A robust algorithm for $k$-point grid generation and symmetry reduction

Understand

We develop an algorithm for i) computing generalized regular $k$-point grids, ii) reducing the grids to their symmetrically distinct points, and iii) mapping the reduced grid points into the Brillouin zone.

• The algorithm exploits the connection between integer matrices and finite groups to achieve a computational complexity that is linear with the number of $k$-points.
• The favorable scaling means that, at a given $k$-point density, all possible commensurate grids can be generated (as suggested by Moreno and Soler) and quickly reduced to identify the grid with the fewest symmetrically unique $k$-points.
• These optimal grids provide significant speed-up compared to Monkhorst-Pack $k$-point grids; they have better symmetry reduction resulting in fewer irreducible $k$-points at a given grid density.