{
  "model": "Hubbard",
  "lattice": "rectangular-8x16",
  "n_sites": 128,
  "boundary": "P",
  "params": {
    "Nf": 56,
    "U": 8,
    "t1": 1,
    "t2": -0.2
  },
  "instance_id": "Hubbard/rectangular-8x16_128_P_56_8_t12_-0.2",
  "rows": [
    {
      "energy": -94.9632,
      "sigma": null,
      "energy_variance": null,
      "dof": 112,
      "einf": 196,
      "v_score": null,
      "method": "Transformer backflow",
      "method_detail": "MARCH",
      "method_as_published": "Transformer backflow + MARCH optimizer",
      "family": "transformer / ViT",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)",
      "reference": "Gu et al., Solving the Hubbard model with neural quantum states, Nat. Commun. 17, 7838 (2026), arXiv:2507.02644, doi:10.1038/s41467-026-74028-6",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-22",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-22",
        "method": "source text of arXiv:2507.02644 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": "−0.7419",
        "note": "Supplementary Table 5, row 16 × 8, t' = -0.2; dl/gu-si.txt line 195; no error bar | Energy per site in units of t. PBC per the Supplementary Table 5 caption ('Raw data for the energies of Hubbard model under PBC'); U = 8 and hole doping 1/8 per the main text (Fig. 4 caption: 'U = 8 and hole doping δ = 1/8 under PBC'; the t' = -0.2 study: 'Lx × 8 with Lx ranging from 8 to 32, with t' = -0.2 and δ = 1/8'). H has -t' sum_<<ij>>, so t2 = t'. The 16 x 4 and 16 x 16 values at t' = 0 equal the arXiv text's -0.76298 and -0.75630 to the printed digits. No error bar printed. Version read: journal Supplementary Information (41467_2026_74028_MOESM1_ESM.pdf); the arXiv text has no raw-data table.",
        "secondary_of": null
      }
    },
    {
      "energy": -94.1312,
      "sigma": null,
      "energy_variance": null,
      "dof": 112,
      "einf": 196,
      "v_score": null,
      "method": "Tensor backflow",
      "method_detail": "NN backflow, direct optimization",
      "method_as_published": "Tensor-Backflow (NN backflow, direct optimization, Ep = 0)",
      "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": "Liang, Investigating the Fermi-Hubbard model by the tensor-backflow method, Phys. Rev. B 113, 155121 (2026), arXiv:2507.01856",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-22",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-22",
        "method": "source text of arXiv:2507.01856 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": "−0.7354",
        "note": "Table V, row 8 × 16, t' = -0.2, NN backflow; dl/liang-prb.txt line 492, column Ep=0; no error bar | Energy per site in units of t at n = 0.875, U = 8, t' = -0.2 (Table V caption: 'n = 0.875 and U = 8'; Table VI column 'n, U, t′ = 0.875, 8, −0.2'). PBC: the 8 x 16 NNN Ep = 1 value -0.7410 and the 12 x 12 U_UHF = 4 Ep = 1 value -0.7420 equal Table II's PBC rows. Ep = 0 is the tensor-backflow state, Ep = 1 the same state after one Lanczos step; both are variational. No error bar printed. Version read: Phys. Rev. B 113, 155121 (publisher PDF); arXiv:2507.01856v4 (sources/2507.01856.txt) prints the same numbers.",
        "secondary_of": null
      }
    },
    {
      "energy": -94.7968,
      "sigma": null,
      "energy_variance": null,
      "dof": 112,
      "einf": 196,
      "v_score": null,
      "method": "Tensor backflow",
      "method_detail": "NN backflow, direct optimization, 1 Lanczos step",
      "method_as_published": "Tensor-Backflow + 1 Lanczos step (NN backflow, direct optimization, Ep = 1)",
      "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": "Liang, Investigating the Fermi-Hubbard model by the tensor-backflow method, Phys. Rev. B 113, 155121 (2026), arXiv:2507.01856",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-22",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-22",
        "method": "source text of arXiv:2507.01856 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": "−0.7406",
        "note": "Table V, row 8 × 16, t' = -0.2, NN backflow; dl/liang-prb.txt line 492, column Ep=1; no error bar | Energy per site in units of t at n = 0.875, U = 8, t' = -0.2 (Table V caption: 'n = 0.875 and U = 8'; Table VI column 'n, U, t′ = 0.875, 8, −0.2'). PBC: the 8 x 16 NNN Ep = 1 value -0.7410 and the 12 x 12 U_UHF = 4 Ep = 1 value -0.7420 equal Table II's PBC rows. Ep = 0 is the tensor-backflow state, Ep = 1 the same state after one Lanczos step; both are variational. No error bar printed. Version read: Phys. Rev. B 113, 155121 (publisher PDF); arXiv:2507.01856v4 (sources/2507.01856.txt) prints the same numbers.",
        "secondary_of": null
      }
    },
    {
      "energy": -94.1952,
      "sigma": null,
      "energy_variance": null,
      "dof": 112,
      "einf": 196,
      "v_score": null,
      "method": "Tensor backflow",
      "method_detail": "NNN backflow, direct optimization",
      "method_as_published": "Tensor-Backflow (NNN backflow, direct optimization, Ep = 0)",
      "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": "Liang, Investigating the Fermi-Hubbard model by the tensor-backflow method, Phys. Rev. B 113, 155121 (2026), arXiv:2507.01856",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-22",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-22",
        "method": "source text of arXiv:2507.01856 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": "−0.7359",
        "note": "Table V, row 8 × 16, t' = -0.2, NNN backflow; dl/liang-prb.txt line 493, column Ep=0; no error bar | Energy per site in units of t at n = 0.875, U = 8, t' = -0.2 (Table V caption: 'n = 0.875 and U = 8'; Table VI column 'n, U, t′ = 0.875, 8, −0.2'). PBC: the 8 x 16 NNN Ep = 1 value -0.7410 and the 12 x 12 U_UHF = 4 Ep = 1 value -0.7420 equal Table II's PBC rows. Ep = 0 is the tensor-backflow state, Ep = 1 the same state after one Lanczos step; both are variational. No error bar printed. Version read: Phys. Rev. B 113, 155121 (publisher PDF); arXiv:2507.01856v4 (sources/2507.01856.txt) prints the same numbers.",
        "secondary_of": null
      }
    },
    {
      "energy": -94.848,
      "sigma": null,
      "energy_variance": null,
      "dof": 112,
      "einf": 196,
      "v_score": null,
      "method": "Tensor backflow",
      "method_detail": "NNN backflow, direct optimization, 1 Lanczos step",
      "method_as_published": "Tensor-Backflow + 1 Lanczos step (NNN backflow, direct optimization, Ep = 1)",
      "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": "Liang, Investigating the Fermi-Hubbard model by the tensor-backflow method, Phys. Rev. B 113, 155121 (2026), arXiv:2507.01856",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-22",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-22",
        "method": "source text of arXiv:2507.01856 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": "−0.7410",
        "note": "Table V, row 8 × 16, t' = -0.2, NNN backflow; dl/liang-prb.txt line 493, column Ep=1; no error bar | Energy per site in units of t at n = 0.875, U = 8, t' = -0.2 (Table V caption: 'n = 0.875 and U = 8'; Table VI column 'n, U, t′ = 0.875, 8, −0.2'). PBC: the 8 x 16 NNN Ep = 1 value -0.7410 and the 12 x 12 U_UHF = 4 Ep = 1 value -0.7420 equal Table II's PBC rows. Ep = 0 is the tensor-backflow state, Ep = 1 the same state after one Lanczos step; both are variational. No error bar printed. Version read: Phys. Rev. B 113, 155121 (publisher PDF); arXiv:2507.01856v4 (sources/2507.01856.txt) prints the same numbers.",
        "secondary_of": null
      }
    }
  ],
  "url": "https://qmbl.org/i/Hubbard/rectangular-8x16_128_P_56_8_t12_-0.2/",
  "per_site_divisor": 128,
  "per_site_label": "E/site",
  "record": {
    "energy": -94.9632,
    "sigma": null,
    "method": "Transformer backflow",
    "method_detail": "MARCH",
    "reference": "Gu et al., Solving the Hubbard model with neural quantum states, Nat. Commun. 17, 7838 (2026), arXiv:2507.02644, doi:10.1038/s41467-026-74028-6",
    "bound_type": "variational",
    "energy_per_site": -0.7419
  },
  "no_record_reason": null
}