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

Correlations and entanglement in flat band models with variable Chern numbers