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We investigate the possibility of exactly flat non-trivial Chern bands intight binding models with local (strictly short-ranged) hopping parameters. Wedemonstrate that while any two of three criteria can be simultaneously realized(exactly flat band, non-zero Chern number, local hopping), it is not possibleto simultaneously satisfy all three. Our theorem covers both the case of asingle flat band, for which we give a rather elementary proof, as well as thecase of multiple degenerate flat bands. In the latter case, our result isobtained as an application of $K$-theory. We also introduce a class of modelson the Lieb lattice with nearest and next-nearest neighbor hopping parameters,which have an isolated exactly flat band of zero Chern number but, in general,non-zero Berry curvature.

The impossibility of exactly flat non-trivial Chern bands in strictly local periodic tight binding models