{
  "abstract": "Onsager solved the two-dimensional Ising model in zero field, but no solution is known in a magnetic field. We prove that in any dimension $d\\ge2$, at every fixed dimensionless field outside a countable set, the free energy obeys no algebraic differential equation in the coupling, even locally. The obstruction arises from droplet geometry: the maximal droplet volume grows as the $d/(d-1)$ power of its boundary size, while an integer band in the pressure coefficients protects the extremal contributions against cancellation. Our result excludes Onsager-type solutions and closed forms in elliptic, hypergeometric, or Painlevé functions of the coupling at these fields.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20569",
  "author_total": 2,
  "authors": [
    {
      "name": "Ran Luo"
    },
    {
      "name": "Jiahua Tian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d24gbh8rnb0vx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20569",
  "fulltext_url": null,
  "id": "d24gbh8rnb0vx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20569",
  "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.stat-mech"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
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      "arxiv",
      "nlin.SI"
    ],
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      "openalex",
      "3104|Theoretical and Computational Physics"
    ]
  ],
  "openalex": "W7222166161",
  "pdf_url": "https://arxiv.org/pdf/2610.20569",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d24gbh8rnb0vx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:31Z",
      "native_id": "2610.20569",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20569"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Droplets obstruct Onsager-type solutions of the Ising model in a magnetic field",
  "title_text": "Droplets obstruct Onsager-type solutions of the Ising model in a magnetic field",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/d24gbh8rnb0vx",
  "venue": "arXiv",
  "versions": []
}