{
  "model": "Hubbard",
  "lattice": "square",
  "n_sites": "inf",
  "boundary": "TDL",
  "params": {
    "U": 6,
    "n": 0.875
  },
  "instance_id": "Hubbard/square_inf_0.875_6",
  "rows": [
    {
      "energy": -0.863,
      "sigma": 0.013,
      "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": "-0.863 0.013  [table gives value and uncertainty in separate columns]",
        "note": "Table V, row DMET, column U = 6; 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": -0.854,
      "sigma": 0.002,
      "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": "-0.854 0.002  [table gives value and uncertainty in separate columns]",
        "note": "Table V, row FN, column U = 6; 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": -0.8679,
      "sigma": 0.0007,
      "method": "DMET",
      "method_detail": "2 × 2",
      "method_as_published": "DMET, size 2 × 2, stripe wavelength d-wave",
      "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": "Zheng et al., Stripe order in the underdoped region of the two-dimensional Hubbard model, Science 358, 1155-1160 (2017), arXiv:1701.00054",
      "peer_reviewed": true,
      "source": "tdl-2026-10-25",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-25",
        "method": "arXiv:1701.00054: arXiv text and the journal's Supplementary Materials (pypdf layout and plain, read locally; version: 1701.00054v3 (main text); Supplementary Materials = Science SM file aan7127_Zheng_SM.pdf from CaltechAUTHORS record 73807 (the arXiv v3 PDF does not contain the SM; arXiv v1/v2 contain an older SM with Tables 1-7, different from Science Tables S1-S10)); journal version not read (publisher returned 403 to scripted access); single reading",
        "reported_as": "−0.8679 (error 0.0007); table columns 'Energy (t)' and 'Error (t)'",
        "note": "Table S3 (U/t = 6), row 'DMET 2 × 2'; line 192 of sources/zheng_science_SM.plain.txt (layout txt has the same digits) (sources/zheng_science_SM.plain.txt) | per site; units t; Main text Eq. (1) (sources/zheng.plain.txt, p. 2): H = -sum_{<ij>, sigma in {up,down}} t a^dag_{i sigma} a_{j sigma} + U sum_i n_{i up} n_{i down}; t, U are 'the kinetic and repulsion energies'; the Hermitian-conjugate term is not printed, bond counting not stated | The minus sign of the Science SM energies is extracted as a NUL byte (not as a minus character); it is present in the rendered PDF page (checked visually on pages 6-9: all energies are negative) and in the arXiv v2 SM text for rows that exist there. Value digits as in extracted text. Doping 1/8 is from main text (Fig. 6 caption: 'at 1/8 doping for weaker and stronger couplings'); the SM table caption does not state the doping.",
        "secondary_of": null
      }
    },
    {
      "energy": -0.85867,
      "sigma": 0.00004,
      "method": "DMET",
      "method_detail": "3 × 2",
      "method_as_published": "DMET, size 3 × 2, stripe wavelength 3",
      "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": "Zheng et al., Stripe order in the underdoped region of the two-dimensional Hubbard model, Science 358, 1155-1160 (2017), arXiv:1701.00054",
      "peer_reviewed": true,
      "source": "tdl-2026-10-25",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-25",
        "method": "arXiv:1701.00054: arXiv text and the journal's Supplementary Materials (pypdf layout and plain, read locally; version: 1701.00054v3 (main text); Supplementary Materials = Science SM file aan7127_Zheng_SM.pdf from CaltechAUTHORS record 73807 (the arXiv v3 PDF does not contain the SM; arXiv v1/v2 contain an older SM with Tables 1-7, different from Science Tables S1-S10)); journal version not read (publisher returned 403 to scripted access); single reading",
        "reported_as": "−0.85867 (error 0.00004); table columns 'Energy (t)' and 'Error (t)'",
        "note": "Table S3 (U/t = 6), row 'DMET 3 × 2'; line 193 of sources/zheng_science_SM.plain.txt (layout txt has the same digits) (sources/zheng_science_SM.plain.txt) | per site; units t; Main text Eq. (1) (sources/zheng.plain.txt, p. 2): H = -sum_{<ij>, sigma in {up,down}} t a^dag_{i sigma} a_{j sigma} + U sum_i n_{i up} n_{i down}; t, U are 'the kinetic and repulsion energies'; the Hermitian-conjugate term is not printed, bond counting not stated | The minus sign of the Science SM energies is extracted as a NUL byte (not as a minus character); it is present in the rendered PDF page (checked visually on pages 6-9: all energies are negative) and in the arXiv v2 SM text for rows that exist there. Value digits as in extracted text. Doping 1/8 is from main text (Fig. 6 caption: 'at 1/8 doping for weaker and stronger couplings'); the SM table caption does not state the doping.",
        "secondary_of": null
      }
    },
    {
      "energy": -0.8589,
      "sigma": 0.00004,
      "method": "DMET",
      "method_detail": "4 × 2",
      "method_as_published": "DMET, size 4 × 2, stripe wavelength 4",
      "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": "Zheng et al., Stripe order in the underdoped region of the two-dimensional Hubbard model, Science 358, 1155-1160 (2017), arXiv:1701.00054",
      "peer_reviewed": true,
      "source": "tdl-2026-10-25",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-25",
        "method": "arXiv:1701.00054: arXiv text and the journal's Supplementary Materials (pypdf layout and plain, read locally; version: 1701.00054v3 (main text); Supplementary Materials = Science SM file aan7127_Zheng_SM.pdf from CaltechAUTHORS record 73807 (the arXiv v3 PDF does not contain the SM; arXiv v1/v2 contain an older SM with Tables 1-7, different from Science Tables S1-S10)); journal version not read (publisher returned 403 to scripted access); single reading",
        "reported_as": "−0.85890 (error 0.00004); table columns 'Energy (t)' and 'Error (t)'",
        "note": "Table S3 (U/t = 6), row 'DMET 4 × 2'; line 194 of sources/zheng_science_SM.plain.txt (layout txt has the same digits) (sources/zheng_science_SM.plain.txt) | per site; units t; Main text Eq. (1) (sources/zheng.plain.txt, p. 2): H = -sum_{<ij>, sigma in {up,down}} t a^dag_{i sigma} a_{j sigma} + U sum_i n_{i up} n_{i down}; t, U are 'the kinetic and repulsion energies'; the Hermitian-conjugate term is not printed, bond counting not stated | The minus sign of the Science SM energies is extracted as a NUL byte (not as a minus character); it is present in the rendered PDF page (checked visually on pages 6-9: all energies are negative) and in the arXiv v2 SM text for rows that exist there. Value digits as in extracted text. Doping 1/8 is from main text (Fig. 6 caption: 'at 1/8 doping for weaker and stronger couplings'); the SM table caption does not state the doping.",
        "secondary_of": null
      }
    },
    {
      "energy": -0.86836,
      "sigma": 0.00001,
      "method": "DMET",
      "method_detail": "5 × 2",
      "method_as_published": "DMET, size 5 × 2, stripe wavelength 5",
      "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": "Zheng et al., Stripe order in the underdoped region of the two-dimensional Hubbard model, Science 358, 1155-1160 (2017), arXiv:1701.00054",
      "peer_reviewed": true,
      "source": "tdl-2026-10-25",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-25",
        "method": "arXiv:1701.00054: arXiv text and the journal's Supplementary Materials (pypdf layout and plain, read locally; version: 1701.00054v3 (main text); Supplementary Materials = Science SM file aan7127_Zheng_SM.pdf from CaltechAUTHORS record 73807 (the arXiv v3 PDF does not contain the SM; arXiv v1/v2 contain an older SM with Tables 1-7, different from Science Tables S1-S10)); journal version not read (publisher returned 403 to scripted access); single reading",
        "reported_as": "−0.86836 (error 0.00001); table columns 'Energy (t)' and 'Error (t)'",
        "note": "Table S3 (U/t = 6), row 'DMET 5 × 2'; line 195 of sources/zheng_science_SM.plain.txt (layout txt has the same digits) (sources/zheng_science_SM.plain.txt) | per site; units t; Main text Eq. (1) (sources/zheng.plain.txt, p. 2): H = -sum_{<ij>, sigma in {up,down}} t a^dag_{i sigma} a_{j sigma} + U sum_i n_{i up} n_{i down}; t, U are 'the kinetic and repulsion energies'; the Hermitian-conjugate term is not printed, bond counting not stated | The minus sign of the Science SM energies is extracted as a NUL byte (not as a minus character); it is present in the rendered PDF page (checked visually on pages 6-9: all energies are negative) and in the arXiv v2 SM text for rows that exist there. Value digits as in extracted text. Doping 1/8 is from main text (Fig. 6 caption: 'at 1/8 doping for weaker and stronger couplings'); the SM table caption does not state the doping.",
        "secondary_of": null
      }
    },
    {
      "energy": -0.87247,
      "sigma": 0.00001,
      "method": "DMET",
      "method_detail": "6 × 2 (SF)",
      "method_as_published": "DMET, size 6 × 2 (SF), stripe wavelength 6",
      "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": "Zheng et al., Stripe order in the underdoped region of the two-dimensional Hubbard model, Science 358, 1155-1160 (2017), arXiv:1701.00054",
      "peer_reviewed": true,
      "source": "tdl-2026-10-25",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-25",
        "method": "arXiv:1701.00054: arXiv text and the journal's Supplementary Materials (pypdf layout and plain, read locally; version: 1701.00054v3 (main text); Supplementary Materials = Science SM file aan7127_Zheng_SM.pdf from CaltechAUTHORS record 73807 (the arXiv v3 PDF does not contain the SM; arXiv v1/v2 contain an older SM with Tables 1-7, different from Science Tables S1-S10)); journal version not read (publisher returned 403 to scripted access); single reading",
        "reported_as": "−0.87247 (error 0.00001); table columns 'Energy (t)' and 'Error (t)'",
        "note": "Table S3 (U/t = 6), row 'DMET 6 × 2 (SF)'; line 196 of sources/zheng_science_SM.plain.txt (layout txt has the same digits) (sources/zheng_science_SM.plain.txt) | per site; units t; Main text Eq. (1) (sources/zheng.plain.txt, p. 2): H = -sum_{<ij>, sigma in {up,down}} t a^dag_{i sigma} a_{j sigma} + U sum_i n_{i up} n_{i down}; t, U are 'the kinetic and repulsion energies'; the Hermitian-conjugate term is not printed, bond counting not stated | The minus sign of the Science SM energies is extracted as a NUL byte (not as a minus character); it is present in the rendered PDF page (checked visually on pages 6-9: all energies are negative) and in the arXiv v2 SM text for rows that exist there. Value digits as in extracted text. Doping 1/8 is from main text (Fig. 6 caption: 'at 1/8 doping for weaker and stronger couplings'); the SM table caption does not state the doping.",
        "secondary_of": null
      }
    },
    {
      "energy": -0.87363,
      "sigma": 0.00002,
      "method": "DMET",
      "method_detail": "7 × 2",
      "method_as_published": "DMET, size 7 × 2, stripe wavelength 7",
      "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": "Zheng et al., Stripe order in the underdoped region of the two-dimensional Hubbard model, Science 358, 1155-1160 (2017), arXiv:1701.00054",
      "peer_reviewed": true,
      "source": "tdl-2026-10-25",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-25",
        "method": "arXiv:1701.00054: arXiv text and the journal's Supplementary Materials (pypdf layout and plain, read locally; version: 1701.00054v3 (main text); Supplementary Materials = Science SM file aan7127_Zheng_SM.pdf from CaltechAUTHORS record 73807 (the arXiv v3 PDF does not contain the SM; arXiv v1/v2 contain an older SM with Tables 1-7, different from Science Tables S1-S10)); journal version not read (publisher returned 403 to scripted access); single reading",
        "reported_as": "−0.87363 (error 0.00002); table columns 'Energy (t)' and 'Error (t)'",
        "note": "Table S3 (U/t = 6), row 'DMET 7 × 2'; line 197 of sources/zheng_science_SM.plain.txt (layout txt has the same digits) (sources/zheng_science_SM.plain.txt) | per site; units t; Main text Eq. (1) (sources/zheng.plain.txt, p. 2): H = -sum_{<ij>, sigma in {up,down}} t a^dag_{i sigma} a_{j sigma} + U sum_i n_{i up} n_{i down}; t, U are 'the kinetic and repulsion energies'; the Hermitian-conjugate term is not printed, bond counting not stated | The minus sign of the Science SM energies is extracted as a NUL byte (not as a minus character); it is present in the rendered PDF page (checked visually on pages 6-9: all energies are negative) and in the arXiv v2 SM text for rows that exist there. Value digits as in extracted text. Doping 1/8 is from main text (Fig. 6 caption: 'at 1/8 doping for weaker and stronger couplings'); the SM table caption does not state the doping.",
        "secondary_of": null
      }
    },
    {
      "energy": -0.87667,
      "sigma": 0.0007,
      "method": "DMET",
      "method_detail": "8 × 2 (SF)",
      "method_as_published": "DMET, size 8 × 2 (SF), stripe wavelength 8",
      "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": "Zheng et al., Stripe order in the underdoped region of the two-dimensional Hubbard model, Science 358, 1155-1160 (2017), arXiv:1701.00054",
      "peer_reviewed": true,
      "source": "tdl-2026-10-25",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-25",
        "method": "arXiv:1701.00054: arXiv text and the journal's Supplementary Materials (pypdf layout and plain, read locally; version: 1701.00054v3 (main text); Supplementary Materials = Science SM file aan7127_Zheng_SM.pdf from CaltechAUTHORS record 73807 (the arXiv v3 PDF does not contain the SM; arXiv v1/v2 contain an older SM with Tables 1-7, different from Science Tables S1-S10)); journal version not read (publisher returned 403 to scripted access); single reading",
        "reported_as": "−0.87667 (error 0.0007); table columns 'Energy (t)' and 'Error (t)'",
        "note": "Table S3 (U/t = 6), row 'DMET 8 × 2 (SF)'; line 198 of sources/zheng_science_SM.plain.txt (layout txt has the same digits) (sources/zheng_science_SM.plain.txt) | per site; units t; Main text Eq. (1) (sources/zheng.plain.txt, p. 2): H = -sum_{<ij>, sigma in {up,down}} t a^dag_{i sigma} a_{j sigma} + U sum_i n_{i up} n_{i down}; t, U are 'the kinetic and repulsion energies'; the Hermitian-conjugate term is not printed, bond counting not stated | The minus sign of the Science SM energies is extracted as a NUL byte (not as a minus character); it is present in the rendered PDF page (checked visually on pages 6-9: all energies are negative) and in the arXiv v2 SM text for rows that exist there. Value digits as in extracted text. Doping 1/8 is from main text (Fig. 6 caption: 'at 1/8 doping for weaker and stronger couplings'); the SM table caption does not state the doping.",
        "secondary_of": null
      }
    }
  ],
  "url": "https://qmbl.org/instances/#x-Hubbard_square_inf_0_875_6",
  "per_site_divisor": 1,
  "per_site_label": "E/site",
  "best": {
    "exact": null,
    "variational": null,
    "projected": null,
    "extrapolated": {
      "energy": -0.854,
      "sigma": 0.002,
      "method": "FN-DMC",
      "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": "extrapolated",
      "energy_per_site": -0.854
    },
    "approximate": {
      "energy": -0.87667,
      "sigma": 0.0007,
      "method": "DMET",
      "ansatz": null,
      "method_detail": "8 × 2 (SF)",
      "reference": "Zheng et al., Stripe order in the underdoped region of the two-dimensional Hubbard model, Science 358, 1155-1160 (2017), arXiv:1701.00054",
      "bound_type": "approximate",
      "energy_per_site": -0.87667
    },
    "below_exact": []
  },
  "no_best_reason": null,
  "record": {
    "energy": -0.854,
    "sigma": 0.002,
    "method": "FN-DMC",
    "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": "extrapolated",
    "energy_per_site": -0.854
  },
  "no_record_reason": null
}