Individual eigenvalue distributions for the Wilson Dirac operator

TY - JOUR

T1 - Individual eigenvalue distributions for the Wilson Dirac operator

AU - Akemann, G.

AU - Ipsen, Asger Cronberg

PY - 2012/4/1

Y1 - 2012/4/1

N2 - 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 DW. 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 v this expansion truncates after at most v 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 v.

AB - 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 DW. 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 v this expansion truncates after at most v 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 v.

U2 - 10.1007/jhep04(2012)102

DO - 10.1007/jhep04(2012)102

M3 - Journal article

SN - 1126-6708

VL - 2012

SP - 102

JO - Journal of High Energy Physics (Online)

JF - Journal of High Energy Physics (Online)

IS - 4

ER -