{
  "model": "Hubbard",
  "lattice": "square",
  "n_sites": "inf",
  "boundary": "TDL",
  "params": {
    "U": 4,
    "n": 0.875
  },
  "instance_id": "Hubbard/square_inf_0.875_4",
  "rows": [
    {
      "energy": -1.026,
      "sigma": 0.001,
      "method": "CP-AFQMC",
      "method_as_published": "AFQMC (Sec. IX: Table II lists TL energies from AFQMC, DMRG, DMET and FN)",
      "bound_type": "extrapolated",
      "bound_type_reason": "extrapolation to an infinite ansatz or to infinite size; not an upper bound (RULES.md 14.4)",
      "how_tdl": "finite-size extrapolation",
      "reference": "LeBlanc et al., Solutions of the Two Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms, Phys. Rev. X 5, 041041 (2015), arXiv:1505.02290",
      "peer_reviewed": true,
      "source": "tdl-2026-10-25",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-25",
        "method": "arXiv:1505.02290: journal version: Phys. Rev. X 5, 041041 (2015), open-access PDF from journals.aps.org (journal/leblanc-prx.pdf), pypdf layout and plain, value found; arXiv text compared (arXiv text (pypdf layout and plain, read locally; version: 1505.02290v2)); single reading, journal check, second reading 2026-10-25",
        "reported_as": "-1.026 0.001  [table gives value and uncertainty in separate columns]",
        "note": "Table V, row AFQMC, column U = 4; line 2791 of sources/leblanc.plain.txt (sources/leblanc.plain.txt) | per site (E/t; text: 'we systematically compute the energy per site'); units t; Eq. (1): H = -sum_{i,j,sigma} t_ij (c^dag_{i sigma} c_{j sigma} + H.c.) + U sum_i n_{i up} n_{i down}; t_ij = t nearest neighbour, t' second neighbour; t sets the energy scale; repulsive U > 0 (Sec. II, lines before Table I). Sum convention over (i,j) not spelled out beyond 'sum_{i,j,sigma}'. No chemical-potential term; energies at fixed density n. | Value and uncertainty are separate columns in the table (second number = one-sigma-type uncertainty 'best available within each algorithm', no systematic error: Sec. IX). n = 0.875 AFQMC is constrained path (caption); main text: rectangular supercells (mostly 8x32).",
        "secondary_of": null
      }
    },
    {
      "energy": -1.031,
      "sigma": 0.003,
      "method": "DMET",
      "method_as_published": "DMET",
      "bound_type": "approximate",
      "bound_type_reason": "series expansion, spin-wave theory or another fixed approximation; error of unknown sign (RULES.md 14.4)",
      "how_tdl": "series or approximate theory",
      "reference": "LeBlanc et al., Solutions of the Two Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms, Phys. Rev. X 5, 041041 (2015), arXiv:1505.02290",
      "peer_reviewed": true,
      "source": "tdl-2026-10-25",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-25",
        "method": "arXiv:1505.02290: journal version: Phys. Rev. X 5, 041041 (2015), open-access PDF from journals.aps.org (journal/leblanc-prx.pdf), pypdf layout and plain, value found; arXiv text compared (arXiv text (pypdf layout and plain, read locally; version: 1505.02290v2)); single reading, journal check, second reading 2026-10-25",
        "reported_as": "-1.031 0.003  [table gives value and uncertainty in separate columns]",
        "note": "Table V, row DMET, column U = 4; line 2792 of sources/leblanc.plain.txt (sources/leblanc.plain.txt) | per site (E/t; text: 'we systematically compute the energy per site'); units t; Eq. (1): H = -sum_{i,j,sigma} t_ij (c^dag_{i sigma} c_{j sigma} + H.c.) + U sum_i n_{i up} n_{i down}; t_ij = t nearest neighbour, t' second neighbour; t sets the energy scale; repulsive U > 0 (Sec. II, lines before Table I). Sum convention over (i,j) not spelled out beyond 'sum_{i,j,sigma}'. No chemical-potential term; energies at fixed density n. | Value and uncertainty are separate columns in the table (second number = one-sigma-type uncertainty 'best available within each algorithm', no systematic error: Sec. IX).",
        "secondary_of": null
      }
    },
    {
      "energy": -1.0225,
      "sigma": 0.0015,
      "method": "FN-DMC",
      "method_as_published": "FN (fixed-node diffusion Monte Carlo with VMC nodes)",
      "bound_type": "extrapolated",
      "bound_type_reason": "extrapolation to an infinite ansatz or to infinite size; not an upper bound (RULES.md 14.4)",
      "how_tdl": "finite-size extrapolation",
      "reference": "LeBlanc et al., Solutions of the Two Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms, Phys. Rev. X 5, 041041 (2015), arXiv:1505.02290",
      "peer_reviewed": true,
      "source": "tdl-2026-10-25",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-25",
        "method": "arXiv:1505.02290: journal version: Phys. Rev. X 5, 041041 (2015), open-access PDF from journals.aps.org (journal/leblanc-prx.pdf), pypdf layout and plain, value found; arXiv text compared (arXiv text (pypdf layout and plain, read locally; version: 1505.02290v2)); single reading, journal check, second reading 2026-10-25",
        "reported_as": "-1.0225 0.0015  [table gives value and uncertainty in separate columns]",
        "note": "Table V, row FN, column U = 4; line 2794 of sources/leblanc.plain.txt (sources/leblanc.plain.txt) | per site (E/t; text: 'we systematically compute the energy per site'); units t; Eq. (1): H = -sum_{i,j,sigma} t_ij (c^dag_{i sigma} c_{j sigma} + H.c.) + U sum_i n_{i up} n_{i down}; t_ij = t nearest neighbour, t' second neighbour; t sets the energy scale; repulsive U > 0 (Sec. II, lines before Table I). Sum convention over (i,j) not spelled out beyond 'sum_{i,j,sigma}'. No chemical-potential term; energies at fixed density n. | Value and uncertainty are separate columns in the table (second number = one-sigma-type uncertainty 'best available within each algorithm', no systematic error: Sec. IX).",
        "secondary_of": null
      }
    },
    {
      "energy": -1.026,
      "sigma": 0.001,
      "method": "CP-AFQMC",
      "method_detail": "supercells 4x16 ... 8x48; U = 8 energies per site −0.7674(7), −0.7658(3), −0.7657(2), −0.7657(4), −0.7660(3) respectively",
      "method_as_published": "constrained-path AFQMC, TABC, supercells 4x16, 8x16, 16x16, 8x32, 8x48 (commensurate with SDW), GHF-type single-determinant trial wave function",
      "bound_type": "extrapolated",
      "bound_type_reason": "extrapolation to an infinite ansatz or to infinite size; not an upper bound (RULES.md 14.4)",
      "how_tdl": "finite-size extrapolation",
      "reference": "Qin et al., 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": "tdl-2026-10-25",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-25",
        "method": "arXiv:1605.09421: journal version: Phys. Rev. B 94, 085103 (2016), PDF from journals.aps.org over the ETH network (journal/qin-prb.pdf), pypdf layout and plain, value found; arXiv text compared (arXiv text (pypdf layout and plain, read locally; version: 1605.09421v2 (the task listed 1605.08794, which is a different (math.PR) paper))); single reading, journal check, second reading 2026-10-25",
        "reported_as": "−1.026(1)",
        "note": "Sec. V.D n = 0.875, text; line 924 of sources/qin.plain.txt (sentence spans lines 923-924) (sources/qin.plain.txt) | per site; units t; Eq. (1): H = K + V = -sum_{i,j,s} t_ij (c^dag_{i,s} c_{j,s} + H.c.) + U sum_i n_{i up} n_{i down}; only nearest-neighbour uniform hopping t_ij = t (t' = 0), t is the energy unit (Sec. II.A, lines after Eq. 1) | Sentence: 'A conservative estimate the ground state energy in the TDL is −0.766(1). Similarly, the TDL value for U = 4 is estimated to be −1.026(1).' Constrained-path (not exact).",
        "secondary_of": null
      }
    },
    {
      "energy": -1.032,
      "sigma": 0.001,
      "method": "CP-AFQMC",
      "method_detail": "supercells up to 16x16 at half filling (finite-size data in Appendix Table IV); see notes",
      "method_as_published": "constrained-path AFQMC, TABC, free-electron trial wave function",
      "bound_type": "extrapolated",
      "bound_type_reason": "extrapolation to an infinite ansatz or to infinite size; not an upper bound (RULES.md 14.4)",
      "how_tdl": "finite-size extrapolation",
      "reference": "Qin et al., 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": "tdl-2026-10-25",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-25",
        "method": "arXiv:1605.09421: journal version: Phys. Rev. B 94, 085103 (2016), PDF from journals.aps.org over the ETH network (journal/qin-prb.pdf), pypdf layout and plain, value found; arXiv text compared (arXiv text (pypdf layout and plain, read locally; version: 1605.09421v2 (the task listed 1605.08794, which is a different (math.PR) paper))); single reading, journal check, second reading 2026-10-25",
        "reported_as": "−1.032(1)",
        "note": "Sec. V.D n = 0.875, bracketed sentence; line 926 of sources/qin.plain.txt (sources/qin.plain.txt) | per site; units t; Eq. (1): H = K + V = -sum_{i,j,s} t_ij (c^dag_{i,s} c_{j,s} + H.c.) + U sum_i n_{i up} n_{i down}; only nearest-neighbour uniform hopping t_ij = t (t' = 0), t is the energy unit (Sec. II.A, lines after Eq. 1) | Bracketed sentence: 'The corresponding energy results using free-electron trial wave-functions are −0.773(1) and −1.032(1), respectively ... expected to be less accurate than the GHF results.' Listed for completeness; the authors prefer −0.766(1) and −1.026(1).",
        "secondary_of": null
      }
    }
  ],
  "url": "https://qmbl.org/instances/#x-Hubbard_square_inf_0_875_4",
  "per_site_divisor": 1,
  "per_site_label": "E/site",
  "best": {
    "exact": null,
    "variational": null,
    "projected": null,
    "extrapolated": {
      "energy": -1.032,
      "sigma": 0.001,
      "method": "CP-AFQMC",
      "ansatz": null,
      "method_detail": "supercells up to 16x16 at half filling (finite-size data in Appendix Table IV); see notes",
      "reference": "Qin et al., Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421",
      "bound_type": "extrapolated",
      "energy_per_site": -1.032
    },
    "approximate": {
      "energy": -1.031,
      "sigma": 0.003,
      "method": "DMET",
      "ansatz": null,
      "reference": "LeBlanc et al., Solutions of the Two Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms, Phys. Rev. X 5, 041041 (2015), arXiv:1505.02290",
      "bound_type": "approximate",
      "energy_per_site": -1.031
    },
    "below_exact": []
  },
  "no_best_reason": null,
  "record": {
    "energy": -1.032,
    "sigma": 0.001,
    "method": "CP-AFQMC",
    "ansatz": null,
    "method_detail": "supercells up to 16x16 at half filling (finite-size data in Appendix Table IV); see notes",
    "reference": "Qin et al., Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421",
    "bound_type": "extrapolated",
    "energy_per_site": -1.032
  },
  "no_record_reason": null
}