{
  "id": "2301.00824",
  "version": "v1",
  "published": "2023-01-02T19:00:02.000Z",
  "updated": "2023-01-02T19:00:02.000Z",
  "title": "A Generic Topological Criterion for Flat Bands in Two Dimensions",
  "authors": [
    "Alireza Parhizkar",
    "Victor Galitski"
  ],
  "comment": "4.5 pages, 3 figures",
  "categories": [
    "cond-mat.mes-hall",
    "cond-mat.str-el",
    "hep-th",
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-02T19:00:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "flat band",
      "generic topological criterion",
      "external magnetic field splits",
      "resulting two-dimensional euclidean theory contains",
      "magnetic field-dependent magic angles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}