{
  "abstract": "Massive discrete Dirac-type operators on periodic planar graphs are investigated. Using Floquet--Bloch decomposition, we first show that neglecting the faces leads to spurious flat bands that are unrelated to the graph's geometry. Then, for the correct Dirac--Hodge operator $D_m$, we establish a spectral reduction expressing $σ(D_m)$ through the graph Laplacian on vertices $-Δ_V$ and the Laplacian on faces $-Δ_F$. As a consequence, $D_m$ carries a flat band precisely when one of these scalar Laplacians does. We finally apply this reduction to the study of flat bands of the operator $D_m$ on Archimedean lattices and contrast the outcome with that on the Lieb lattice.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10964",
  "author_total": 3,
  "authors": [
    {
      "name": "Koichi Kato"
    },
    {
      "name": "Daniel Parra"
    },
    {
      "name": "Serge Richard"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a6ak04bhmysnv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10964",
  "fulltext_url": null,
  "id": "a6ak04bhmysnv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10964",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.SP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.10964",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a6ak04bhmysnv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:54Z",
      "native_id": "2610.10964",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10964"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Dirac operators on periodic planar graphs: the right definition advocated by flat bands",
  "title_text": "Dirac operators on periodic planar graphs: the right definition advocated by flat bands",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/a6ak04bhmysnv",
  "venue": "arXiv",
  "versions": []
}