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：我们提出了适用于非零但可能极小展宽因子η的布里渊区积分的高效方法，重点研究了在能通过万尼尔插值高效计算降阶哈密顿量的情形。我们描述了鲁棒且高阶精度的算法，能够自动收敛至用户指定的误差容限ε，并强调在η下实现高效的计算缩放。在分析适用于大展宽情形的标准等间距积分方法后，我们介绍了一种在小η区间有效的简单迭代自适应积分算法。在三维情况下，该算法计算复杂度随η→0⁺缩放为O(log³(η⁻¹))，而等间距积分的复杂度为O(η⁻³)。我们论证了相比之下，基于树的自适应积分方法对典型布里渊区积分仅能达到O(log(η⁻¹)/η²)的缩放。除其优越的缩放特性外，迭代自适应算法实现简单，特别适用于不可约布里渊区积分——在此场景中可避免基于树方案所需的多面体网格。我们通过计算SrVO₃在毫电子伏特展宽下的谱函数对算法进行验证。