In principle, the Berry-curvature can be related to the degeneracy of some underlying energy levels, using the adiabatic picture and expanding the Berry's expression in the language of instantaneous energy levels.
In particular, the Berry (or Zak) curvature is usually described as being responsible for the anomalous quantum Hall effect in graphene. Despite the graphene is usually presented as an example of gapless electrons (or massless Dirac fermions), actual samples present a gap.
Does the anomalous quantum Hall effect persists in the case of gaped graphene ? If yes, how is the anomalous quantum Hall effect related to the Berry curvature when the graphene sample presents a gap ?
More generally, is the Berry curvature necessarily related to a degeneracy of (instantaneous) energy levels (at the adiabatic level) ?