arXiv subjects

J. J. M. Verbaarschot

Publications and source records attributed to J. J. M. Verbaarschot.

At least 91 records · Page 5Linked to original sources

arXiv1999

Chiral Symmetry and the Low-Energy Spectrum of the QCD Dirac Operator

The order parameter of the chiral phase transition is directly related to the infrared part of the spectrum of the QCD Dirac operator. This part of the spectrum follows from the low energy limit of QCD which is given by a partition function of weakly interacting Goldstone modes. We find that the slope of the Dirac spectrum is determined by the pion decay constant whereas for $λ\ll 1/L^2 Λ_{\rm QCD}$ the correlations of the Dirac eigenvalues are given by a random matrix theory with the global symmetries of the QCD partition function. A possible observation of these continuum results in lattice QCD with staggered fermions is discussed.

hep-ph↗

arXiv1999

Random Matrices and the Convergence of Partition Function Zeros in Finite Density QCD

We apply the Glasgow method for lattice QCD at finite chemical potential to a schematic random matrix model (RMM). In this method the zeros of the partition function are obtained by averaging the coefficients of its expansion in powers of the chemical potential. In this paper we investigate the phase structure by means of Glasgow averaging and demonstrate that the method converges to the correct analytically known result. We conclude that the statistics needed for complete convergence grows exponentially with the size of the system, in our case, the dimension of the Dirac matrix. The use of an unquenched ensemble at $μ=0$ does not give an improvement over a quenched ensemble. We elucidate the phenomenon of a faster convergence of certain zeros of the partition function. The imprecision affecting the coefficients of the polynomial in the chemical potential can be interpeted as the appearance of a spurious phase. This phase dominates in the regions where the exact partition function is exponentially small, introducing additional phase boundaries, and hiding part of the true ones. The zeros along the surviving parts of the true boundaries remain unaffected.

hep-lat↗

arXiv1999

The Infrared Limit of the QCD Dirac Spectrum

The distribution of the low-lying QCD Dirac spectrum is analyzed by means of partial quenched chiral perturbation theory. We identify an energy scale below which the valence quark mass dependence of the QCD partition function is given by chiral Random Matrix Theory. This is the the domain where the low-energy QCD partition function is dominated by the fluctuations of the mass term of the Goldstone bosons. In mesoscopic physics this domain is known the ergodic domain and the corresponding energy scale is the Thouless energy.

hep-lat↗

arXiv1999

The Spectral Density of the QCD Dirac Operator and Patterns of Chiral Symmetry Breaking

We study the spectrum of the QCD Dirac operator for two colors with fermions in the fundamental representation and for two or more colors with adjoint fermions. For $N_f$ flavors, the chiral flavor symmetry of these theories is spontaneously broken according to $SU(2N_f)\to Sp(2N_f)$ and $SU(N_f)\to O(N_f)$, respectively, rather than the symmetry breaking pattern $SU(N_f) \times SU(N_f) \to SU(N_f)$ for QCD with three or more colors and fundamental fermions. In this paper we study the Dirac spectrum for the first two symmetry breaking patterns. Following previous work for the third case we find the Dirac spectrum in the domain $λ\ll Λ_{\rm QCD}$ by means of partially quenched chiral perturbation theory. In particular, this result allows us to calculate the slope of the Dirac spectrum at $λ= 0$. We also show that for $λ\ll 1/L^2 Λ_{QCD}$ (with $L$ the linear size of the system) the Dirac spectrum is given by a chiral Random Matrix Theory with the symmetries of the Dirac operator.

hep-th↗

arXiv1999

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

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 phase diagram of the chRMT partition function, as well as the properties of Yang-Lee zeros 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.

hep-ph↗

arXiv1998

The Microscopic Spectral Density of the QCD Dirac Operator

We derive the microscopic spectral density of the Dirac operator in $SU(N_c\geq 3)$ Yang-Mills theory coupled to $N_f$ fermions in the fundamental representation. An essential technical ingredient is an exact rewriting of this density in terms of integrations over the super Riemannian manifold $Gl(N_f+1|1)$. The result agrees exactly with earlier calculations based on Random Matrix Theory.

hep-th↗

arXiv1998

Thouless Energy and Correlations of QCD Dirac Eigenvalues

Eigenvalues and eigenfunctions of the QCD Dirac operator are studied for an instanton liquid partition function. We find that for energy differences $δE$ below an energy scale $E_c$, identified as the Thouless energy, the eigenvalue correlations are given by Random Matrix Theory. The value of $E_c$ shows a weak volume dependence for eigenvalues near zero and is consistent with a scaling of $E_c \sim 1/L^2$ in the bulk of the spectrum in agreement with estimates from chiral perturbation theory, that $E_c/Δ\approx F_π^2 L^2/π$ (with average level spacing $Δ$). For $δE> E_c$ the number variance shows a linear dependence. For the wave functions we find a small nonzero multifractality index.

hep-ph↗

arXiv1998

On the Phase Diagram of QCD

We analyze the phase diagram of QCD with two massless quark flavors in the space of temperature, T, and chemical potential of the baryon charge, mu, using available experimental knowledge of QCD, insights gained from various models, as well as general and model independent arguments including continuity, universality, and thermodynamic relations. A random matrix model is used to describe the chiral symmetry restoration phase transition at finite T and mu. In agreement with general arguments, this model predicts a tricritical point in the T mu plane. Certain critical properties at such a point are universal and can be relevant to heavy ion collision experiments.

hep-ph↗

arXiv1998

Characterization of Universal Behavior in QCD Dirac Spectra

In this lecture we discuss correlations of the QCD Dirac eigenvalues. We find that below a scale of $E_c\sim Λ/L^2$ they are given by chiral Random Matrix Theory. This follows from analytical arguments based on partially quenched Chiral Perturbation Theory and is substantiated by lattice QCD and instanton liquid simulations.

hep-th↗

arXiv1998

Random Matrix Theory and QCD at nonzero chemical potential

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\ne 0 $ we identify universal properties of the Dirac eigenvalues and study the effect of quenching on the distribution of Yang-Lee zeros.

hep-ph↗

arXiv1998

From chiral Random Matrix Theory to chiral Perturbation Theory

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 ($m_v \ll E_c$) and the diffusive domain ($m_v \gg E_c$). These domains are separated by a mass scale $E_c \sim F^2/Σ_0 L^2$ (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 superunitary 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.

hep-th↗

arXiv1998

Thouless Energy and Correlations of QCD Dirac Eigenvalues

Eigenvalues and eigenfunctions of the QCD Dirac operator are studied for gauge field configurations given by a liquid of instantons. We find that for energy differences $δE$ below an energy scale $E_c$ the eigenvalue correlations are given by Random Matrix Theories with the chiral symmetries of the QCD partition function. For eigenvalues near zero this energy scale shows a weak volume dependence that is not consistent with $E_c \sim 1/L^2$ which might be expected from the pion Compton wavelength and from the behavior of the Thouless energy in mesoscopic systems. However, the numerical value of $E_c$ for our largest volumes is in rough agreement with estimates from the pion Compton wavelength. A scaling behaviour consistent with $E_c\sim 1/L^2$ is found in the bulk of the spectrum. For $δE> E_c$ the number variance shows a linear dependence with a slope which is larger than the nonzero multifractality index of the wave functions. Finally, the average spectral density and the scalar susceptibilities are discussed in the context of quenched chiral perturbation theory. We argue that a nonzero value of the disconnected scalar susceptibility requires a linear dependence of the number variance on $δE$.

hep-ph↗

arXiv1998

Universality in Chiral Random Matrix Theory at $β=1$ and $β=4$

In this paper the kernel for the spectral correlation functions of the invariant chiral random matrix ensembles with real ($β=1$) and quaternion real ($β= 4$) matrix elements is expressed in terms of the kernel of the corresponding complex Hermitean random matrix ensembles ($β=2$). Such identities are exact in case of a Gaussian probability distribution and, under certain smoothness assumptions, they are shown to be valid asymptotically for an arbitrary finite polynomial potential. They are proved by means of a construction proposed by Brézin and Neuberger. Universal behavior at the hard edge of the spectrum for all three chiral ensembles then follows from microscopic universality for $β=2$ as shown by Akemann, Damgaard, Magnea and Nishigaki.

hep-th↗

arXiv1997

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.

hep-lat↗

arXiv1997

Universal Behavior in Dirac Spectra

In these lectures we review recent results on universal fluctuations of QCD Dirac spectra and applications of Random Matrix Theory (RMT) to QCD. We review general properties of Dirac spectra and discuss the relation between chiral symmetry breaking and correlations of Dirac eigenvalues. In particular, we will focus on the microscopic spectral density density, i.e. the spectral density near zero virtuality on the scale of a typical level spacing. The relation with Leutwyler-Smilga sum-rules will be discussed. The success of applications of RMT to spectra of 'complex' systems leads us to the introduction of a chiral Random Matrix Theory (chRMT) with the global symmetries of the QCD partition function. Our central conjecture is that it decribes correlations of QCD Dirac spectra. We will review recent universality proofs supporting this conjecture. Lattice QCD results for the microscopic spectral density and for correlations in the bulk of the spectrum are shown to be in perfect agreement with chRMT. We close with a review of chRMT at nonzero chemical potential. Novel features of spectral universality in nonhermitean matrices will be discussed. As an illustration of mathematical methods used in RMT several important recent results will be derived in all details. We mention the derivation of the microscopic spectral density, the universality proof by Akemann, Damgaard, Magnea and Nishigaki, the spectral density of a chRMT at nonzero temperature and the Stephanov solution for chRMT at nonzero chemical potential.

hep-th↗

arXiv1997

Random-matrix universality in the small-eigenvalue spectrum of the lattice Dirac operator

We analyze complete spectra of the lattice Dirac operator in SU(2) gauge theory and demonstrate that the distribution of low-lying eigenvalues is described by random matrix theory. We present possible practical applications of this random-matrix universality. In particular, reliable extrapolations of lattice gauge data to the thermodynamic limit are discussed.

hep-lat↗

arXiv1997

Universal Fluctuations in Dirac Spectra

In these two lectures given at the 1997 Zakopane workshop on "New Developments in Quantum Field Theory" we review recent results on universal fluctuations in QCD Dirac spectra. We start the first lecture with a review of some general properties of Dirac spectra. It will be argued that there is an intimate relation between chiral symmetry breaking and correlations of Dirac eigenvalues. In particular, we will focus on the microscopic spectral density density, i.e. the spectral density near zero virtuality on the scale of a typical level spacing. The relation with Leutwyler-Smilga sum-rules will be discussed. Standard methods for the statistical analysis of quantum spectra will be reviewed. Recent results on the application of Random Matrix Theory to spectra of 'complex' systems will be summarized. This leads to the introduction of a chiral Random Matrix Theory (chRMT) with the global symmetries of the QCD partition function. In the second lecture the chiral random matrix model will be compared to QCD and some of its properties will be discussed. Our central conjecture is that correlations of QCD Dirac spectra are described by chRMT. We will review recent results showing that the microscopic spectral density and eigenvalue correlations near zero virtuality are strongly universal. Lattice QCD results for the microscopic spectral density and for correlations in the bulk of the spectrum will be presented. In all cases that have been considered, the correlations are in perfect agreement with chRMT. We will end the second lecture with a review of chiral Random Matrix Theory at nonzero chemical potential. New features of spectral universality in nonhermitean matrices will be discussed.

hep-th↗

arXiv1997

Random matrix triality at nonzero chemical potential

We introduce three universality classes of chiral random matrix ensembles with a nonzero chemical potential and real, complex or quaternion real matrix elements. In the thermodynamic limit we find that the distribution of the eigenvalues in the complex plane does not depend on the Dyson index, and is given by the solution proposed by Stephanov. For a finite number of degrees of freedom, $N$, we find an accumulation of eigenvalues on the imaginary axis for real matrices, whereas for quaternion real matrices we find a depletion of eigenvalues in this domain. This effect is of order $1/\sqrt N$. In particular for the real case the resolvent shows a discontinuity of order $1/\sqrt N$. These results are in agreement with lattice QCD simulations with staggered fermions and recent instanton liquid simulations both for two colors and nonzero chemical potential.

hep-lat↗