Spectrum of Dirac eigenvalues across QCD chiral transition
Here, we show that the universal scaling properties of the chiral phase transition in quantum chromodynamics (QCD) at the macroscale are encoded in the microscopic energy levels of the quarks in the background of gluon fields, defined as eigenvalues of the Dirac operator. We introduce a novel relation between the cumulants of the chiral condensate, and the correlations among the energy levels of quarks, i.e., the eigenspectra of the massless QCD Dirac operator. This relation elucidates how the fluctuations of the chiral condensate arise from the correlations within the infrared part of the energy spectra of quarks, and naturally leads to a generalization of the Banks-Casher relation for the cumulants of the chiral condensate. Then, through (2+1)-flavor lattice QCD calculations using HISQ action with varying light quark masses around the chiral phase transition temperature, we demonstrate that the correlations among the infrared part of the Dirac eigenvalue spectra exhibit same universal scaling behaviors as expected of the cumulants of the chiral condensate. We find that these universal scaling behaviors extend up to the physical values of the up and down quark masses.