{
  "abstract": "Replica symmetry breaking (RSB) is the organizing principle of mean-field spin-glass theory, yet whether a continuous Parisi hierarchy can arise in a genuinely short-range system in finite spatial dimension remains unresolved. Here we introduce a disordered model of $N$-component vector spins on a $D$-dimensional hypercubic lattice, with random matrix couplings on nearest-neighbor links and random tensor couplings on elementary plaquettes. In the large-$N$ limit, the model supports thermodynamically stable one-step, full, and one-full RSB phases, alongside replica-symmetric paramagnetic and ferromagnetic phases. We find that spatial locality constrains these ordered phases through two distinct infrared mechanisms. ($i$) Quenched random anisotropy destroys uniform ferromagnetism for $D\\leq4$ through an Imry-Ma mechanism. ($ii$) A continuous Parisi hierarchy gives rise to replica Goldstone modes whose gaplessness is protected by a Ward identity. At leading order in $1/N$, these modes behave as a band of free spin waves, producing infrared-divergent fluctuations in $D\\leq2$ and a Mermin-Wagner instability of the continuously broken phase. Remarkably, the Parisi hierarchy also acquires a direct real-space interpretation. For two pure states with mutual overlap $q$, a characteristic length $ξ(q)$ describes how far the differences between their frozen magnetization patterns remain correlated. Throughout a continuous sector of the Parisi hierarchy, $ξ(q)$ is independent of temperature and decreases monotonically with $q$. The ultrametric organization of pure states is therefore accompanied by an ultrametric hierarchy of real-space correlation lengths: states that separate near the root of the Parisi tree differ through long-wavelength frozen patterns, whereas states that separate farther from the root differ only on progressively shorter scales.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12441",
  "author_total": 3,
  "authors": [
    {
      "name": "Akash Vijay"
    },
    {
      "name": "Jaewon Kim"
    },
    {
      "name": "Jong Yeon Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8s10t69sq0kgd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12441",
  "fulltext_url": null,
  "id": "8s10t69sq0kgd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12441",
  "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.dis-nn"
    ],
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "openalex",
      "3104|Theoretical and Computational Physics"
    ]
  ],
  "openalex": "W7221191849",
  "pdf_url": "https://arxiv.org/pdf/2610.12441",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8s10t69sq0kgd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:54Z",
      "native_id": "2610.12441",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12441"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.dis-nn",
      "subfield": "condensed-matter"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Short-Range Vector Spin Glasses: Parisi Hierarchies and Real-Space Ultrametricity",
  "title_text": "Short-Range Vector Spin Glasses: Parisi Hierarchies and Real-Space Ultrametricity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:54Z",
  "url": "https://research.draconitic.com/paper/8s10t69sq0kgd",
  "venue": "arXiv",
  "versions": []
}