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

Abstract

We investigate the possibility of exactly flat non-trivial Chern bands in tight binding models with local (strictly short-ranged) hopping parameters. We demonstrate that while any two of three criteria can be simultaneously realized (exactly flat band, non-zero Chern number, local hopping), it is not possible to simultaneously satisfy all three. Our theorem covers both the case of a single flat band, for which we give a rather elementary proof, as well as the case of multiple degenerate flat bands. In the latter case, our result is obtained as an application of $K$-theory. We also introduce a class of models on 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.

Published Version. Some references and additional comments added

References in corpus (11)

• High temperature fractional quantum Hall states
• Fractional quantum Hall states at zero magnetic field
• Nearly-flat bands with nontrivial topology
• Fractional quantum Hall effect in the absence of Landau levels
• Topological Flat Band Models and Fractional Chern Insulators
• Fractional Chern Insulators in Topological Flat bands with Higher Chern Number
• Topological flat band models with arbitrary Chern numbers
• Fractional Quantum Hall Effect in Topological Flat Bands with Chern Number Two
• Flat bands with higher Chern number in pyrochlore slabs
• Momentum-space instantons and maximally localized flat-band topological Hamiltonians
• Landau Levels in Lattices with Long Range Hopping

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