{
  "abstract": "We consider the Ising spin-glass model on Migdal--Kadanoff (MK) hierarchical lattices, characterized by the fractal dimension $d=1+\\log s/\\log b$, where $s$ and $b$ denote the number of parallel branches and the number of bonds in series within each branch, respectively. For independent Gaussian couplings, we rigorously establish a necessary and sufficient condition for the existence of finite-temperature spin-glass order for arbitrary integers $b,s\\ge2$. We prove that a finite-temperature spin-glass phase exists whenever \\[ \\frac{b}{\\sqrt{s}} \\left( \\frac{b\\sqrt{s(s-1)}}{bs-1} \\right)^{s-1}<1. \\] In particular, this condition holds for all three-dimensional MK hierarchical lattices ($d=3$, or equivalently $s=b^2$), regardless of the choice of $b$. Conversely, we prove the absence of finite-temperature spin-glass order for fractal dimensions $d<1+\\log17/\\log6\\approx2.58$, as well as in the large-$b$ limit for any fixed $d<3$. In the intermediate regime $1+\\log17/\\log6\\le d<3$, we provide a finite-step procedure to determine the presence or absence of spin-glass order, which depends on $b$ and $s$. Our results provide a complete characterization of finite-temperature spin-glass order on MK hierarchical lattices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32291",
  "author_total": 2,
  "authors": [
    {
      "name": "Manaka Okuyama"
    },
    {
      "name": "Masayuki Ohzeki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9fpy820heeahh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32291",
  "fulltext_url": null,
  "id": "9fpy820heeahh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32291",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.dis-nn"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "3104|Theoretical and Computational Physics"
    ]
  ],
  "openalex": "W7223650997",
  "pdf_url": "https://arxiv.org/pdf/2610.32291",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9fpy820heeahh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:15Z",
      "native_id": "2610.32291",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32291"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.dis-nn",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Existence and Absence of Spin-Glass Phases on Migdal--Kadanoff Hierarchical Lattices: A Complete Classification",
  "title_text": "Existence and Absence of Spin-Glass Phases on Migdal--Kadanoff Hierarchical Lattices: A Complete Classification",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/9fpy820heeahh",
  "venue": "arXiv",
  "versions": []
}