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

TL;DR

This paper introduces efficient, high-order adaptive algorithms for Brillouin zone integration that are particularly effective for small broadening factors, improving computational scaling and simplicity over traditional methods.

Contribution

The authors develop and analyze novel adaptive integration algorithms with superior scaling properties for small broadening, applicable to Brillouin zone calculations using Wannier interpolation.

Findings

Adaptive algorithms scale as O(log^3(η^{-1})) for small η

Equispaced integration scales as O(η^{-3})

Tree-based adaptive methods scale as O(log(η^{-1})/η^2)

Abstract

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.