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 $\delta$-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 $\delta$-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.

中文翻译：

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谱函数非耗散部分的收敛四面体法计算

固态物理学中的许多物理量是从$ķ$-空间求和。对于谱函数，频率相关因子可以分解为能量守恒$\delta$-函数部分和非耗散主值部分。一个非常有用的方案$ķ$-空间求和是四面体方法。四面体方法已被广泛用于计算能量守恒的总和$\delta$-函数部分，例如介电函数的虚部。另一方面，对于非耗散部分，如介电函数的实部，相应的四面体法并没有被大量使用。在本文中，我们解决了非耗散部分的四面体方法中的技术难题，并提出了一种易于实现的稳定方法来克服这些困难。我们通过计算铂的静态和动态自旋霍尔电导率来展示我们的方法。我们的方法可广泛应用于计算线性静态或动态电导率、电子自能和电极化率等。