Converging tetrahedron method calculations for the nondissipative parts of spectral functions

Abstract

Many physical quantities in solid-state physics are calculated from $k$-space summation. For spectral functions, the frequency-dependent factor can be decomposed into the energy-conserving $\ensuremath{\delta}$-function part and the nondissipative principal value part. A very useful scheme for this $k$-space summation is the tetrahedron method. Tetrahedron methods have been widely used to calculate the summation of the energy-conserving $\ensuremath{\delta}$-function part, such as the imaginary part of the dielectric function. On the other hand, the corresponding tetrahedron method for the nondissipative part, such as the real part of the dielectric function has not been used much. In this paper, we address the technical difficulties in the tetrahedron method for the nondissipative part and present an easy-to-implement stable method to overcome those difficulties. We demonstrate our method by calculating the static and dynamical spin Hall conductivity of platinum. Our method can be widely applied to calculate linear static or dynamical conductivity, self-energy of an electron, and electric polarizability, to name a few.