{
  "model": "Hubbard",
  "lattice": "square",
  "n_sites": 64,
  "boundary": "P",
  "params": {
    "Nf": 28,
    "U": 8
  },
  "instance_id": "Hubbard/square_64_P_28_8",
  "rows": [
    {
      "energy": -47.501,
      "sigma": 0.002,
      "energy_variance": 6.62,
      "dof": 56,
      "einf": 98,
      "v_score": 0.017511125482456442,
      "method": "VMC with uniform BCS pairing (+Jastrow and backflow)",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_28_8/VMC-uniform/vmc_hubbard.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -47.954,
      "sigma": 0.001,
      "energy_variance": null,
      "dof": 56,
      "einf": 98,
      "v_score": null,
      "method": "FN on the state above",
      "bound_type": "projected",
      "bound_type_reason": "fixed[- ]node|^fn\\b|\\bfn",
      "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_28_8/FN-uniform/fn_hubbard.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -47.7312,
      "sigma": 0.0384,
      "energy_variance": 3.36,
      "dof": 56,
      "einf": 98,
      "v_score": 0.008859765401290735,
      "method": "Jastrow-backflow (JBf), 8x8 torus",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)",
      "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. B 113, 245104 (2026), arXiv:2510.11710",
      "peer_reviewed": true,
      "source": "sweep-tables-2026-09-14",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-14",
        "method": "arXiv HTML parsed locally, no LLM transcription",
        "reported_as": "E_0/M = -0.7458 (+/- 0.0006), sigma^2/M = 0.0525",
        "note": "Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Row \"JBf (8x8 Torus)\".",
        "secondary_of": null
      }
    },
    {
      "energy": -47.7056,
      "sigma": 0.0576,
      "energy_variance": 3.6032,
      "dof": 56,
      "einf": 98,
      "v_score": 0.009504382557134799,
      "method": "Hidden-fermion determinant state (HFDS), 8x8 torus",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)",
      "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. B 113, 245104 (2026), arXiv:2510.11710",
      "peer_reviewed": true,
      "source": "sweep-tables-2026-09-14",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-14",
        "method": "arXiv HTML parsed locally, no LLM transcription",
        "reported_as": "E_0/M = -0.7454 (+/- 0.0009), sigma^2/M = 0.0563",
        "note": "Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Row \"HFDS (8x8 Torus)\".",
        "secondary_of": null
      }
    },
    {
      "energy": -47.5008,
      "sigma": 0.0192,
      "energy_variance": 4.4672,
      "dof": 56,
      "einf": 98,
      "v_score": 0.011816603445266605,
      "method": "JBf-ViT (vision transformer backflow), 8x8 torus",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)",
      "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. B 113, 245104 (2026), arXiv:2510.11710",
      "peer_reviewed": true,
      "source": "sweep-tables-2026-09-14",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-14",
        "method": "arXiv HTML parsed locally, no LLM transcription",
        "reported_as": "E_0/M = -0.7422 (+/- 0.0003), sigma^2/M = 0.0698",
        "note": "Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Right-hand table, row \"JBf-ViT\"; same lattice and parameters per the caption.",
        "secondary_of": null
      }
    },
    {
      "energy": -47.104,
      "sigma": 0.064,
      "energy_variance": 6.72,
      "dof": 56,
      "einf": 98,
      "v_score": 0.017873044264928566,
      "method": "HFDS-ViT, 8x8 torus",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)",
      "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. B 113, 245104 (2026), arXiv:2510.11710",
      "peer_reviewed": true,
      "source": "sweep-tables-2026-09-14",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-14",
        "method": "arXiv HTML parsed locally, no LLM transcription",
        "reported_as": "E_0/M = -0.736 (+/- 0.001), sigma^2/M = 0.105",
        "note": "Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Right-hand table, row \"HFDS-ViT\".",
        "secondary_of": null
      }
    },
    {
      "energy": -46.656,
      "sigma": 0.256,
      "energy_variance": 8.896,
      "dof": 56,
      "einf": 98,
      "v_score": 0.02380728645124025,
      "method": "HFDS Symm-ViT-Ti, 8x8 torus",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)",
      "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. B 113, 245104 (2026), arXiv:2510.11710",
      "peer_reviewed": true,
      "source": "sweep-tables-2026-09-14",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-14",
        "method": "arXiv HTML parsed locally, no LLM transcription",
        "reported_as": "E_0/M = -0.729 (+/- 0.004), sigma^2/M = 0.139",
        "note": "Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Right-hand table, row \"HFDS Symm-ViT-Ti\".",
        "secondary_of": null
      }
    },
    {
      "energy": -46.464,
      "sigma": 0.576,
      "energy_variance": 10.752,
      "dof": 56,
      "einf": 98,
      "v_score": 0.02885081004475789,
      "method": "JBf Symm-ViT-Ti, 8x8 torus",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)",
      "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. B 113, 245104 (2026), arXiv:2510.11710",
      "peer_reviewed": true,
      "source": "sweep-tables-2026-09-14",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-14",
        "method": "arXiv HTML parsed locally, no LLM transcription",
        "reported_as": "E_0/M = -0.726 (+/- 0.009), sigma^2/M = 0.168",
        "note": "Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Right-hand table, row \"JBf Symm-ViT-Ti\".",
        "secondary_of": null
      }
    }
  ],
  "url": "https://qmbl.org/i/Hubbard/square_64_P_28_8/",
  "per_site_divisor": 64,
  "per_site_label": "E/site",
  "record": {
    "energy": -47.7312,
    "sigma": 0.0384,
    "method": "Jastrow-backflow (JBf), 8x8 torus",
    "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. B 113, 245104 (2026), arXiv:2510.11710",
    "energy_per_site": -0.7458
  },
  "no_record_reason": null
}