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A Generic Topological Criterion for Flat Bands in Two Dimensions

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What is the main accomplishment of this work?

AI: The main accomplishment of this work is the derivation of a topological criterion for finding flat bands in a variety of material systems, including but not limited to moiré bilayers. This is achieved through the study of the continuum limit of models of mutually distorted layers of graphene, which are shown to be equivalent to a (2+1)-dimensional field theory of Dirac fermions coupled to two classical gauge fields. The existence of a flat band implies an effective dimensional reduction in the field theory, and the associated Atiyah-Singer index theorem provides a self-consistency condition for the existence of flat bands. The criterion is applied to the study of twisted bilayer graphene in the chiral limit and the effect of an external magnetic field on the magic angles associated with flat moiré-Landau bands.
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Does the argument presented in this paper generalize to higher dimensional models?

Authors

Alireza Parhizkar, Victor Galitski

Abstract

Mutually distorted layers of graphene give rise to a moiré 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 moiré-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 moiré 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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