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        "note": "Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row Det, Symm -; sources/2604.21978.txt line 190; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.",
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        "note": "Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row Det-PH, Symm T+R; sources/2604.21978.txt line 195; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.",
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    },
    {
      "energy": -47.8112,
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      "dof": 56,
      "einf": 98,
      "v_score": null,
      "method": "Transformer backflow",
      "method_detail": "Pfaffian",
      "method_as_published": "Pfaffian (transformer backflow), no projection",
      "family": "transformer / ViT",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)",
      "reference": "L. L. Viteritti, R. Rende, C. Roth, A. Sengupta, G. Carleo & A. Georges, Beyond Variational Bias: Resolving Intertwined Orders in the Hubbard Model, arXiv:2604.21978",
      "peer_reviewed": false,
      "source": "sweep-allresults-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "method": "source text of arXiv:2604.21978 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "-0.74705(1)",
        "note": "Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row Pfaffian, Symm -; sources/2604.21978.txt line 196; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.",
        "secondary_of": null
      }
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    {
      "energy": -47.96928,
      "sigma": 0.00064,
      "energy_variance": null,
      "dof": 56,
      "einf": 98,
      "v_score": null,
      "method": "Transformer backflow",
      "method_detail": "Pfaffian; projected: translations",
      "method_as_published": "Pfaffian (transformer backflow), T",
      "family": "transformer / ViT",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)",
      "reference": "L. L. Viteritti, R. Rende, C. Roth, A. Sengupta, G. Carleo & A. Georges, Beyond Variational Bias: Resolving Intertwined Orders in the Hubbard Model, arXiv:2604.21978",
      "peer_reviewed": false,
      "source": "sweep-allresults-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "method": "source text of arXiv:2604.21978 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "-0.74952(1)",
        "note": "Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row Pfaffian, Symm T; sources/2604.21978.txt line 197; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.",
        "secondary_of": null
      }
    },
    {
      "energy": -48.03712,
      "sigma": 0.00064,
      "energy_variance": null,
      "dof": 56,
      "einf": 98,
      "v_score": null,
      "method": "Transformer backflow",
      "method_detail": "Pfaffian; projected: translations, C4",
      "method_as_published": "Pfaffian (transformer backflow), T+R",
      "family": "transformer / ViT",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)",
      "reference": "L. L. Viteritti, R. Rende, C. Roth, A. Sengupta, G. Carleo & A. Georges, Beyond Variational Bias: Resolving Intertwined Orders in the Hubbard Model, arXiv:2604.21978",
      "peer_reviewed": false,
      "source": "sweep-allresults-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "method": "source text of arXiv:2604.21978 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "-0.75058(1)",
        "note": "Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row Pfaffian, Symm T+R; sources/2604.21978.txt line 198; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.",
        "secondary_of": null
      }
    },
    {
      "energy": -48.256,
      "sigma": 0.0256,
      "energy_variance": null,
      "dof": 56,
      "einf": 98,
      "v_score": null,
      "method": "Transformer backflow",
      "method_detail": "Pfaffian; extrapolated: variance → 0 (linear)",
      "method_as_published": "Pfaffian (transformer backflow), zero-variance extrapolation",
      "family": "transformer / ViT",
      "bound_type": "extrapolated",
      "bound_type_reason": "zero-variance or bond-dimension extrapolation; not an upper bound (assigned during source reading)",
      "reference": "L. L. Viteritti, R. Rende, C. Roth, A. Sengupta, G. Carleo & A. Georges, Beyond Variational Bias: Resolving Intertwined Orders in the Hubbard Model, arXiv:2604.21978",
      "peer_reviewed": false,
      "source": "sweep-allresults-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "method": "source text of arXiv:2604.21978 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "-0.7540(4)",
        "note": "Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row 'Zero-Var.Extr.'; sources/2604.21978.txt line 199; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page); the table prints -0.7540(4), the text -0.7540(1) three times; the table's error bar is kept | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.",
        "secondary_of": null
      }
    }
  ],
  "url": "https://qmbl.org/i/Hubbard/square_64_P_28_8_t12_-0.2/",
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  "per_site_label": "E/site",
  "record": {
    "energy": -48.256,
    "sigma": 0.0256,
    "method": "Transformer backflow",
    "method_detail": "Pfaffian; extrapolated: variance → 0 (linear)",
    "reference": "L. L. Viteritti, R. Rende, C. Roth, A. Sengupta, G. Carleo & A. Georges, Beyond Variational Bias: Resolving Intertwined Orders in the Hubbard Model, arXiv:2604.21978",
    "bound_type": "extrapolated",
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  },
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}