{
  "model": "Heisenberg",
  "lattice": "square",
  "n_sites": 36,
  "boundary": "P",
  "params": {},
  "instance_id": "Heisenberg/square_36_P",
  "rows": [
    {
      "energy": -97.7575895972357,
      "sigma": null,
      "energy_variance": null,
      "dof": 36,
      "einf": 0,
      "v_score": null,
      "method": "Exact diagonalization",
      "bound_type": "exact",
      "bound_type_reason": "exact diagonalization|ex",
      "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_36_P/ed_lattice_symmetries.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -97.72162414290578,
      "sigma": null,
      "energy_variance": 1.385137434483113,
      "dof": 36,
      "einf": 0,
      "v_score": 0.005221725221940988,
      "method": "DMRG (bond dimension = 2048)",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_36_P/dmrg.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -96.2702,
      "sigma": 0.0037,
      "energy_variance": 13.9167,
      "dof": 36,
      "einf": 0,
      "v_score": 0.05405738311978833,
      "method": "RBM (alpha = 1)",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_36_P/vmc_rbm.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -96.4427,
      "sigma": 0.0038,
      "energy_variance": 14.7488,
      "dof": 36,
      "einf": 0,
      "v_score": 0.05708479772811618,
      "method": "Jastrow baseline",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_36_P/vmc_jastrow.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -97.756992,
      "sigma": 0.000288,
      "energy_variance": null,
      "dof": 36,
      "einf": 0,
      "v_score": null,
      "method": "RBM + Lanczos recursion",
      "bound_type": "variational",
      "bound_type_reason": "assigned during source verification (RULES.md 4)",
      "reference": "Chen, Hendry, Weinberg & Feiguin, Systematic improvement of neural network quantum states using a Lanczos recursion, arXiv:2206.14307 (2022)",
      "peer_reviewed": null,
      "source": "sweep-tables-2026-09-14",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-14",
        "method": "found in the citing paper's comparison table, then confirmed in the primary source's own PDF (pypdf); no LLM transcription",
        "reported_as": "-0.678868 (+/- 0.000002) per site in S.S units",
        "note": "Read from Table 5 of arXiv:2502.17144 (SLAHM, E/N in S.S units), parsed from the arXiv HTML. Row \"RBM+Lanczos [40]\" = arXiv:2206.14307. CONFIRMED IN PRIMARY SOURCE - arXiv:2206.14307 Table 4: \"6x6 -0.678873(4) 2.51799(6) -0.678868(2) 2.51(2)\" - Energy(QMC) beside Energy(RBM).",
        "secondary_of": null,
        "found_via": "Moss, Wiersema, Hibat-Allah, Carrasquilla & Melko, Phys. Rev. B (2025), arXiv:2502.17144"
      }
    },
    {
      "energy": -97.75728,
      "sigma": 0.00288,
      "energy_variance": null,
      "dof": 36,
      "einf": 0,
      "v_score": null,
      "method": "2D RNN wavefunction (best variational)",
      "bound_type": "variational",
      "bound_type_reason": "assigned during source verification (RULES.md 4)",
      "reference": "Moss, Wiersema, Hibat-Allah, Carrasquilla & Melko, Phys. Rev. B (2025), arXiv:2502.17144",
      "peer_reviewed": true,
      "source": "sweep-tables-2026-09-14",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-14",
        "method": "arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription",
        "reported_as": "-0.67887 (+/- 0.00002) per site in S.S units",
        "note": "Read from Table 5 of arXiv:2502.17144 (SLAHM, E/N in S.S units), parsed from the arXiv HTML. Row \"This work (best RNN)\".",
        "secondary_of": null,
        "found_via": null
      }
    },
    {
      "energy": -97.75008,
      "sigma": 0.00144,
      "energy_variance": null,
      "dof": 36,
      "einf": 0,
      "v_score": null,
      "method": "CNN",
      "bound_type": "variational",
      "bound_type_reason": "assigned during source verification (RULES.md 4)",
      "reference": "Choo, Neupert & Carleo, Two-dimensional frustrated J1-J2 model studied with neural network quantum states, Phys. Rev. B 100, 125124 (2019)",
      "peer_reviewed": null,
      "source": "sweep-tables-2026-09-14",
      "provenance": "secondary",
      "verified": {
        "checked_on": "2026-09-14",
        "method": "arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription",
        "reported_as": "-0.67882 (+/- 0.00001) per site in S.S units",
        "note": "Read from Table 5 of arXiv:2502.17144 (SLAHM, E/N in S.S units), parsed from the arXiv HTML. Row \"CNN [33]\" = Phys. Rev. B 100, 125124 (2019).",
        "secondary_of": "Moss, Wiersema, Hibat-Allah, Carrasquilla & Melko, Phys. Rev. B (2025), arXiv:2502.17144",
        "found_via": null
      }
    },
    {
      "energy": -97.75753488,
      "sigma": 0.00001008,
      "energy_variance": null,
      "dof": 36,
      "einf": 0,
      "v_score": null,
      "method": "2D RNN wavefunction, zero-variance extrapolation",
      "bound_type": "extrapolated",
      "bound_type_reason": "assigned during source verification (RULES.md 4)",
      "reference": "Moss, Wiersema, Hibat-Allah, Carrasquilla & Melko, Phys. Rev. B (2025), arXiv:2502.17144",
      "peer_reviewed": true,
      "source": "sweep-tables-2026-09-14",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-14",
        "method": "arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription",
        "reported_as": "-0.67887177 (+/- 7e-8) per site in S.S units",
        "note": "Read from Table 5 of arXiv:2502.17144 (SLAHM, E/N in S.S units), parsed from the arXiv HTML. Row \"This work (zero-variance)\". The paper states in the Table 3 caption that these extrapolated values can fall below the reference energies and that this is \"an artifact of the zero-variance extrapolation\" - not an achieved energy, so NOT a bound (RULES.md 4).",
        "secondary_of": null,
        "found_via": null
      }
    },
    {
      "energy": -97.7575896,
      "sigma": 0.00000288,
      "energy_variance": null,
      "dof": 36,
      "einf": 0,
      "v_score": null,
      "method": "SSE QMC (stochastic series expansion), T -> 0 converged at beta/L = 32 and 64",
      "bound_type": "exact",
      "bound_type_reason": "sign-problem-free SSE QMC on a bipartite lattice, T -> 0 converged; numerically exact (RULES.md 4)",
      "reference": "Sandvik, High-precision ground state parameters of the two-dimensional spin-1/2 Heisenberg model on the square lattice, arXiv:2601.20189 (2026)",
      "peer_reviewed": false,
      "source": "exact-2026-09-15",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-15",
        "method": "arXiv HTML parsed locally, one value per table cell; no LLM transcription",
        "reported_as": "e0 = -0.67887215(2) per spin (S.S), L = 6, periodic",
        "note": "Table 1, \"Results for different system sizes computed at inverse temperature beta/L = 64 for L > 30 and averaged over beta/L = 32 and beta/L = 64 simulations for L <= 30\", column e0, row L = 6. Pauli total = 4 N e0. The L = 6 value agrees with the exact-diagonalization row on square_36_P to 1.6e-8.",
        "secondary_of": null
      }
    }
  ],
  "url": "https://qmbl.org/i/Heisenberg/square_36_P/",
  "per_site_divisor": 144,
  "per_site_label": "E/N (S.S)",
  "record": {
    "energy": -97.75728,
    "sigma": 0.00288,
    "method": "2D RNN wavefunction (best variational)",
    "reference": "Moss, Wiersema, Hibat-Allah, Carrasquilla & Melko, Phys. Rev. B (2025), arXiv:2502.17144",
    "energy_per_site": -0.67887
  },
  "no_record_reason": null
}