A Generic Topological Criterion for Flat Bands in Two Dimensions

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A Generic Topological Criterion for Flat Bands in Two Dimensions


Alireza Parhizkar, Victor Galitski


Mutually distorted layers of graphene give rise to a moir\'e pattern and a variety of non-trivial phenomena. We show that the continuum limit of this class of models is equivalent to a (2+1)-dimensional field theory of Dirac fermions coupled to two classical gauge fields. We further show that the existence of a flat band implies an effective dimensional reduction in the field theory, where the time dimension is ``removed.'' The resulting two-dimensional Euclidean theory contains the chiral anomaly. The associated Atiyah-Singer index theorem provides a self-consistency condition for the existence of flat bands. In particular, it reproduces a series of quantized magic angles known to exist in twisted bilayer graphene in the chiral limit where there is a particle-hole symmetry. We also use this criterion to prove that an external magnetic field splits this series into pairs of magnetic field-dependent magic angles associated with flat moir\'e-Landau bands. The topological criterion we derive provides a generic practical method for finding flat bands in a variety of material systems including but not limited to moir\'e bilayers.

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It would be interesting to see if the argument based on the chiral anomaly in 2+0 dimensions generalizes to higher dimensions. It seems that flat bands are possible in higher dimensions as well.

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