Numerical quadrature in the Brillouin zone for periodic Schrodinger operators

TL;DR

This paper analyzes and compares various numerical quadrature methods for integrating over the Brillouin zone in periodic Schrödinger operators, providing error bounds and practical guidance for metals with discontinuous integrands.

Contribution

It offers a rigorous error analysis of common quadrature and smearing methods for Brillouin zone integration, clarifying assumptions and guiding parameter choices.

Findings

Error bounds for quadrature rules are established.

Discontinuities in metals affect numerical integration accuracy.

Numerical experiments validate theoretical error estimates.

Abstract

As a consequence of Bloch's theorem, the numerical computation of the fermionic ground state density matrices and energies of periodic Schrodinger operators involves integrals over the Brillouin zone. These integrals are difficult to compute numerically in metals due to discontinuities in the integrand. We perform an error analysis of several widely-used quadrature rules and smearing methods for Brillouin zone integration. We precisely identify the assumptions implicit in these methods and rigorously prove error bounds. Numerical results for two-dimensional periodic systems are also provided. Our results shed light on the properties of these numerical schemes, and provide guidance as to the appropriate choice of numerical parameters.