  The Berry curvature in Chern insulators appears to be a non-gauge-invariant
quantity and does not immediately allow local length characterization. However,
in two examples of 2- and 3-band models that we discuss, we find high-symmetry
points in the Brillouin zone that have Berry curvature invariant under diagonal
gauge transformations, and may serve as expansion points of geometrical
description. On the basis of the geometrical description, in the case of Dirac
based 2-band Chern insulators like Haldane model we conclude that the
characteristic length based on the value of the Berry curvature at the
expansion point plays the role of the magnetic length in the expression for the
Hall viscosity. In the case of 2-band models the characteristic "cyclotron"
spin is equal to 1/2, while in the 3-band kagome case this spin is likely
non-quantized and non-universal.
