Spectral properties of the Wilson-Dirac operator and random matrix theory

Abstract

Random matrix theory has been successfully applied to lattice quantum chromodynamics. In particular, a great deal of progress has been made on the understanding, numerically as well as analytically, of the spectral properties of the Wilson-Dirac operator. In this paper, we study the infrared spectrum of the Wilson-Dirac operator via random matrix theory including the three leading order a 2 correction terms that appear in the corresponding chiral Lagrangian. A derivation of the joint probability density of the eigenvalues is presented. This result is used to calculate the density of the complex eigenvalues, the density of the real eigenvalues, and the distribution of the chiralities over the real eigenvalues. A detailed discussion of these quantities shows how each low-energy constant affects the spectrum. Especially we consider the limit of small and large (which is almost the mean field limit) lattice spacing. Comparisons with Monte Carlo simulations of the random matrix theory show a perfect agreement with the analytical predictions. Furthermore we present some quantities which can be easily used for comparison of lattice data and the analytical results.

Universal distributions from non-Hermitian perturbation of zero modes

“…This is not true for a direct sum, where we get a shift by a diagonal matrix consisting of blocks of different scalars in front of the identity matrix when we do not have the trace condition. This has indeed physical effects as known from the Wilson-Dirac random matrix model [15][16][17]19] and its corresponding lattice QCD-operator [41].…”

“…For the sake of generality, we choose S R and S I being independently a direct sum of operators in one of the ten symmetry classes rather than just a single one. The direct sum is highly important because some ensembles in the Magnea-Bernard-LeClair classification [30][31][32] have real and imaginary parts that decompose in direct sums, e.g., the Wilson-Dirac operator [15][16][17]19]. This happens exactly then when S has a pseudo-Hermiticity property, meaning S † = γ 5 Sγ 5 with γ 5 = γ † 5 = γ −1 5 .…”

“…This poses a question: Is it possible in the presence of the perturbation to distinguish modes that have their origin in the topological zero modes of the unperturbed system? This question is highly relevant for the study of Majorana modes [6,[8][9][10][11][12][13][14] as well as in QCD [15][16][17][18][19]. A first answer was found in [15][16][17] in the context of QCD where it was realized that indeed it is possible to distinguish would-be zero modes, since their distribution have very specific spectral statistics and scale differently with the volume than the rest of the small eigenvalues.…”

Progress on Complex Langevin simulations of a finite density matrix model for QCD

“…RMT has provided a great deal of analytical results for non-perturbative aspects of QCD, including finite density, lattice cutoff effects and topology [6][7][8][9][10][11][12][13]. First results of our studies were presented in [14].…”

Extracting the electromagnetic pion form factor from QCD in a finite volume

“…When the simulated quark mass is as small as the typical breaking scale of the chiral (flavor) symmetry (it is typically ∼Λ 3 QCD a 2 for the improved Wilson or staggered fermions, where Λ QCD is the QCD scale and a denotes the lattice spacing), it is known that the chiral logarithm is largely distorted. The low-lying Dirac eigenvalue spectrum, for example, is a quantity sensitive to such discretization effects [7].…”