Gauge-covariant derivatives of the Berry curvature and orbital moment by Wannier interpolation

The momentum-space derivatives of the Berry curvature Ω and intrinsic orbital magnetic moment m of the Bloch states arise in multiple problems, such as the nonlinear anomalous Hall effect [1] and magneto-transport within the Boltzmann-equation formalism [2]. To study these properties from first principles, we developed a Wannier interpolation scheme for evaluating "generalized derivatives" of the non-Abelian Ω and m matrices for a group of bands of interest confined by some energy region.
Unlike the simple derivative, the generalized derivative does not involve couplings within the group, and preserves the gauge covariance of the Ω and m matrices. This formulation leads to robust “Fermi-sea” formulas for the Berry curvature dipole [1] and kinetic magnetoelectric effect tensor [2], which converge much faster with the density of the integration k-grid than the “Fermi-surface” formulas implemented earlier [3] in the Wannier90 code. The implementation is done in our newly-developed code WannierBerri [4]. We demonstrate the method with ab initio calculations on real materials, as well as tight-binding toy models.

[1] Sodemann et al, PRL 115 (2015): 216806.
[2] Zhong et al, PRL 116 (2016): 077201.
[3] Tsirkin et al, PRB 97 (2018): 035158.
[4] https://wannier-berri.org