{
  "model": "Hubbard",
  "lattice": "square",
  "n_sites": 256,
  "boundary": "PA",
  "params": {
    "Nf": 128,
    "U": 8
  },
  "instance_id": "Hubbard/square_256_PA_128_8",
  "rows": [
    {
      "energy": -132.86,
      "sigma": 0.0044,
      "energy_variance": 8.1,
      "dof": 256,
      "einf": 512,
      "v_score": 0.0049864802850897744,
      "method": "mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos, (U=8)",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_PA_128_8/mVMC/mVMC.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -134.2125,
      "sigma": 0.00923,
      "energy_variance": 1,
      "dof": 256,
      "einf": 512,
      "v_score": 0.0006130406263201699,
      "method": "VAFQMC",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)",
      "source": "varbench@2024-10-22",
      "provenance": "imported"
    },
    {
      "energy": -134.25,
      "sigma": 0.03,
      "energy_variance": null,
      "dof": 256,
      "einf": 512,
      "v_score": null,
      "method": "AFQMC",
      "bound_type": null,
      "bound_type_reason": "ambiguous: bare method string, sign-problem-free vs constrained undetermined",
      "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.94.085103)",
      "source": "varbench@2024-10-22",
      "provenance": "imported"
    },
    {
      "energy": -134.25,
      "sigma": 0.03,
      "energy_variance": null,
      "dof": 256,
      "einf": 512,
      "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 = -134.25(3) (total), 16 x 16, U = 8, 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 16 x 16, column U = 8 PBC-APBC. Half filling: N_up = N_dn = 128. 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_256_PA_128_8/",
  "per_site_divisor": 256,
  "per_site_label": "E/site",
  "record": {
    "energy": -134.2125,
    "sigma": 0.00923,
    "method": "VAFQMC",
    "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)",
    "energy_per_site": -0.524267578125
  },
  "no_record_reason": null
}