{
  "abstract": "A recent remarkable paper by OpenAI showed that the spin-one Heisenberg chain with periodic boundary conditions has a spectral gap in the thermodynamic limit. The proof elegantly deduces this gap from the Hamiltonian's spectrum on a finite system. This basic idea is an inequality involving the purity of the original Gibbs state and a dual Gibbs state that is defined via space-time rotation. Here we consider a few generalizations of this purity bootstrap argument. We first extend it to arbitrary spatial dimensions. Our main result uses the space-time duality between symmetry generators and symmetry twists to obtain upper bounds on the splitting of nearly degenerate low-energy states, together with a nonzero gap above them, from finite-system information. As applications, we show that finite-system calculations for the 2+1-D toric code in a field and the 2+1-D transverse-field Ising model establish parameter regimes with four and two low-lying states whose splittings are bounded from above by $e^{-cL}$ and $e^{-cL^2}$ respectively for arbitrarily large system size $L$, while the gap above these states remains nonzero in the thermodynamic limit. Next, we extend it to fermionic systems and use a finite calculation to establish a symmetric mass generation phase in a $2+1$-D fermionic model. Finally, we apply the bootstrap to tensor-network states and establish exponential decay of connected correlations of all local operators in the symmetric square-lattice spin-one state introduced by Jian and Zaletel.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07400",
  "author_total": 1,
  "authors": [
    {
      "name": "Tarun Grover"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7cajdrvp1c1t1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07400",
  "fulltext_url": null,
  "id": "7cajdrvp1c1t1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07400",
  "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",
      "cond-mat.str-el"
    ],
    [
      "arxiv",
      "hep-th"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07400",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7cajdrvp1c1t1.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:30Z",
      "native_id": "2611.07400",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07400"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.str-el",
      "subfield": "condensed-matter"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Extensions of the Haldane gap proof: higher dimensions, near degeneracy, fermions and tensor-network states",
  "title_text": "Extensions of the Haldane gap proof: higher dimensions, near degeneracy, fermions and tensor-network states",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/7cajdrvp1c1t1",
  "venue": "arXiv",
  "versions": []
}