Individual eigenvalue distributions for the Wilson Dirac operator

Abstract

We derive the distributions of individual eigenvalues for the Hermitian Wilson Dirac Operator D 5 as well as for real eigenvalues of the Wilson Dirac Operator D W . The framework we provide is valid in the epsilon regime of chiral perturbation theory for any number of flavours N f and for non-zero low energy constants W 6,7,8 . It is given as a perturbative expansion in terms of the k-point spectral density correlation functions and integrals thereof, which in some cases reduces to a Fredholm Pfaffian. For the real eigenvalues of D W at fixed chirality ν this expansion truncates after at most ν terms for small lattice spacing a. Explicit examples for the distribution of the first and second eigenvalue are given in the microscopic domain as a truncated expansion of the Fredholm Pfaffian for quenched D 5 , where all k-point densities are explicitly known from random matrix theory. For the real eigenvalues of quenched D W at small a we illustrate our method by the finite expansion of the corresponding Fredholm determinant of size ν.

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

“…First comparisons of the analytical predictions with lattice data show a promising agreement [4,6,7]. Good fits of the low-energy constants are expected for the distributions of individual eigenvalues [9,10,39].…”

“…(37) and (39) from the expansion in the first column of the determinant in the joint probability density. Equation (84) is a complicated expression which is quite difficult to numerically evaluate.…”

Determination of low-energy constants of Wilson chiral perturbation theory

“…In refs. [25][26][27][28] they have been studied by matching the analytical predictions [27,[29][30][31][32][33][34][35][36] for the spectrum of the Wilson Dirac operator -with fixed index in a finite volume -to lattice data. 1 Determinations of the Wilson LECs have also been carried out via the spectral density of the Hermitian Wilson-Dirac operator [38][39][40].…”

Preserving topology while breaking chirality: from chiral orthogonal to anti-symmetric Hermitian ensemble

“…Let us highlight one peculiarity that is distinct from the other models, which is its corresponding symmetry group. Whereas most models, e.g., in [2,3,18,19,20,21,26,27,28,29,32,33,36], are usually invariant with respect to a unitary group in one or another limit, our model always satisfies an orthogonal symmetry, regardless of the value of a, including infinity. This difference is remarkable because of the group integral that has to be solved to obtain the joint probability density function (jpdf) of the eigenvalues.…”