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

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 $\epsilon$, 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.

References in corpus (8)

• Quantum ESPRESSO: a modular and open-source software project for quantum simulations of materials
• Spectral and Fermi surface properties from Wannier interpolation
• Improved tetrahedron method for the Brillouin-zone integration applicable to response functions
• Non-Drude universal scaling laws for the optical response of local Fermi liquids
• Nonuniform grids for Brillouin zone integration and interpolation
• A general algorithm for calculating irreducible Brillouin zones
• Efficient extraction of resonant states in systems with defects
• Converging tetrahedron method calculations for the nondissipative parts of spectral functions

Cited by in corpus (4)

• The Wannier Function Software Ecosystem for Materials Simulations
• Verification and Validation of zero-point electron-phonon renormalization of the bandgap, mass enhancement, and spectral functions
• Fermi-Liquid $T^2$ Resistivity: Dynamical Mean-Field Theory Meets Experiment
• Low-temperature transport in high-conductivity correlated metals: a density-functional plus dynamical mean-field study of cubic perovskites