Constant Berry curvature, GMP algebra and Chern insulators

Band geometry, especially the Berry curvature, has played an important role in topological systems. A non-vanishing Berry curvature leads to coordinates that do not commute and resembles the effects of a magnetic field. Motivated by the analog between Chern insulators and Landau levels, constant Berry curvature and the GMP algebra have been suggested as good conditions to realise fractional Chern insulators. We show that while in two-band models, there is a topological obstruction to make the Berry curvature exactly flat, it is possible to do so with three or more degrees of freedom per unit cell. However, through numerical calculations, we find that constant Berry curvature does not always improve realising bosonic fractional Chern insulator states. Moreover, we show that the GMP algebra, the Landau-level commutation relation for density operators, cannot be realised in lattice models with finite bands.