Thouless Energy and Correlations of QCD Dirac Eigenvalues

Abstract

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 $\delta E$ 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 $\delta E>E_c$ 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 $\delta E$.

19 pages, Latex and 8 postscript figures; explained the meaning of associated Goldstone bosons and corrected typos

References in corpus (4)

• Universal Spectral Correlators and Massive Dirac Operators
• The microscopic spectrum of the QCD Dirac operator with finite quark masses
• Universality in Chiral Random Matrix Theory at $\beta =1$ and $\beta =4$
• Statistical properties at the spectrum edge of the QCD Dirac operator

Cited by in corpus (44)

• From chiral Random Matrix Theory to chiral Perturbation Theory
• Scale-invariance of parity-invariant three-dimensional QED
• Correlated Dirac Eigenvalues and Axial Anomaly in Chiral Symmetric QCD
• No evidence for bilinear condensate in parity-invariant three-dimensional QED with massless fermions
• Chiral phase transition in lattice QCD as a metal-insulator transition
• Matrix Models and QCD with Chemical Potential
• Small eigenvalues of the SU(3) Dirac operator on the lattice and in Random Matrix Theory
• The Spectral Density of the QCD Dirac Operator and Patterns of Chiral Symmetry Breaking
• Eigenvalue spectrum of massless Dirac operators on the lattice
• Effective Low Energy Theories and QCD Dirac Spectra
• The Chiral Condensate in a Finite Volume
• Modeling Finite-Volume Effects and Chiral Symmetry Breaking in Two-Flavor QCD Thermodynamics
• Chiral phase transition and Anderson localization in the Instanton Liquid Model for QCD
• Universal Scaling of the Chiral Condensate in Finite-Volume Gauge Theories
• Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature
• Statistical properties of the spectrum of the QCD Dirac operator at low energy
• Anderson transition and multifractals in the spectrum of the Dirac operator of Quantum Chromodynamics at high temperature
• Spectral ergodicity and normal modes in ensembles of sparse matrices
• Calorons and localization of quark eigenvectors in lattice QCD
• Critical statistics for non-Hermitian matrices
• Localization of Dirac Fermions in Finite-Temperature Gauge Theory
• Deconfinement, chiral transition and localisation in a QCD-like model
• Spectral Universality of Real Chiral Random Matrix Ensembles
• Random Matrix Theory, Chiral Perturbation Theory, and Lattice Data
• Eigenvalues and Eigenvectors of the Staggered Dirac Operator at Finite Temperature
• Spectral correlations of the massive QCD Dirac operator at finite temperature
• Spectrum of the U(1) staggered Dirac operator in four dimensions
• Staggered chiral random matrix theory
• Stochastic field theory for a Dirac particle propagating in gauge field disorder
• Long range disorder and Anderson transition in systems with chiral symmetry
• The QCD vacuum as a disordered medium: A simplified model for the QCD Dirac operator
• Spectrum of the SU(3) Dirac operator on the lattice: Transition from random matrix theory to chiral perturbation theory
• Spectral properties of a generalized chGUE
• Metal or Insulator? Dirac operator spectrum in holographic QCD
• Dirac eigenvalues and eigenvectors at finite temperature
• Universal and non-universal behavior in Dirac spectra
• The zeros of the QCD partition function
• Hydrodynamics of the Chiral Dirac Spectrum
• Gluino Condensation in an Interacting Instanton Ensemble
• Universal scaling of higher-order spacing ratios in Gaussian random matrices
• Random Matrix Model for Wilson Fermions on the Lattice
• Randomness on the Lattice
• Beyond the Thouless energy
• Topology and chiral symmetry breaking in QCD