  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.
