{
  "model": "Hubbard",
  "lattice": "square",
  "n_sites": 16,
  "boundary": "P",
  "params": {
    "Nf": 4,
    "U": 2.15443469
  },
  "instance_id": "Hubbard/square_16_P_4_2.15443469",
  "rows": [
    {
      "energy": -18.4053208,
      "sigma": 8e-7,
      "energy_variance": null,
      "dof": 8,
      "einf": 2.15443469,
      "v_score": null,
      "method": "ED",
      "method_detail": "",
      "method_as_published": "Exact diagonalization",
      "family": "other",
      "bound_type": "exact",
      "bound_type_reason": "numerically exact reference as stated by the source (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-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "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": "-18.4053208(8)",
        "note": "Appendix H Table II, U=2.15, row ED; sources/2111.10420.txt line 1193; 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": -18.06,
      "sigma": 0.0592,
      "energy_variance": null,
      "dof": 8,
      "einf": 2.15443469,
      "v_score": null,
      "method": "Hartree-Fock",
      "method_detail": "unrestricted Slater determinant",
      "method_as_published": "USD (Table II)",
      "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-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "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": "-18.06(6)",
        "note": "Appendix H Table II, U=2.15, row USD; sources/2111.10420.txt line 1194; 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": -18.306,
      "sigma": 0.00304,
      "energy_variance": null,
      "dof": 8,
      "einf": 2.15443469,
      "v_score": null,
      "method": "Slater-Jastrow",
      "method_detail": "unrestricted Slater determinant + Gutzwiller factor",
      "method_as_published": "USD (Gutzwiller) (Table II)",
      "family": "classic VMC",
      "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-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "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": "-18.306(3)",
        "note": "Appendix H Table II, U=2.15, row USD (Gutzwiller); sources/2111.10420.txt line 1195; 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": -18.3173,
      "sigma": 0.000208,
      "energy_variance": null,
      "dof": 8,
      "einf": 2.15443469,
      "v_score": null,
      "method": "Slater-Jastrow",
      "method_detail": "unrestricted Slater determinant + Jastrow factor",
      "method_as_published": "USD (Jastrow) (Table II)",
      "family": "classic VMC",
      "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-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "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": "-18.3173(2)",
        "note": "Appendix H Table II, U=2.15, row USD (Jastrow); sources/2111.10420.txt line 1196; 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": -18.3742,
      "sigma": 0.0000704,
      "energy_variance": null,
      "dof": 8,
      "einf": 2.15443469,
      "v_score": null,
      "method": "RBM",
      "method_detail": "unrestricted Slater determinant × RBM",
      "method_as_published": "USD (RBM) (Table II)",
      "family": "RBM",
      "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-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "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": "-18.37420(7)",
        "note": "Appendix H Table II, U=2.15, row USD (RBM); sources/2111.10420.txt line 1197; 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": -18.32346,
      "sigma": 0.00008,
      "energy_variance": null,
      "dof": 8,
      "einf": 2.15443469,
      "v_score": null,
      "method": "HFDS",
      "method_detail": "Gutzwiller-inspired constraint function",
      "method_as_published": "Gutzwiller inspired CF (Table II)",
      "family": "classic VMC",
      "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-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "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": "-18.32346(8)",
        "note": "Appendix H Table II, U=2.15, row Gutzwiller inspired CF; sources/2111.10420.txt line 1198; 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": -18.33683,
      "sigma": 0.0000704,
      "energy_variance": null,
      "dof": 8,
      "einf": 2.15443469,
      "v_score": null,
      "method": "HFDS",
      "method_detail": "unrestricted hidden-spin constraint function",
      "method_as_published": "Unrestricted hidden spin (Table II)",
      "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-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "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": "-18.33683(7)",
        "note": "Appendix H Table II, U=2.15, row Unrestricted hidden spin; sources/2111.10420.txt line 1199; 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": -17.8293,
      "sigma": 0.000208,
      "energy_variance": null,
      "dof": 8,
      "einf": 2.15443469,
      "v_score": null,
      "method": "HFDS",
      "method_detail": "Jastrow-inspired constraint function",
      "method_as_published": "Jastrow inspired CF (Table II)",
      "family": "classic VMC",
      "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-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "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": "-17.8293(2)",
        "note": "Appendix H Table II, U=2.15, row Jastrow inspired CF; sources/2111.10420.txt line 1200; 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": -18.37803,
      "sigma": 0.0000496,
      "energy_variance": null,
      "dof": 8,
      "einf": 2.15443469,
      "v_score": null,
      "method": "HFDS",
      "method_detail": "unrestricted hidden-spin constraint function + RBM in the augmented space",
      "method_as_published": "Unrestricted hid. spin(RBM) (Table II)",
      "family": "RBM",
      "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-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "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": "-18.37803(5)",
        "note": "Appendix H Table II, U=2.15, row Unrestricted hid. spin(RBM); sources/2111.10420.txt line 1202; 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
      }
    }
  ],
  "url": "https://qmbl.org/i/Hubbard/square_16_P_4_2.15443469/",
  "per_site_divisor": 16,
  "per_site_label": "E/site",
  "record": {
    "energy": -18.4053208,
    "sigma": 8e-7,
    "method": "ED",
    "method_detail": "",
    "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",
    "bound_type": "exact",
    "energy_per_site": -1.15033255
  },
  "no_record_reason": null
}