Statistical analysis and the equivalent of a Thouless energy in lattice QCD Dirac spectra

Abstract

Random Matrix Theory (RMT) is a powerful statistical tool to model spectral fluctuations. This approach has also found fruitful application in Quantum Chromodynamics (QCD). Importantly, RMT provides very efficient means to separate different scales in the spectral fluctuations. We try to identify the equivalent of a Thouless energy in complete spectra of the QCD Dirac operator for staggered fermions from SU(2) lattice gauge theory for different lattice size and gauge couplings. We focus on the bulk of the spectrum. In disordered systems, the Thouless energy sets the universal scale for which RMT applies. This relates to recent theoretical studies which suggest a strong analogy between QCD and disordered systems. The wealth of data allows us to analyze several statistical measures in the bulk of the spectrum with high quality. We find deviations which allows us to give an estimate for this universal scale. Other deviations than these are seen whose possible origin is discussed. Moreover, we work out higher order correlators as well, in particular three-point correlation functions.

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.…”

Equivalent of a thouless energy in lattice QCD Dirac spectra

“…3. Independently of the spectral region considered [7] and of β we find that the point where the deviation sets in, scales as…”

“…4). In order to find out, if these deviations are due to a Thouless energy, we performed a Fourier analysis of the oscillations of the staircase function [7]. From it we concluded that the deviations of the data obtained with polynomial unfolding are due to a non-polynomial-like part in the average level density and not to an equivalent of the Thouless energy.…”

“…Furthermore there is configuration unfolding, where N (λ) is fitted for each configuration to a polynomial of degree n. Strong coupling expansions for SU c (2) with staggered fermions [5] and 1/N c expansion of the QCD level density [6] motivate this ansatz. For technical details and further unfolding procedures see [7]. Whatever approach one uses, the mean number of rescaled levels in an interval of length L in units of the mean level spacing should equal L. This assures that the unfolded spectrum has mean level density unity.…”

“…Again we see good agreement (see Figs. 11 and 12 in [7]). With ensemble unfolding of the data we obtain for Σ 2 (L) the curves plotted in Fig.…”