Crossover to non-universal microscopic spectral fluctuations in lattice gauge theory

Abstract

The spectrum of the Dirac operator near zero virtuality obtained in lattice gauge simulations is known to be universally described by chiral random matrix theory. We address the question of the maximum energy for which this universality persists. For this purpose, we analyze large ensembles of complete spectra of the Euclidean Dirac operator for staggered fermions. We calculate the disconnected scalar susceptibility and the microscopic number variance for the chiral symplectic ensemble of random matrices and compare the results with lattice Dirac spectra for quenched SU(2). The crossover to a non-universal regime is clearly identified and found to scale with the square of the linear lattice size and with f 2 π , in agreement with theoretical expectations.Recently, it has been shown by several authors that chiral random matrix theory (chRMT) is able to reproduce quantitatively spectral properties of the Dirac operator obtained from QCD lattice data. This statement is valid both for fluctuation properties in the bulk of the spectrum and for microscopic spectral properties near zero virtuality, see the reviews [1,2] and Refs. [3][4][5][6]. This result implies that the spectral fluctuation properties of the Dirac operator are universal, i.e., determined solely by the underlying symmetry of

Chiral random matrix model for critical statistics

“…In particular, this has been confirmed by the analysis of the low-energy effective theory [4,5,6,7], universality studies [8,9,10,11,12], lattice QCD simulations [13,4,14,15,17,18,16,19,20,21,22] and the study two-sublattice theories with disorder [23,24,25,26]. This means that the dynamical details of QCD are not important on energy scales of the order of the average level spacing.…”

“…Spectral correlations have been studied in great detail for both lattice QCD simulations [4,14,15,19,17,20,21] and instanton-liquid simulations [5] (see [2] for a recent review). In lattice QCD they were studied by means of the disconnected scalar susceptibility [22], and complete agreement with partially quenched chiral perturbation theory was found.…”

“…It can be obtained from the n-particle matrix element (16) by integration over all coordinates (eigenvalues) except x 1 and x 2 . The remaining sum over ω π (1) and ω π(2) can be rewritten as…”

Long range disorder and Anderson transition in systems with chiral symmetry

“…from chiral perturbation theory [43] and lattice simulations [44]. The range of applicability of this chRBM was found not to be restricted to the above mentioned zero temperature case.…”

Universal and non-universal behavior in Dirac spectra

“…Recently, some progress was made in identifying the equivalent of the Thouless energy in QCD [19][20][21][22][23]. Again, we shall concentrate on the low-lying eigenvalues.…”