Eigenspectrum Calculation of the O(a)-Improved Wilson-Dirac Operator in Lattice QCD Using the Sakurai-Sugiura Method

Abstract

We have developed a computer code to find eigenvalues and eigenvectors of non-Hermitian sparse matrices arising in lattice quantum chromodynamics (lattice QCD). The Sakurai-Sugiura (SS) method (Sakurai and Sugiura, J Comput Appl Math 159:119, 2003) is employed here, which is based on a contour integral, allowing us to obtain desired eigenvalues located inside a given contour of the complex plane. We apply the method here to calculating several low-lying eigenvalues of the non-Hermitian O(a)-improved Wilson-Dirac operator D (Sakurai et al., Comput Phys Commun 181:113, 2010). Evaluation of the low-lying eigenvalues is crucial since they determine the sign of its determinant detD, important quantity in lattice QCD. We are particularly interested in such cases as finding the lowest eigenvalues to be equal or close to zero in the complex plane. Our implementation is tested for the Wilson-Dirac operator in free case, for which the eigenvalues are analytically known. We also carry out several numerical experiments using different sets of gauge field configurations obtained in quenched approximation as well as in full QCD simulation almost at the physical point. Various lattice sizes L _x L _y L _z L _t are considered from 8³ × 16 to 96⁴, amounting to the matrix order 12L _x L _y L _z L _t from 98,304 to 1,019,215,872.