ensembles of complete spectra of the Euclidean Dirac operator for
staggered fermions are calculated for SU(2) lattice gauge theory. The
accumulation of eigenvalues near zero is analyzed as a signal of chiral
symmetry breaking and compared with parameter-free predictions from chiral
random matrix theory. Excellent agreement for the distribution of the smallest
eigenvalue and the microscopic spectral density is found. This provides direct
evidence for the conjecture that these quantities are universal functions.Comment: 4 pages, 3 figures (included), REVTeX 3.1; updated version to appear
in Phys. Rev. Let
Random Matrix Theory and Chiral Symmetry in QCD
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.Comment: invited review article for Ann. Rev. Nucl. Part. Sci., 61 pages, 11
figures, uses ar.sty (included); references added and typos correcte
The microscopic spectrum of the QCD Dirac operator with finite quark masses
We compute the microscopic spectrum of the QCD Dirac operator in the presence
of dynamical fermions in the framework of random-matrix theory for the chiral
Gaussian unitary ensemble. We obtain results for the microscopic spectral
correlators, the microscopic spectral density, and the distribution of the
smallest eigenvalue for an arbitrary number of flavors, arbitrary quark masses,
and arbitrary topological charge.Comment: 11 pages, RevTeX, 2 figures (included), minor typos corrected and
discussion extended, version to appear in Phys. Rev.
Spectral sum rules and finite volume partition function in gauge theories with real and pseudoreal fermions
Based on the chiral symmetry breaking pattern and the corresponding
low-energy effective lagrangian, we determine the fermion mass dependence of
the partition function and derive sum rules for eigenvalues of the QCD Dirac
operator in finite Euclidean volume. Results are given for and for
Yang-Mills theory coupled to several light adjoint Majorana fermions. They
coincide with those derived earlier in the framework of random matrix theory.Comment: 22p., SUNY-NTG-94/18, TPI-MINN-94/10-
Effective Lagrangians and Chiral Random Matrix Theory
Recently, sum rules were derived for the inverse eigenvalues of the Dirac
operator. They were obtained in two different ways: i) starting from the
low-energy effective Lagrangian and ii) starting from a random matrix theory
with the symmetries of the Dirac operator. This suggests that the effective
theory can be obtained directly from the random matrix theory. Previously, this
was shown for three or more colors with fundamental fermions. In this paper we
construct the effective theory from a random matrix theory for two colors in
the fundamental representation and for an arbitrary number of colors in the
adjoint representation. We construct a fermionic partition function for
Majorana fermions in Euclidean space time. Their reality condition is
formulated in terms of complex conjugation of the second kind.Comment: 27 page
Replica Limit of the Toda Lattice Equation
In a recent breakthrough Kanzieper showed that it is possible to obtain exact
nonperturbative Random Matrix results from the replica limit of the
corresponding Painlev\'e equation. In this article we analyze the replica limit
of the Toda lattice equation and obtain exact expressions for the resolvent of
the chiral Unitary Ensemble both in the quenched limit and in the presence of
additional massive flavors. This derivation explains in a natural way the
appearance of both compact and noncompact integrals, the hallmark of the
supersymmetric method, in the replica limit of the expression for the
resolvent. We also show that the supersymmetric partition function and the
partition function with fermionic replicas are related through the Toda lattice
equation.Comment: 4 pages, latex. One reference added. Discussion of GUE now in the
main text. Note added. Version to appear in Phys. Rev. Let
Random matrices and the replica method
Recent developments [Kamenev and Mezard, cond-mat/9901110, cond-mat/9903001;
Yurkevich and Lerner, cond-mat/9903025; Zirnbauer, cond-mat/9903338] have
revived a discussion about applicability of the replica approach to description
of spectral fluctuations in the context of random matrix theory and beyond. The
present paper, concentrating on invariant non-Gaussian random matrix ensembles
with orthogonal, unitary and symplectic symmetries, aims to demonstrate that
both the bosonic and the fermionic replicas are capable of reproducing
nonperturbative fluctuation formulas for spectral correlation functions in
entire energy scale, including the self-correlation of energy levels, provided
no sigma-model mapping is used.Comment: 12 pages (latex), presentation clarified, misprints fixe
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