{
  "model": "Hubbard",
  "lattice": "square",
  "n_sites": 144,
  "boundary": "P",
  "params": {
    "Nf": 72,
    "U": 6
  },
  "instance_id": "Hubbard/square_144_P_72_6",
  "rows": [
    {
      "energy": -94.66,
      "sigma": 0.02,
      "energy_variance": null,
      "dof": 144,
      "einf": 216,
      "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 = -94.66(2) (total), 12 x 12, U = 6, PBC",
        "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 12 x 12, column U = 6 PBC. Half filling: N_up = N_dn = 72. 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_144_P_72_6/",
  "per_site_divisor": 144,
  "per_site_label": "E/site",
  "record": null,
  "no_record_reason": "solved exactly"
}