{
  "abstract": "Polyakov predicted that the classical Heisenberg model on $\\mathbb{Z}^2$ has exponentially decaying correlations at every temperature, and a proof of this has recently been posted. We show that the massive phase survives small anisotropy. For every inverse temperature $β$ there is $δ(β)>0$ such that the $S^2$-valued model with couplings $\\sum_iJ^iσ_x^iσ_y^i$ has exponentially decaying spin two-point functions, uniformly in finite subgraphs with free boundary conditions and in edge strengths up to $β$, whenever $|J^i-1|\\leδ(β)$ for $i=1,2,3$. For the easy-plane model with couplings $σ^xσ^x+σ^yσ^y+λσ^zσ^z$ this shows that the onset of the Berezinskii-Kosterlitz-Thouless phase, proved by Garban and de Montgolfier for every $λ\\in(-1,1)$, moves to zero temperature as $λ\\uparrow1$: in free-boundary limits, correlations decay exponentially for every $β\\le c |\\log(1-λ)|^{1/2}$. The physical prediction is of order $|\\log(1-λ)|$. The same holds for aligned single-ion anisotropy, which includes a confining potential $\\varepsilon\\sum(σ^z)^2$, and for nearly isotropic $O(n)$ models with $n\\ge3$. The proof keeps the sign-cluster argument of the isotropic proof, using Ginibre's correlation inequality for anisotropic planar energies, and uses the rotation symmetry only on one finite annulus, where the anisotropy is a bounded perturbation of the Hamiltonian.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01532",
  "author_total": 1,
  "authors": [
    {
      "name": "Yan Ru Pei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cd69e88efyqgr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01532",
  "fulltext_url": null,
  "id": "cd69e88efyqgr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01532",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01532",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cd69e88efyqgr.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:59Z",
      "native_id": "2611.01532",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01532"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "The massive phase of the two-dimensional Heisenberg model is stable under small anisotropy",
  "title_text": "The massive phase of the two-dimensional Heisenberg model is stable under small anisotropy",
  "type": "preprint",
  "updated_at": "2026-11-04T15:21:05Z",
  "url": "https://research.draconitic.com/paper/cd69e88efyqgr",
  "venue": "arXiv",
  "versions": []
}