We present efficient methods for Brillouin zone integration with a non-zero but possibly very small broadening factor $\eta$, focusing on cases in which downfolded Hamiltonians can be evaluated efficiently using Wannier interpolation. We describe robust, high-order accurate algorithms automating convergence to a user-specified error tolerance $\varepsilon$, emphasizing an efficient computational scaling with respect to $\eta$. After analyzing the standard equispaced integration method, applicable in the case of large broadening, we describe a simple iterated adaptive integration algorithm effective in the small $\eta$ regime. Its computational cost scales as $\mathcal{O}(\log^3(\eta^{-1}))$ as $\eta \to 0^+$ in three dimensions, as opposed to $\mathcal{O}(\eta^{-3})$ for equispaced integration. We argue that, by contrast, tree-based adaptive integration methods scale only as $\mathcal{O}(\log(\eta^{-1})/\eta^{2})$ for typical Brillouin zone integrals. In addition to its favorable scaling, the iterated adaptive algorithm is straightforward to implement, particularly for integration on the irreducible Brillouin zone, for which it avoids the tetrahedral meshes required for tree-based schemes. We illustrate the algorithms by calculating the spectral function of SrVO$_3$ with broadening on the meV scale.

翻译：本文介绍了布里渊区积分的有效方法，对于具有非零但可能非常小的展宽因子$\eta$进行了重点研究，注重利用Wannier插值可以高效计算被压缩的哈密顿量的情况。我们描述了强大的、高阶准确的算法，使其收敛到用户指定的误差容限$\varepsilon$，并强调了与$\eta$的计算缩放的效率。在分析适用于较大展宽的标准均匀间距积分方法后，我们描述了一种简单的迭代自适应积分算法，可以在小$\eta$区域内产生有效结果。该算法在三维中的计算成本为$\mathcal{O}(\log^3(\eta^{-1}))$，当$\eta \to 0^+$时，而均匀间距积分的计算成本为$\mathcal{O}(\eta^{-3})$。相比之下，我们认为基于树结构的自适应积分方法对于典型的布里渊区积分只能以$\mathcal{O}(\log(\eta^{-1})/\eta^{2})$的速度缩放。除了具有有利的计算缩放之外，迭代自适应算法容易实现，特别用于在不可约布里渊区上的积分，其避免了需要树型网格的树形方案。我们通过在meV区域对SrVO$_3$的光谱函数进行计算来说明这些算法。