{
  "abstract": "Momentum-dependent symmetry actions can turn the even-integer constraint of a $2\\mathbb{Z}$ classification of topological insulators and superconductors into an odd-integer constraint without changing the symmetry algebra. For internal and order-two crystalline symmetries in up to four spatial dimensions, we derive parity relations between the Chern or winding number of a gapped Hamiltonian and topological invariants of its momentum-dependent antiunitary symmetry matrices. Whenever the symmetry action enforces odd parity, any symmetry-preserving gapped phase must be topologically nontrivial. Using cellwise actions, which are represented by momentum-independent matrices in a fixed unit-cell basis, as a reference, we determine how locality constrains the realization of odd topological numbers. Explicit model constructions and no-go theorems distinguish three cases: odd values can be realized by finite-range symmetry actions, can be realized by exponentially decaying quasilocal actions but not by finite-range actions, or are impossible in finite-dimensional Bloch systems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37106",
  "author_total": 1,
  "authors": [
    {
      "name": "Ken Shiozaki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/96gn2tjffkaf2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37106",
  "fulltext_url": null,
  "id": "96gn2tjffkaf2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37106",
  "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",
      "cond-mat.mes-hall"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "3107|Topological Materials and Phenomena"
    ]
  ],
  "openalex": "W7214891829",
  "pdf_url": "https://arxiv.org/pdf/2609.37106",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/96gn2tjffkaf2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:31Z",
      "native_id": "2609.37106",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37106"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.mes-hall",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Symmetry-enforced topological parity: revisiting $2\\mathbb{Z}$ classifications beyond cellwise symmetry actions",
  "title_text": "Symmetry-enforced topological parity: revisiting 2ℤ classifications beyond cellwise symmetry actions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/96gn2tjffkaf2",
  "venue": "arXiv",
  "versions": []
}