{
  "model": "Hubbard",
  "lattice": "square",
  "n_sites": 16,
  "boundary": "P",
  "params": {
    "Nf": 5,
    "U": 2.15443469
  },
  "instance_id": "Hubbard/square_16_P_5_2.15443469",
  "rows": [
    {
      "energy": -21.21223577699911,
      "sigma": null,
      "energy_variance": null,
      "dof": 10,
      "einf": 3.366304203125,
      "v_score": null,
      "method": "ED",
      "method_detail": "",
      "method_as_published": "Exact diagonalization",
      "family": "other",
      "bound_type": "exact",
      "bound_type_reason": "exact diagonalization|ex",
      "reference": "[paper](https://www.pnas.org/doi/full/10.1073/pnas.2122059119)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": false,
      "corrections": [
        {
          "field": "reference",
          "to": "[paper](https://www.pnas.org/doi/full/10.1073/pnas.2122059119)",
          "reported_as": "ED -21.2122357(7)",
          "location": "SI Table III, row ED, column U = 2.15",
          "version_read": "arXiv:2111.10420v2 PDF, pypdf layout mode (Robledo Moreno et al., PNAS 119, e2122059119, 2022)",
          "conversion": "none",
          "checked_on": "2026-09-17",
          "reason": "The linked lattice-symmetries script passes the U in VarBench's instance name and cannot have produced this energy, which is the ground state at the grid U (checks/hubbard-u-labels/). The number was uploaded by Robledo Moreno on 2022-08-23 as \"Lanczos (Quspin + Scipy)\"; the script link was attached on 2024-07-23 (VarBench 3604bb4). It is printed in the paper that produced it, which the row now cites (ruling 2026-09-17, Tristan).",
          "source_entry": "RB1-varbench-code-hubbard#2, ruling 3a (qmbl-verify 2026-09-17)",
          "from": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_16_P_5_2.1544/ed_lattice_symmetries.sh)"
        },
        {
          "field": "baseline",
          "to": false,
          "reported_as": "ED -21.2122357(7)",
          "location": "SI Table III, row ED, column U = 2.15",
          "version_read": "arXiv:2111.10420v2 PDF, pypdf layout mode (Robledo Moreno et al., PNAS 119, e2122059119, 2022)",
          "conversion": "none",
          "checked_on": "2026-09-17",
          "reason": "Collected from the authors' paper, not computed by VarBench: see the reference correction on this row (RULES.md 8.2).",
          "source_entry": "RB1-varbench-code-hubbard#2, ruling 3a (qmbl-verify 2026-09-17)",
          "from": true
        }
      ],
      "verified": {
        "checked_on": "2026-09-16",
        "scope": "value",
        "method": "recomputed: Lanczos ED over the full fixed-(N_up, N_dn) space by two independent codes (checks/hubbard-u-labels/)",
        "location": "checks/hubbard-u-labels/",
        "reported_as": "-21.21223577699904 (ed_check.py) | -21.212235776995612 (ed_lib.mjs) | -21.212235776999105 (skeptic_ed.py)",
        "conversion": "none",
        "note": "Reproduced at the grid U the instance now carries, not at the U of VarBench's name (relabels.mjs).",
        "source_entry": "B2-baseline-hubbard and E1-exact-recompute (qmbl-verify 2026-09-16)",
        "secondary_of": null
      }
    },
    {
      "energy": -21.211896,
      "sigma": 0.000002,
      "energy_variance": 0.027,
      "dof": 10,
      "einf": 3.366304203125,
      "v_score": 0.00044695480363968985,
      "method": "HFDS",
      "method_detail": "10 hidden fermions, 1-layer FFN with α = 64, C4 and K = 0 projections",
      "method_as_published": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 10. Single hidden layer fully connected net with alpha = 64). C4 and K = 0 projections",
      "family": "backflow / Pfaffian",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[paper](https://www.pnas.org/doi/full/10.1073/pnas.2122059119)",
      "source": "varbench@2024-10-22",
      "provenance": "imported"
    }
  ],
  "coverage": [
    {
      "checked_on": "2026-09-15",
      "method": "table harvest of cached arXiv sources (match_tables.mjs --all), every matched paper read for this instance",
      "screened": [
        "arXiv:2502.17144",
        "arXiv:2507.10705",
        "arXiv:2511.07566",
        "arXiv:2511.12551",
        "arXiv:2606.00924"
      ],
      "found": 0,
      "note": "Only papers whose tables matched the instance by size were read; not a citation screen."
    },
    {
      "checked_on": "2026-09-15",
      "method": "citation screen of Robledo Moreno et al. 2022 (PNAS, HFDS): citing works from OpenAlex and Semantic Scholar (scripts/cited_by.mjs)",
      "screened_count": 29,
      "found": 0,
      "note": "142 citing works, 118 on arXiv, 118 fetched; 29 read in full, 89 excluded by a text filter (no energy to four decimals, or no QMBL model named)."
    }
  ],
  "relabelled": {
    "from": "Hubbard/square_16_P_5_2.1544",
    "checked_on": "2026-09-16",
    "reason": "The name's 2.1544 cuts off the coupling of this Hamiltonian, 2.15443469 (10^(3/9)). The exact diagonalization, -21.21223577699911, is the ground state there and 3.65e-5 above the ground state at U = 2.1544.",
    "evidence": "checks/hubbard-u-labels/"
  },
  "url": "https://qmbl.org/i/Hubbard/square_16_P_5_2.15443469/",
  "per_site_divisor": 16,
  "per_site_label": "E/site",
  "record": {
    "energy": -21.21223577699911,
    "sigma": null,
    "method": "ED",
    "method_detail": "",
    "reference": "[paper](https://www.pnas.org/doi/full/10.1073/pnas.2122059119)",
    "bound_type": "exact",
    "energy_per_site": -1.3257647360624443
  },
  "no_record_reason": null
}