Orbital magnetization and Chern number in a supercell framework: Single k-point formula

TL;DR

This paper derives a simplified single k-point formula for calculating orbital magnetization and Chern number in large supercell systems, facilitating efficient computations in noncrystalline and supercell frameworks.

Contribution

It introduces a single k-point limit formula for orbital magnetization and Chern number, validated on crystalline systems, aiding large supercell and noncrystalline system simulations.

Findings

Single k-point formula accurately computes orbital magnetization.

Chern number can be evaluated with a single Hamiltonian diagonalization.

Validated approach for large supercells and noncrystalline systems.

Abstract

The key formula for computing the orbital magnetization of a crystalline system has been recently found [D. Ceresoli, T. Thonhauser, D. Vanderbilt, R. Resta, Phys. Rev. B {\bf 74}, 024408 (2006)]: it is given in terms of a Brillouin-zone integral, which is discretized on a reciprocal-space mesh for numerical implementation. We find here the single ${\bf k}$-point limit, useful for large enough supercells, and particularly in the framework of Car-Parrinello simulations for noncrystalline systems. We validate our formula on the test case of a crystalline system, where the supercell is chosen as a large multiple of the elementary cell. We also show that--somewhat counterintuitively--even the Chern number (in 2d) can be evaluated using a single Hamiltonian diagonalization.