Correlations and entanglement in flat band models with variable Chern numbers

We discuss a number of illuminating results for tight binding models supporting a band with variable Chern number, and illustrate them explicitly for a simple class of two-banded models. First, for models with a fixed number of bands, we show that the minimal hopping range needed to achieve a given Chern number $C$ is increasing with $C$, and that the band flattening requires an exponential tail of long-range processes. We further verify that the entanglement spectrum corresponding to a real-space partitioning contains $C$ chiral modes and thereby complies with the archetypal correspondence between the bulk entanglement and the edge energetics. Finally, we address the issue of interactions and study the problem of two interacting particles projected to the flattened band as a function of the Chern number. Our results provide valuable insights for the full interacting problem of a partially filled Chern band at variable filling fractions and Chern numbers.