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    "title": "Higher-dimensional spin constraints rule out a unique ground state with a size-independent gap",
    "summary": "Hastings extended the Lieb–Schultz–Mattis argument to higher-dimensional local quantum spin systems by combining separated twists with locality estimates. For the specified lattice families with a half-odd-integer spin per transverse column, a unique ground state has an excitation gap bounded by a constant times log(L)/L. Nachtergaele and Sims later gave a detailed rigorous proof with explicit geometric, interaction and symmetry hypotheses. The result restricts the possible low-energy structure of these many-body systems.",
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    "area": "Condensed matter theory / quantum spin constraints",
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        "description": "The bound concerns particular finite-size sequences with even periodic length and the required odd transverse spin parity; it is not a theorem for every width, boundary condition or long-range interaction. The later proof assumes its F1/F2 and LSM1–6 conditions, including U(1) symmetry, zero total spin component of the ground state and a real-basis condition used to control twist derivatives. A vanishing first gap can reflect near-degenerate ground states and does not by itself prove a continuous gapless spectrum, magnetic order or a particular spin-liquid phase.",
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      "sources": [
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          "citation": "Lieb-Schultz-Mattis in higher dimensions (arXiv cond-mat/0305505v6)",
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          "citation": "A Multi-Dimensional Lieb-Schultz-Mattis Theorem (arXiv math-ph/0608046v2)",
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      "statement": "# Higher-dimensional spin constraints rule out a unique ground state with a size-independent gap\n\nHastings extended the Lieb–Schultz–Mattis argument to higher-dimensional local quantum spin systems by combining separated twists with locality estimates. For the specified lattice families with a half-odd-integer spin per transverse column, a unique ground state has an excitation gap bounded by a constant times log(L)/L. Nachtergaele and Sims later gave a detailed rigorous proof with explicit geometric, interaction and symmetry hypotheses. The result restricts the possible low-energy structure of these many-body systems.\n\n## Scientific scope\n\nThe bound concerns particular finite-size sequences with even periodic length and the required odd transverse spin parity; it is not a theorem for every width, boundary condition or long-range interaction. The later proof assumes its F1/F2 and LSM1–6 conditions, including U(1) symmetry, zero total spin component of the ground state and a real-basis condition used to control twist derivatives. A vanishing first gap can reflect near-degenerate ground states and does not by itself prove a continuous gapless spectrum, magnetic order or a particular spin-liquid phase.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nM. B. Hastings; Bruno Nachtergaele; Robert Sims\n\nThese are the listed authors of the cited contribution sources. Paper authorship does not quantify individual effort. An incomplete list is explicitly marked; same names are not automatically merged.\n\n\n\n## Original sources\n\n- [Lieb-Schultz-Mattis in higher dimensions (arXiv cond-mat/0305505v6)](https://arxiv.org/pdf/cond-mat/0305505v6)\n- [A Multi-Dimensional Lieb-Schultz-Mattis Theorem (arXiv math-ph/0608046v2)](https://arxiv.org/pdf/math-ph/0608046v2)\n",
      "summary": "Hastings extended the Lieb–Schultz–Mattis argument to higher-dimensional local quantum spin systems by combining separated twists with locality estimates. For the specified lattice families with a half-odd-integer spin per transverse column, a unique ground state has an excitation gap bounded by a constant times log(L)/L. Nachtergaele and Sims later gave a detailed rigorous proof with explicit geometric, interaction and symmetry hypotheses. The result restricts the possible low-energy structure of these many-body systems.",
      "title": "Higher-dimensional spin constraints rule out a unique ground state with a size-independent gap"
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    "title": "Higher-dimensional spin constraints rule out a unique ground state with a size-independent gap",
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    "what_changed": "Hastings extended the Lieb–Schultz–Mattis argument to higher-dimensional local quantum spin systems by combining separated twists with locality estimates. For the specified lattice families with a half-odd-integer spin per transverse column, a unique ground state has an excitation gap bounded by a constant times log(L)/L. Nachtergaele and Sims later gave a detailed rigorous proof with explicit geometric, interaction and symmetry hypotheses. The result restricts the possible low-energy structure of these many-body systems.",
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