{
  "model": "Heisenberg",
  "lattice": "checkerboard",
  "n_sites": "inf",
  "boundary": "TDL",
  "params": {
    "S": 1
  },
  "instance_id": "Heisenberg/checkerboard_inf_S1",
  "rows": [
    {
      "energy": -6.132,
      "sigma": 0.008,
      "method": "NLCE",
      "method_detail": "numerical linked-cluster expansion at T = 0 built on the hexagons (squares on the checkerboard), second order",
      "method_as_published": "NLCE2: second-order NLCE; error = difference between the second and the third order (S II)",
      "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": "Schäfer et al., Abundance of Hard-Hexagon Crystals in the Quantum Pyrochlore Antiferromagnet, Phys. Rev. Lett. 131, 096702 (2023), arXiv:2210.07235",
      "peer_reviewed": true,
      "source": "tdl-2026-10-26",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-26",
        "method": "arXiv:2210.07235: journal version: Phys. Rev. Lett. 131, 096702 (2023), PDF from journals.aps.org over the ETH network (journal/schaefer-prl.pdf), value found; arXiv text (pypdf plain, read locally; version: 2210.07235v3 of 1 Sep 2023, with the Supplemental Material); Fig. 1 read from the rendered page; single reading, second reading 2026-10-27 (verify-schaefer.json)",
        "reported_as": "NLCE2 −1.533(2)",
        "note": "Table I, Checkerboard, S = 1 (line 152 of sources/2210.07235.txt) | per site; units J, Table I caption 'Ground-state energies in units of J'; Heisenberg model on the checkerboard lattice (planar pyrochlore: squares, every other one crossed), all bonds J (Fig. 1(c)); spin S = 1; the printed error is a truncation estimate",
        "secondary_of": null
      }
    },
    {
      "energy": -6.128,
      "sigma": 0.004,
      "method": "DMRG",
      "method_detail": "snake-path DMRG (ITensor) on periodic clusters N = 64, 100, 144 (L = 4, 5, 6; Table S3), extrapolated linearly in 1/chi, then to N -> infinity with a quadratic fit (Fig. S4)",
      "method_as_published": "DMRG (Table I); 'We first extrapolate to infinite bond dimension and then to the thermodynamic limit' (S III B)",
      "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": "Schäfer et al., Abundance of Hard-Hexagon Crystals in the Quantum Pyrochlore Antiferromagnet, Phys. Rev. Lett. 131, 096702 (2023), arXiv:2210.07235",
      "peer_reviewed": true,
      "source": "tdl-2026-10-26",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-26",
        "method": "arXiv:2210.07235: journal version: Phys. Rev. Lett. 131, 096702 (2023), PDF from journals.aps.org over the ETH network (journal/schaefer-prl.pdf), value found; arXiv text (pypdf plain, read locally; version: 2210.07235v3 of 1 Sep 2023, with the Supplemental Material); Fig. 1 read from the rendered page; single reading, second reading 2026-10-27 (verify-schaefer.json)",
        "reported_as": "DMRG −1.532(1)",
        "note": "Table I, Checkerboard, S = 1 (line 152 of sources/2210.07235.txt) | per site; units J, Table I caption 'Ground-state energies in units of J'; Heisenberg model on the checkerboard lattice (planar pyrochlore: squares, every other one crossed), all bonds J (Fig. 1(c)); spin S = 1; error from the difference between the largest cluster and the extrapolated value (S III B)",
        "secondary_of": null
      }
    },
    {
      "energy": -6.1356,
      "sigma": 0.002,
      "method": "variational wave function",
      "method_detail": "dressed hard-hexagon valence-bond crystal exp(-alpha0 V)|Psi0>, one parameter alpha0 (Table S1), energy of the infinite lattice evaluated by a linked-cluster expansion in the edge terms V (order seven, S I C)",
      "method_as_published": "E_alpha0: variational wave function Eq. (2) with optimization of the parameter alpha; 'a strict upper bound on the ground-state energy of Eq. (1) in the thermodynamic limit'",
      "bound_type": "variational",
      "bound_type_reason": "variational state of the infinite system at a fixed ansatz size; energy per site is an upper bound up to the stated contraction (RULES.md 14.4)",
      "how_tdl": "infinite system",
      "reference": "Schäfer et al., Abundance of Hard-Hexagon Crystals in the Quantum Pyrochlore Antiferromagnet, Phys. Rev. Lett. 131, 096702 (2023), arXiv:2210.07235",
      "peer_reviewed": true,
      "source": "tdl-2026-10-26",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-26",
        "method": "arXiv:2210.07235: journal version: Phys. Rev. Lett. 131, 096702 (2023), PDF from journals.aps.org over the ETH network (journal/schaefer-prl.pdf), value found; arXiv text (pypdf plain, read locally; version: 2210.07235v3 of 1 Sep 2023, with the Supplemental Material); Fig. 1 read from the rendered page; single reading, second reading 2026-10-27 (verify-schaefer.json)",
        "reported_as": "E_alpha0 −1.5339(5)",
        "note": "Table I, Checkerboard, S = 1 (line 152 of sources/2210.07235.txt) | per site; units J, Table I caption 'Ground-state energies in units of J'; Heisenberg model on the checkerboard lattice (planar pyrochlore: squares, every other one crossed), all bonds J (Fig. 1(c)); spin S = 1; error = difference between the largest and second-largest expansion order at alpha0 (S I C); some top-order coefficients are approximate (Table S2 asterisks)",
        "secondary_of": null
      }
    }
  ],
  "url": "https://qmbl.org/instances/#x-Heisenberg_checkerboard_inf_S1",
  "per_site_divisor": 4,
  "per_site_label": "E/N (S.S)",
  "best": {
    "exact": null,
    "variational": {
      "energy": -6.1356,
      "sigma": 0.002,
      "method": "variational wave function",
      "ansatz": null,
      "method_detail": "dressed hard-hexagon valence-bond crystal exp(-alpha0 V)|Psi0>, one parameter alpha0 (Table S1), energy of the infinite lattice evaluated by a linked-cluster expansion in the edge terms V (order seven, S I C)",
      "reference": "Schäfer et al., Abundance of Hard-Hexagon Crystals in the Quantum Pyrochlore Antiferromagnet, Phys. Rev. Lett. 131, 096702 (2023), arXiv:2210.07235",
      "bound_type": "variational",
      "energy_per_site": -1.5339
    },
    "projected": null,
    "extrapolated": {
      "energy": -6.128,
      "sigma": 0.004,
      "method": "DMRG",
      "ansatz": null,
      "method_detail": "snake-path DMRG (ITensor) on periodic clusters N = 64, 100, 144 (L = 4, 5, 6; Table S3), extrapolated linearly in 1/chi, then to N -> infinity with a quadratic fit (Fig. S4)",
      "reference": "Schäfer et al., Abundance of Hard-Hexagon Crystals in the Quantum Pyrochlore Antiferromagnet, Phys. Rev. Lett. 131, 096702 (2023), arXiv:2210.07235",
      "bound_type": "extrapolated",
      "energy_per_site": -1.532
    },
    "approximate": {
      "energy": -6.132,
      "sigma": 0.008,
      "method": "NLCE",
      "ansatz": null,
      "method_detail": "numerical linked-cluster expansion at T = 0 built on the hexagons (squares on the checkerboard), second order",
      "reference": "Schäfer et al., Abundance of Hard-Hexagon Crystals in the Quantum Pyrochlore Antiferromagnet, Phys. Rev. Lett. 131, 096702 (2023), arXiv:2210.07235",
      "bound_type": "approximate",
      "energy_per_site": -1.533
    },
    "below_exact": []
  },
  "no_best_reason": null,
  "record": {
    "energy": -6.1356,
    "sigma": 0.002,
    "method": "variational wave function",
    "ansatz": null,
    "method_detail": "dressed hard-hexagon valence-bond crystal exp(-alpha0 V)|Psi0>, one parameter alpha0 (Table S1), energy of the infinite lattice evaluated by a linked-cluster expansion in the edge terms V (order seven, S I C)",
    "reference": "Schäfer et al., Abundance of Hard-Hexagon Crystals in the Quantum Pyrochlore Antiferromagnet, Phys. Rev. Lett. 131, 096702 (2023), arXiv:2210.07235",
    "bound_type": "variational",
    "energy_per_site": -1.5339
  },
  "no_record_reason": null
}