Many quantum materials of interest, ex., bilayer graphene, possess a number of closely spaced but not fully degenerate bands near the Fermi level, where the coupling to the far detuned remote bands can induce Berry curvatures of the non-Abelian character in this active multiple-band manifold for transport effects. Under finite electric fields, non-adiabatic interband transition processes are expected to play significant roles in the associated Hall conduction. Here through an exemplified study on the valley Hall conduction in AB-stacked bilayer graphene, we show that the contribution arising from non-adiabatic transitions around the bands near the Fermi energy to the Hall current is not only quantitatively about an order-of-magnitude larger than the contribution due to adiabatic inter-manifold transition with the non-Abelian Berry curvatures. Due to the trigonal warping, the former also displays an anisotropic response to the orientation of the applied electric field that is qualitatively distinct from that of the latter. We further show that these anisotropic responses also reveal the essential differences between the diagonal and off-diagonal elements of the non-Abelian Berry curvature matrix in terms of their contributions to the Hall currents. We provide a physically intuitive understanding of the origin of distinct anisotropic features from different Hall current contributions, in terms of band occupations and interband coherence. This then points to the generalization beyond the specific example of bilayer graphenes.
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