Covariant derivatives of Berry-type quantities: Application to nonlinear transport

The derivatives of the Berry curvature $\Omega$ and intrinsic orbital magnetic moment m in momentum space are relevant to various problems, including the nonlinear anomalous Hall effect and magneto-transport within the Boltzmann-equation formalism. To investigate these properties using first-principles methods, we have developed a Wannier interpolation scheme that evaluates the ''covariant derivatives'' of the non-Abelian $\Omega$ and m matrices for a group of bands within a specific energy range of interest. Unlike the simple derivative, the covariant derivative does not involve couplings within the groups and preserves the gauge covariance of the $\Omega$ and m matrices. In the simulation of nonlinear anomalous Hall conductivity, the resulting ''Fermi-sea'' formula for the Berry curvature dipole are more robust and converges faster with the density of the integration k-grid than the ''Fermi-surface'' formula implemented earlier. The developed methodology is made available via the open-source code WannierBerri and we demonstrate the efficiency of this method through first-principles calculations on trigonal Tellurium.