Application of Gauss - Lanczos Algorithm to Determine Low Modes Density of Dirac Operator

Abstract

There are numerous applications in physics, especially in Lattice QCD, where is required to bound entries and the trace of the inverse and the determinant of a large sparse matrix. This paper review one of the most popular methods which are used in lattice QCD to compute the determinant of the lattice Dirac operator: Gaussian integral representation. A modified algorithm can be used for other purposes too, for example for the determination of the density of eigenvalues of the Dirac operator near the origin. This because in Lattice QCD, low-lying Dirac modes are a suitable tool to understand chiral symmetry since they encode the nature of quark propagation as well as the condensate itself in the chiral regime. The formation of a non-zero chiral condensate is an effect of the accumulation of the low modes of the Dirac operator near zero. We review the development in Krylov subspace evaluation of matrix functions and we develop a practical numerical algorithm to achieve a reliable determination of the density of eigenvalues of the Dirac operator near the origin using the Gauss-Lanczos quadrature. We utilize the optimal properties of Krylov subspaces in approximating the distribution of the eigenvalues of the Dirac operator. In this work we have used the Boriçi - Creutz operator to test our method, as an example of using this algorithm in Lattice QCD.