Universal Fluctuations in Dirac Spectra

Abstract

We have argued that there is an intimate relation between correlations of Dirac eigenvalues and the breaking of chiral symmetry. In the chiral limit, the fermion determinant suppresses gauge field configurations with small Dirac eigenvalues. Correlations counteract this suppression, and are a necessary ingredient of chiral symmetry breaking. From the study of eigenvalue correlations in strongly interacting systems, we have concluded that they are described naturally with by Random Matrix Theory with the global symmetries of the physical system. In QCD, this led to the introduction of chiral Random Matrix Theories. They provided us with an analytical understanding of the statistical properties of the eigenvalues on the scale of a typical level spacing. In particular, impressive agreement between lattice QCD and chiral Random Matrix Theory was found for the microscopic spectral density and for spectral correlations in the bulk of the spectrum. An extension of this model to nonzero chemical potential explains some intriguing properties of previously obtained lattice QCD Dirac spectra and instanton liquid Dirac spectra.