Tetrahedron integration method for strongly varying functions: Application to the GT self-energy

摘要

We develop a tetrahedron method for the Brillouin-zone integration of expressions that vary a lot as a function of energy. The usual tetrahedron method replaces the continuous integral over the Brillouin zone by a weighted sum over a finite number of k points. The weight factors are determined under the assumption that the function to be integrated be linear inside each tetrahedron, so the method works best for functions that vary smoothly over the Brillouin zone. In this paper, we describe a new method that can deal with situations where this condition is not fulfilled. Instead of weight factors, we employ weight functions, defined as piecewise cubic polynomials over energy. Since these polynomials are analytic, any function, also strongly varying ones, can be integrated accurately and piecewise analytically. The method is applied to the evaluation of the GT self-energy using two techniques, analytic continuation and contour deformation. (We also describe a third technique, which is a hybrid of the two. An efficient algorithm for the dilogarithm needed for analytic continuation is formulated in Appendix.) The resulting spectral functions converge very quickly with respect to the k-point sampling.