controls the closing of the gap. (The general procedure is the same as was used in Sec. \[near\_zero\] of this text.) If the first order coefficients do not vanish, the touching point is characterized by a Dirac fermion. Its effective $2\times2$ Hamiltonian is brought into a standard form (essentially the valley-isotropic representation) by means of appropriate $2\times2$ unitary transforms. It turns out that the change in Chern numbers is completely determined by the sign of the mass term in this representation. If the bands touch at several points in $\bm{k}$ space simultaneously, then each touching point gives an independent contribution.

For the graphene system I am considering, this leads to a result $${\rm Ch}_\pm^>-{\rm Ch}_\pm^<=\pm\{\rm{sign}[\mu^>]-\rm{sign}[\mu^<]\}.\label{eq_delta_chern}$$ Here $+$ ($-$) refers to the conduction (valence) band and $\mu^<$ ($\mu^>$) is the mass induced by the scalar potential before (after) the bands touched. There are two contributions to the result, one from each touching point. Since the same mass term appears at both touching points (in the valley-isotropic representation), the contributions from the two touching points are the same. If the sign of the mass were opposite in the two valleys, as is the case for a substrate-induced mass, the two contributions would 