{
  "abstract": "Do phases of Gibbs states admit a classification beyond symmetry breaking? We study the three-dimensional toric code with transverse and longitudinal fields that explicitly break both microscopic higher-form symmetries. We introduce the connected correlation $f_W$ of two locally decoded winding Wilson loops. Nonzero asymptotic $f_W$ rules out shallow local preparation from product states. An asymptotic optimization over the observables and physical recovery channels defines an invariant $\\mathcal{F}$, which we prove constant within phases defined by two-way local channels with product ancillas and sublinear causal radius. The invariant equals one for the canonical zero-temperature state of the field-free toric code. Quantum Monte Carlo tests the conjectured thermodynamic values $f_W=1$ at weak fields and low temperature and $f_W=0$ elsewhere. The fixed-decoder correlation provides a lower bound on $\\mathcal{F}$, so the results suggest an extended region with $\\mathcal{F}=1$. Product states, including the infinite-temperature state and fully polarized infinite-field limits, have $\\mathcal{F}=0$ exactly and therefore lie in a different phase from states with $\\mathcal{F}=1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.00134",
  "author_total": 1,
  "authors": [
    {
      "name": "Haruki Watanabe"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/04xemfjj45ypk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.00134",
  "fulltext_url": null,
  "id": "04xemfjj45ypk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.00134",
  "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.other"
    ],
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "cond-mat.str-el"
    ],
    [
      "openalex",
      "3107|Quantum many-body systems"
    ]
  ],
  "openalex": "W7167073268",
  "pdf_url": "https://arxiv.org/pdf/2607.00134",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/04xemfjj45ypk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:31Z",
      "native_id": "2607.00134",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.00134"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.other",
      "subfield": "condensed-matter"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Gibbs-state phases beyond microscopic symmetry breaking: A phase invariant for the 3D toric code",
  "title_text": "Gibbs-state phases beyond microscopic symmetry breaking: A phase invariant for the 3D toric code",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:31Z",
  "url": "https://research.draconitic.com/paper/04xemfjj45ypk",
  "venue": "arXiv",
  "versions": []
}