Statistical properties at the spectrum edge of the QCD Dirac operator.pdf

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Statistical properties at the spectrum edge of the QCD Dirac operator.pdf

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Statistical properties at the spectrum edge of the QCD Dirac operator

a r X i v : h e p - l a t / 9 7 1 2 0 2 6 v 3 3 0 A p r 1 9 9 8 EPJ manuscript No. (will be inserted by the editor) Statistical properties at the spectrum edge of the QCD Dirac operator Jian–Zhong Ma1, Thomas Guhr1, and Tilo Wettig2 1 Max–Planck–Institut fu?r Kernphysik, Postfach 103980, D-69029 Heidelberg, Germany 2 Institut fu?r Theoretische Physik, Technische Universita?t Mu?nchen, D-85747 Garching, Germany 26 January 1998 Abstract. The statistical properties of the spectrum of the staggered Dirac operator in an SU(2) lattice gauge theory are analyzed both in the bulk of the spectrum and at the spectrum edge. Two commonly used statistics, the number variance and the spectral rigidity, are investigated. While the spectral fluctuations at the edge are suppressed to the same extent as in the bulk, the spectra are more rigid at the edge. To study this effect, we introduce a microscopic unfolding procedure to separate the variation of the microscopic spectral density from the fluctuations. For the unfolded data, the number variance shows oscillations of the same kind as before unfolding, and the average spectral rigidity becomes larger than the one in the bulk. In addition, the short-range statistics at the origin is studied. The lattice data are compared to predictions of chiral random-matrix theory, and agreement with the chiral Gaussian Symplectic Ensemble is found. PACS. 11.15.Ha Lattice gauge theory – 05.45.+b Theory and models of chaotic systems – 11.30.Rd Chiral symmetries – 12.38.Gc Lattice QCD calculations 1 Introduction The spectrum of the Dirac operator is an important as- pect of nonperturbative QCD. In Euclidean space, the Dirac operator reads i/D = i/? + g(λa/2)/Aa, where g is the coupling constant, λa are the generators of SU(N)-color, and Aaμ are the gauge fields. Whether in the continuum or on the lattice, to obtain physical observables one has to perform an average over the ensemble of gauge field configurations. The eigenvalues of i/D fluctuate