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Many quantum materials of interest, ex., bilayer graphene, possess a numberof closely spaced but not fully degenerate bands near the Fermi level, wherethe coupling to the far detuned remote bands can induce Berry curvatures of thenon-Abelian character in this active multiple-band manifold for transporteffects. Under finite electric fields, non-adiabatic interband transitionprocesses are expected to play significant roles in the associated Hallconduction. Here through an exemplified study on the valley Hall conduction inAB-stacked bilayer graphene, we show that the contribution arising fromnon-adiabatic transitions around the bands near the Fermi energy to the Hallcurrent is not only quantitatively about an order-of-magnitude larger than thecontribution due to adiabatic inter-manifold transition with the non-AbelianBerry curvatures. Due to the trigonal warping, the former also displays ananisotropic response to the orientation of the applied electric field that isqualitatively distinct from that of the latter. We further show that theseanisotropic responses also reveal the essential differences between thediagonal and off-diagonal elements of the non-Abelian Berry curvature matrix interms of their contributions to the Hall currents. We provide a physicallyintuitive understanding of the origin of distinct anisotropic features fromdifferent Hall current contributions, in terms of band occupations andinterband coherence. This then points to the generalization beyond the specificexample of bilayer graphenes.

Revealing the non-adiabatic and non-Abelian multiple-band effects via anisotropic valley Hall conduction in bilayer graphene