The dc anomalous Hall conductivity (AHC) is simply given as the Brillouin zone integral of the Berry curvature of occupying energy bands,

\[ \sigma_{xy} = -\frac{e^2}{\hbar} \sum_{n}^{occ} \int_{\text{BZ}} \frac{d\mathbf{k}}{(2\pi)^3} f_n(\mathbf{k})\Omega_{n, z}(\mathbf{k}). \]


An example (refer to folder example/Fe) of calculating the AHC of the fcc-Fe is given here.

The Input file is:

    nspin               4
    package             ABACUS
    fermi_energy        18.18839115931923
    fermi_energy_unit   eV
    HR_route            data-HR-sparse_SPIN0.csr
    SR_route            data-SR-sparse_SPIN0.csr
    rR_route            data-rR-sparse.csr
    HR_unit             Ry
    rR_unit             Bohr

    lattice_constant        5.4235
    lattice_constant_unit   Bohr
     0.5  0.5  0.5
    -0.5  0.5  0.5
    -0.5 -0.5  0.5

    integrate_mode          Grid
    integrate_grid          100 100 100
    adaptive_grid           20 20 20
    adaptive_grid_threshold 100  

integrate_mode: Specifies the mode of integration, which can be grid integration and adaptive integration.

integrate_grid: Specifies a uniform grid for grid integration.

adaptive_grid: Specifies the grid for adaptive densification.

adaptive_grid_threshold: Specifies the cut-off value of adaptive densification, the unit is \(\AA^2\).

Once the calculation is complete, the resulting file ahc.dat will be generated in the Out/AHC directory.