Automatic, high-order, and adaptive algorithms for Brillouin zone integration

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. describe robust, high-order accurate algorithms automating convergence to user-specified error tolerance $\varepsilon$, emphasizing an computational scaling respect $\eta$. After analyzing the standard equispaced method, applicable case of large broadening, we simple iterated adaptive algorithm effective $\eta$ regime. Its cost scales as $\mathcal{O}(\log^3(\eta^{-1}))$ $\eta \to 0^+$ three dimensions, opposed $\mathcal{O}(\eta^{-3})$ integration. argue that, by contrast, tree-based scale only $\mathcal{O}(\log(\eta^{-1})/\eta^{2})$ typical integrals. In addition its favorable scaling, is straightforward implement, particularly irreducible zone, it avoids tetrahedral meshes required schemes. illustrate calculating spectral function SrVO$_3$ meV scale.