{
  "model": "Hubbard",
  "lattice": "square",
  "n_sites": 16,
  "boundary": "P",
  "params": {
    "Nf": 7,
    "U": 8
  },
  "instance_id": "Hubbard/square_16_P_7_8",
  "rows": [
    {
      "energy": -11.8544,
      "sigma": 0.0016,
      "energy_variance": null,
      "dof": 14,
      "einf": 24.5,
      "v_score": null,
      "method": "HFDS",
      "method_detail": "full hidden sub-matrix",
      "method_as_published": "HFDS (fully parametrized hidden sub-matrix)",
      "family": "other",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)",
      "reference": "J. Robledo Moreno, G. Carleo, A. Georges & J. Stokes, Fermionic Wave Functions from Neural-Network Constrained Hidden States, Proc. Natl. Acad. Sci. 119, e2122059119 (2022), arXiv:2111.10420, doi:10.1073/pnas.2122059119",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-22",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-22",
        "method": "source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "-0.7409(1)",
        "note": "Appendix H Table IV, L=4, column \"Energy per site HFDS (PBC-PBC)\"; sources/2111.10420.txt line 1253; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29 | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).",
        "secondary_of": null
      }
    },
    {
      "energy": -11.86832,
      "sigma": 0.0008,
      "energy_variance": null,
      "dof": 14,
      "einf": 24.5,
      "v_score": null,
      "method": "HFPS",
      "method_detail": "GCNN",
      "method_as_published": "HFPS + GCNN (Table IV)",
      "family": "backflow / Pfaffian",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)",
      "reference": "A. Chen, Z.-Q. Wan, A. Sengupta, A. Georges & C. Roth, Neural Network-Augmented Pfaffian Wave-functions for Scalable Simulations of Interacting Fermions, Proc. Natl. Acad. Sci. 123, e2535288123 (2026), arXiv:2507.10705, doi:10.1073/pnas.2535288123",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-22",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-22",
        "method": "source text of arXiv:2507.10705 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "-0.74177(5)",
        "note": "TABLE IV, 4x4 row; sources/2507.10705.txt line 968; version read: arXiv v1 (sources/2507.10705.txt); PNAS copy not reached (Cloudflare 403, 2026-10-16) | Eq. (15): H = -t sum_<ij> + U sum n_up n_dn, t = 1, t' = 0, periodic; all tables per site (E/Nsite).",
        "secondary_of": null
      }
    }
  ],
  "url": "https://qmbl.org/i/Hubbard/square_16_P_7_8/",
  "per_site_divisor": 16,
  "per_site_label": "E/site",
  "record": {
    "energy": -11.86832,
    "sigma": 0.0008,
    "method": "HFPS",
    "method_detail": "GCNN",
    "reference": "A. Chen, Z.-Q. Wan, A. Sengupta, A. Georges & C. Roth, Neural Network-Augmented Pfaffian Wave-functions for Scalable Simulations of Interacting Fermions, Proc. Natl. Acad. Sci. 123, e2535288123 (2026), arXiv:2507.10705, doi:10.1073/pnas.2535288123",
    "bound_type": "variational",
    "energy_per_site": -0.74177
  },
  "no_record_reason": null
}