General

In OpenMX Ver. 3.9, a post-processing code 'calB' is supported to calculate the Chern number and Berry curvature of bands using overlap matrix elements between Kohn-Sham orbitals at neighboring k-points by the Fukui-Hatsugai-Suzuki method [81,85]. The functionality is compatible with only the non-collinear calculations. To acknowledge in any publications by using the functionality, the citation of the reference [84] would be appreciated.

The Chern number is a topological invariant being an integer number, which characterizes the topology of bands for any materials. In systems having a finite Chern number , the anomalous Hall conductivity defined by

is induced. Using the Berry curvature , the Chern number is defined as
In the Fukui-Hatsugai-Suzuki method [81], the overlap matrix defined by

plays a central role to calculate the Berry connection and Berry curvature , which are defined by

As shown in Fig. 73, the Berry curvature can be calculated in each 'plaquette' (plaquette means meshed area in Brillouin zone) on a regular mesh introduced in the first Brillouin zone by the following formula:

By summing up all the contributions of the contour integrals for the Berry curvature, one can calculate the Chern number as