37 research outputs found
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
Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model
We study spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, a variant of the k-body embedded random ensembles studied for several decades in the context of nuclear physics and quantum chaos. We show analytically that the fourth- and sixth-order energy cumulants vanish in the limit of a large number of particles Nââ, which is consistent with a Gaussian spectral density. However, for finite N, the tail of the average spectral density is well approximated by a semicircle law. The specific heat coefficient, determined numerically from the low-temperature behavior of the partition function, is consistent with the value obtained by previous analytical calculations. For energy scales of the order of the mean level spacing we show that level statistics are well described by random matrix theory. Due to the underlying Clifford algebra of the model, the universality class of the spectral correlations depends on N. For larger energy separations we identify an energy scale that grows with N, reminiscent of the Thouless energy in mesoscopic physics, where deviations from random matrix theory are observed. Our results are a further confirmation that the Sachdev-Ye-Kitaev model is quantum chaotic for all time scales. According to recent claims in the literature, this is an expected feature in field theories with a gravity dual.EPSRC, Grant No. EP/I004637/1; U.S. Department of Energy Grant No. DE-FG-88FR4038
Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition
We introduce a random two-matrix model interpolating between a chiral
Hermitian (2n+nu)x(2n+nu) matrix and a second Hermitian matrix without
symmetries. These are taken from the chiral Gaussian Unitary Ensemble (chGUE)
and Gaussian Unitary Ensemble (GUE), respectively. In the microscopic large-n
limit in the vicinity of the chGUE (which we denote by weakly non-chiral limit)
this theory is in one to one correspondence to the partition function of Wilson
chiral perturbation theory in the epsilon regime, such as the related two
matrix-model previously introduced in refs. [20,21]. For a generic number of
flavours and rectangular block matrices in the chGUE part we derive an
eigenvalue representation for the partition function displaying a Pfaffian
structure. In the quenched case with nu=0,1 we derive all spectral correlations
functions in our model for finite-n, given in terms of skew-orthogonal
polynomials. The latter are expressed as Gaussian integrals over standard
Laguerre polynomials. In the weakly non-chiral microscopic limit this yields
all corresponding quenched eigenvalue correlation functions of the Hermitian
Wilson operator.Comment: 27 pages, 4 figures; v2 typos corrected, published versio
The QCD sign problem and dynamical simulations of random matrices
At nonzero quark chemical potential dynamical lattice simulations of QCD are
hindered by the sign problem caused by the complex fermion determinant. The
severity of the sign problem can be assessed by the average phase of the
fermion determinant. In an earlier paper we derived a formula for the
microscopic limit of the average phase for general topology using chiral random
matrix theory. In the current paper we present an alternative derivation of the
same quantity, leading to a simpler expression which is also calculable for
finite-sized matrices, away from the microscopic limit. We explicitly prove the
equivalence of the old and new results in the microscopic limit. The results
for finite-sized matrices illustrate the convergence towards the microscopic
limit. We compare the analytical results with dynamical random matrix
simulations, where various reweighting methods are used to circumvent the sign
problem. We discuss the pros and cons of these reweighting methods.Comment: 34 pages, 3 figures, references added, as published in JHE
Two dimensional fermions in three dimensional YM
Dirac fermions in the fundamental representation of SU(N) live on the surface
of a cylinder embedded in and interact with a three dimensional SU(N)
Yang Mills vector potential preserving a global chiral symmetry at finite .
As the circumference of the cylinder is varied from small to large, the chiral
symmetry gets spontaneously broken in the infinite limit at a typical bulk
scale. Replacing three dimensional YM by four dimensional YM introduces
non-trivial renormalization effects.Comment: 21 pages, 7 figures, 5 table
The epsilon expansion at next-to-next-to-leading order with small imaginary chemical potential
We discuss chiral perturbation theory for two and three quark flavors in the
epsilon expansion at next-to-next-to-leading order (NNLO) including a small
imaginary chemical potential. We calculate finite-volume corrections to the
low-energy constants and and determine the non-universal
modifications of the theory, i.e., modifications that cannot be mapped to
random matrix theory (RMT). In the special case of two quark flavors in an
asymmetric box we discuss how to minimize the finite-volume corrections and
non-universal modifications by an optimal choice of the lattice geometry.
Furthermore we provide a detailed calculation of a special version of the
massless sunset diagram at finite volume.Comment: 21 pages, 5 figure
Nonlinear Sigma Model for Disordered Media: Replica Trick for Non-Perturbative Results and Interactions
In these lectures, given at the NATO ASI at Windsor (2001), applications of
the replicas nonlinear sigma model to disordered systems are reviewed. A
particular attention is given to two sets of issues. First, obtaining
non-perturbative results in the replica limit is discussed, using as examples
(i) an oscillatory behaviour of the two-level correlation function and (ii)
long-tail asymptotes of different mesoscopic distributions. Second, a new
variant of the sigma model for interacting electrons in disordered normal and
superconducting systems is presented, with demonstrating how to reduce it,
under certain controlled approximations, to known ``phase-only'' actions,
including that of the ``dirty bosons'' model.Comment: 25 pages, Proceedings of the NATO ASI "Field Theory of Strongly
Correlated Fermions and Bosons in Low - Dimensional Disordered Systems",
Windsor, August, 2001; to be published by Kluwe
Singular values of the Dirac operator in dense QCD-like theories
We study the singular values of the Dirac operator in dense QCD-like theories
at zero temperature. The Dirac singular values are real and nonnegative at any
nonzero quark density. The scale of their spectrum is set by the diquark
condensate, in contrast to the complex Dirac eigenvalues whose scale is set by
the chiral condensate at low density and by the BCS gap at high density. We
identify three different low-energy effective theories with diquark sources
applicable at low, intermediate, and high density, together with their
overlapping domains of validity. We derive a number of exact formulas for the
Dirac singular values, including Banks-Casher-type relations for the diquark
condensate, Smilga-Stern-type relations for the slope of the singular value
density, and Leutwyler-Smilga-type sum rules for the inverse singular values.
We construct random matrix theories and determine the form of the microscopic
spectral correlation functions of the singular values for all nonzero quark
densities. We also derive a rigorous index theorem for non-Hermitian Dirac
operators. Our results can in principle be tested in lattice simulations.Comment: 3 references added, version published in JHE