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 $C$, the anomalous Hall conductivity defined by

$$\sigma_{xy} = -\frac{e^2}{h}C\ (C=1,2,3,\cdots)$$

is induced. Using the Berry curvature ${\bf F}_{n}=\nabla\times{\bf A}_{n},\ {\bf A}_{n}=-i\langle u_{n{\bf k}}\vert\frac{\partial}{\partial{\bf k}}\vert u_{n{\bf k}}\rangle$, the Chern number is defined as

$${\displaystyle C = \frac{1}{2\pi}\sum_{n}^{\rm occ.}\int F_{nz}dk^2}$$

In the Fukui-Hatsugai-Suzuki method [81], the overlap matrix $U$ defined by

$$U_{\Delta {\bf k}}({\bf k}) =\det \langle u_{n}({\bf k})\vert u_{m}({\bf k}+\Delta {\bf k})\rangle$$

plays a central role to calculate the Berry connection ${\bf A}({\bf k})$ and Berry curvature $F({\bf k})$, which are defined by

$${\bf A}({\bf k}) = {\rm Im}\log U_{\Delta {\bf k}}({\bf k}),$$

$$F({\bf k}) = {\rm Im}\log U_{\Delta k_{1}}({\bf k})U_{\Delta k_{2... ...k_{1})U^{-1}_{\Delta k_{1}}({\bf k}+\Delta k_{2})U^{-1}_{\Delta k_{2}}({\bf k})$$

Figure 73: Computational method of the Berry curvature, where a contour integral is performed in each plaquette.

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:

$$F({\bf k})={\rm Im}\log U_{12}U_{23}U_{34}U_{41}$$

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

$$C = \frac{1}{2\pi}{\displaystyle \sum_{\bf k}^{\rm BZ} F({\bf k})}$$