{
  "abstract": "We study invertible phases protected by non-onsite symmetries carrying nontrivial lattice anomaly indices. While lattice anomalies are usually viewed as obstructions to symmetric short-range-entangled (SRE) states, we show that anomalous lattice symmetries can nevertheless admit invertible phases that are not SRE, and can moreover shift the set of symmetry-compatible invertible phases. In particular, we consider a fermionic $\\mathbb Z_4^F$ symmetry in (2+1) dimensions. For an onsite $\\mathbb Z_4^F$ symmetry, symmetric invertible phases have integer chiral central charge $c_-\\in\\mathbb Z$, whereas a non-onsite symmetry with nontrivial lattice anomaly index obstructs all such phases despite having trivial continuum 't Hooft anomaly. We resolve this by constructing an exact exponentially quasi-local $\\mathbb Z_4^F$ symmetry of a $p+ip$ superconductor with $c_-=1/2$. We compute its lattice anomaly, and find that the symmetry indeed forbids $c_-\\in \\mathbb{Z}$ invertible states. The allowed invertible phases are consequently shifted to $c_-\\in\\mathbb Z+1/2$. By gauging the fermion parity of the $p+ip$ superconductor, one obtains an Ising topological order enriched by a $\\mathbb Z_4$ symmetry. We describe the corresponding symmetry structure in terms of a fusion 2-category. We further show that the $p+ip$ state admits an enlarged non-onsite $U(1)^f$ symmetry, providing an analogue of fractional quantum Hall response in an invertible phase. This non-onsite $U(1)^f$ symmetry again forbids Chern insulators carrying $c_-\\in\\mathbb{Z}$, while being compatible with invertible states with $c_-\\in\\mathbb{Z}+1/2$. This result motivates a systematic study of new classes of invertible phases and spin liquids enriched by non-onsite symmetries.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12042",
  "author_total": 3,
  "authors": [
    {
      "name": "Ryohei Kobayashi"
    },
    {
      "name": "Kansei Inamura"
    },
    {
      "name": "Ken Shiozaki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2hsqsatrqcq3g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12042",
  "fulltext_url": null,
  "id": "2hsqsatrqcq3g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12042",
  "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.str-el"
    ],
    [
      "arxiv",
      "hep-th"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "3107|Topological Materials and Phenomena"
    ]
  ],
  "openalex": "W7213073454",
  "pdf_url": "https://arxiv.org/pdf/2609.12042",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2hsqsatrqcq3g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:07Z",
      "native_id": "2609.12042",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12042"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.str-el",
      "subfield": "condensed-matter"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Remarks on invertible phases with non-onsite symmetry",
  "title_text": "Remarks on invertible phases with non-onsite symmetry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:07Z",
  "url": "https://research.draconitic.com/paper/2hsqsatrqcq3g",
  "venue": "arXiv",
  "versions": []
}