{
  "abstract": "We review mathematical results about the spectrum of spin-$s$ chains with a nearest neighbor interaction given by the projection onto the singlet state of two spin-$s$ degrees of freedom. Up to normalization, these interaction terms generate a Temperley-Lieb algebra at loop parameter $d=2s+1$. We also present several new results. First, we prove a ferromagnetic ordering of energy levels across symmetry sectors of the model, thus generalizing a previous result of Nachtergaele, Spitzer and Starr (2004) for the spin-$\\tfrac12$ $XXZ$ chain. This enables us to determine a uniform lower bound on the finite volume spectral gap for these spin chains. By combining a variational estimate with Knabe's finite size criterion, we show that the exact spectral gap of the GNS Hamiltonian of the fully polarized state is $1-2/d$. Furthermore, this value serves as a lower bound for the gap of the maximally mixed ground state. Finally, we show that the ground state projectors on open chains satisfy the Jones-Wenzl recursion, thereby recovering a known result that the degeneracies are the quantum integers $[L+1]_q$ for $q+q^{-1}=d$. We discuss how the high degeneracy of the ground states allows the model to break a continuous symmetry while remaining gapped, and renders that gap unstable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38074",
  "author_total": 2,
  "authors": [
    {
      "name": "Robert Ferydouni"
    },
    {
      "name": "Bruno Nachtergaele"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bqxqtxdy7ecy3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38074",
  "fulltext_url": null,
  "id": "bqxqtxdy7ecy3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38074",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.stat-mech"
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      "arxiv",
      "cond-mat.str-el"
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      "arxiv",
      "math-ph"
    ],
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      "math.MP"
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      "openalex",
      "3107|Quantum many-body systems"
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  "openalex": "W7214926015",
  "pdf_url": "https://arxiv.org/pdf/2609.38074",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bqxqtxdy7ecy3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:31Z",
      "native_id": "2609.38074",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38074"
    }
  ],
  "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:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "On the Spectrum of Some Temperley-Lieb Spin Chains",
  "title_text": "On the Spectrum of Some Temperley-Lieb Spin Chains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:36Z",
  "url": "https://research.draconitic.com/paper/bqxqtxdy7ecy3",
  "venue": "arXiv",
  "versions": []
}