{
  "model": "Hubbard",
  "lattice": "square",
  "n_sites": 36,
  "boundary": "PA",
  "params": {
    "Nf": 18,
    "U": 2
  },
  "instance_id": "Hubbard/square_36_PA_18_2",
  "rows": [
    {
      "energy": -76.162,
      "sigma": 0.006,
      "energy_variance": 0.14,
      "dof": 36,
      "einf": 18,
      "v_score": 0.0005684328757789593,
      "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel 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",
      "defect": {
        "flag": "wrong-instance",
        "finding": "Energy, sigma and variance (-76.162, 0.006, 0.14(2)) are identical to the 8x8 row on square_64_PA_32_2. The HFDS first author uploaded them to the 6x6 file 67 minutes after the 8x8 file (VarBench commits c810bdcd22, then 21f00e54f6), and they never changed. -76.162 / 64 = -1.19003 is the paper's 8x8, U = 2 value, -1.1900(2) (PNAS SI Table 5, arXiv SI Table V). As a 36-site total it is -2.1156 per site: 16.45 below the non-interacting ground state of this lattice, -59.712813, which a U >= 0 Hamiltonian cannot go below, and 32.7 below the sign-free AFQMC total -43.499(2) (Qin, Shi & Zhang, PRB 94, 085103 (2016), Table IV). The paper's 6x6 value, -1.2079(1), is carried as a separate row."
      }
    },
    {
      "energy": -43.4844,
      "sigma": 0.0036,
      "energy_variance": null,
      "dof": 36,
      "einf": 18,
      "v_score": null,
      "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 8, fully parametrized hidden sub-matrix, hidden-unit density alpha = 78)",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)",
      "reference": "Robledo Moreno, Carleo, Georges & Stokes, Fermionic wave functions from neural-network constrained hidden states, Proc. Natl. Acad. Sci. U.S.A. 119, e2122059119 (2022), arXiv:2111.10420, doi:10.1073/pnas.2122059119",
      "peer_reviewed": true,
      "source": "sweep-worklist-2026-09-15",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-15",
        "method": "PNAS supplementary PDF (via Europe PMC, PMC9371695) extracted locally, cross-read against the arXiv version; no LLM transcription",
        "reported_as": "-1.2079(1) per site, no variance",
        "note": "PNAS SI Table 5 (identical to arXiv SI Table V), \"Variational energy per site in the L x L Hubbard model at half filling with periodic boundary conditions along one of the sides of the square and anti-periodic boundary conditions along the other side\", row L = 6, column U = 2. SI Sec. 6: N_hidden = 8, alpha = 78 for 6x6; no projection or constraint stated, nor what the error bar is. Column alignment checked: the VarBench HFDS rows for 6x6 at U = 4, 6, 8 and 8x8 at U = 2, 4, 8 match the printed digits. As a total, -43.4844 sits 3.4e-4 (relative) above the sign-free AFQMC value -43.499(2) for this lattice (Qin, Shi & Zhang, PRB 94, 085103 (2016), Table IV, PBC-APBC) and above the non-interacting bound -59.712813.",
        "secondary_of": null
      }
    },
    {
      "energy": -43.499,
      "sigma": 0.002,
      "energy_variance": null,
      "dof": 36,
      "einf": 18,
      "v_score": null,
      "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)",
      "bound_type": "exact",
      "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)",
      "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421",
      "peer_reviewed": true,
      "source": "exact-2026-09-15",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-15",
        "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription",
        "reported_as": "E = -43.499(2) (total), 6 x 6, U = 2, PBC-APBC",
        "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 6 x 6, column U = 2 PBC-APBC. Half filling: N_up = N_dn = 18. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.",
        "secondary_of": null
      }
    }
  ],
  "url": "https://qmbl.org/i/Hubbard/square_36_PA_18_2/",
  "per_site_divisor": 36,
  "per_site_label": "E/site",
  "record": {
    "energy": -43.4844,
    "sigma": 0.0036,
    "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 8, fully parametrized hidden sub-matrix, hidden-unit density alpha = 78)",
    "reference": "Robledo Moreno, Carleo, Georges & Stokes, Fermionic wave functions from neural-network constrained hidden states, Proc. Natl. Acad. Sci. U.S.A. 119, e2122059119 (2022), arXiv:2111.10420, doi:10.1073/pnas.2122059119",
    "energy_per_site": -1.2079
  },
  "no_record_reason": null
}