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：我们提出了针对非零但可能极小的展宽因子η的布里渊区积分高效方法，重点聚焦于可通过万尼尔插值有效评估降阶哈密顿量的情形。我们描述了鲁棒的高阶精确算法，可自动收敛至用户指定的误差容限ε，并强调算法在η维度上的高效计算扩展性。在分析适用于大展宽情况的传统等间距积分方法后，我们提出了一种在η极小时有效的简单迭代自适应积分算法。在三维情形下，该算法计算复杂度在η→0⁺时量级为O(log³(η⁻¹))，而等间距积分法复杂度为O(η⁻³)。我们论证了相比之下，基于树结构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)的量级规模。除具有有利的扩展性外，迭代自适应算法还易于实现，特别是在对称约化布里渊区积分中——该方法避免了树结构方案所需的四面体网格。我们通过计算展宽为meV量级的SrVO₃谱函数对算法进行了演示。