{
  "model": "Hubbard",
  "lattice": "rectangular-4x8",
  "n_sites": 32,
  "boundary": "P",
  "params": {
    "Nf": 14,
    "U": 8
  },
  "instance_id": "Hubbard/rectangular-4x8_32_P_14_8",
  "rows": [
    {
      "energy": -24.426,
      "sigma": 0.002,
      "energy_variance": 0.021,
      "dof": 28,
      "einf": 49,
      "v_score": 0.00010906303678744678,
      "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 14). Soft mean-field constraint for lambda = 8 stripe order.",
      "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"
    },
    {
      "energy": -24.5408,
      "sigma": 0.00032,
      "energy_variance": 0.52224,
      "dof": 28,
      "einf": 49,
      "v_score": 0.0027037807032315774,
      "method": "HFPS (hidden-fermion Pfaffian state)",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)",
      "reference": "Chen, Wan, Sengupta & Georges, Neural network-augmented Pfaffian wave-functions for scalable simulations of interacting fermions, Proc. Natl. Acad. Sci. U.S.A. 123, e2535288123 (2026), arXiv:2507.10705",
      "peer_reviewed": true,
      "source": "sweep-worklist-2026-09-15",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-15",
        "method": "arXiv PDF text extracted locally with pypdf in layout mode, no LLM transcription",
        "reported_as": "-0.76690(1) per site, sigma^2/N_site = 0.016320",
        "note": "Table IV of arXiv:2507.10705, \"Raw data of Fig. 7 in the L x 4 lattice with 1/8 doping and U = 8\", row 8 x 4; Sec. III states Fig. 7 shows \"L x 4 PBC lattices\". 32 sites, 28 electrons, U = 8, periodic: this instance. The same table's 16 x 4 value -0.76413(3) is the HFPS row on rectangular-4x16_64_P_28_8, and its 4 x 4 value -0.74177(5) sits 3e-5 above the ED energy -0.7418 quoted by arXiv:2606.00924. New record, 0.47% below the VarBench HFDS row -0.7633125 and below the RHB -0.7641(3) of arXiv:2606.00924 on the same lattice. Variance: sigma^2/N_site stored as a total (x 32). Its V-score, 2.7e-3, is ~25x the VarBench HFDS row's 1.1e-4 at a higher energy; the HFDS variance is the likelier outlier, since on 4x16 the same VarBench ansatz has ten times the per-site variance and this paper puts HFDS 3e-4 above exact already at 4x4. Read from the arXiv version; the PNAS version was not compared (pnas.org refuses this sandbox).",
        "secondary_of": null
      }
    },
    {
      "energy": -24.4512,
      "sigma": 0.0096,
      "energy_variance": null,
      "dof": 28,
      "einf": 49,
      "v_score": null,
      "method": "RHB (hierarchical backflow K = 2 + nonlocal FNN factor, Ndet = 5, Nneuron = 100)",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)",
      "reference": "Zhou, Zhou & Liu, Locality-Induced Hierarchical Backflow Wavefunctions for Correlated Fermions, arXiv:2606.00924",
      "peer_reviewed": false,
      "source": "sweep-worklist-2026-09-15",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-15",
        "method": "arXiv PDF text extracted locally with pypdf in layout mode, no LLM transcription",
        "reported_as": "-0.7641 per site, table-wide sampling error ~0.0003",
        "note": "Table IV of arXiv:2606.00924, \"Energies per site for the Hubbard model at nh = 0.125, U = 8 for different systems under periodic boundary conditions. The energy sampling errors are around 0.0003\", row 4 x 8, column RHB (Ndet = 5, Nneuron = 100, the setting adopted for all large-scale runs after Table III). Instance identity: the same row's HFDS column, -0.7633, is the VarBench HFDS row -0.7633125. sigma is the caption's table-wide \"around 0.0003\". Layout mode shifts this row one column left in sweep-tables.tsv; read here from the printed table. Internal check: the same ansatz gives -0.7410 on 4 x 4 against ED -0.7418 (Table III), above exact as it must be.",
        "secondary_of": null
      }
    }
  ],
  "url": "https://qmbl.org/i/Hubbard/rectangular-4x8_32_P_14_8/",
  "per_site_divisor": 32,
  "per_site_label": "E/site",
  "record": {
    "energy": -24.5408,
    "sigma": 0.00032,
    "method": "HFPS (hidden-fermion Pfaffian state)",
    "reference": "Chen, Wan, Sengupta & Georges, Neural network-augmented Pfaffian wave-functions for scalable simulations of interacting fermions, Proc. Natl. Acad. Sci. U.S.A. 123, e2535288123 (2026), arXiv:2507.10705",
    "energy_per_site": -0.7669
  },
  "no_record_reason": null
}