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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) ?

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The anomalous Hall effect should be present in the Gapped case as well (Although I don't know if it has been experimentally observed in a gapped system).

The reason is that for massive Dirac Fermions, the effective Bloch dynamics is also governed by a Berry gauge field, this time a non-Abelian gauge field. Please see Chen, Pang, Pu and Wang equations (34), (compared to e.g., Horv´athy equations (1) and (2) for the massless case).

The equations in the massive case include a third equation for spin precession absent from the massless case. Since in the massless case the helicity is constant.

The appearence of the non-Abelian (Berry) synthetic gauge field is not a new result, it was observed in other contexts in the past. I don't know who discoved this phenomenon originally, but see for example the following article by Keppler.

An explicit expression of the non-Abelian Berry non-Abelian gauge potential is given in Chen, Pang, Pu and Wang equations (36). It is a $U(2)$ gauge field. It is related to the degeneracy in the spin components of a Dirac field. Please observe that except for the third components all other components can be taken to vanish in the massless limit. The third component is the famous Stephanov and Yin Berry mangetic monopole term present in the effective massless equations of motin.

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