{
  "model": "Heisenberg",
  "lattice": "square",
  "n_sites": 100,
  "boundary": "O",
  "params": {},
  "instance_id": "Heisenberg/square_100_O",
  "rows": [
    {
      "energy": -251.46248,
      "sigma": null,
      "energy_variance": null,
      "dof": 100,
      "einf": 0,
      "v_score": null,
      "method": "QMC",
      "bound_type": null,
      "bound_type_reason": "ambiguous: bare method string, sign-problem-free vs constrained undetermined",
      "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.90.064425)",
      "source": "varbench@2024-10-22",
      "provenance": "imported"
    },
    {
      "energy": -251.443,
      "sigma": 0.001,
      "energy_variance": 0.1102,
      "dof": 100,
      "einf": 0,
      "v_score": 0.00017430205008945538,
      "method": "RNN",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_100_O/vmc_rnn.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -251.4595,
      "sigma": 0.0007,
      "energy_variance": 0.0195,
      "dof": 100,
      "einf": 0,
      "v_score": 0.000030838874246263875,
      "method": "RNN + translational symmetry",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_100_O/vmc_rnn.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -251.4534,
      "sigma": null,
      "energy_variance": 0.3203,
      "dof": 100,
      "einf": 0,
      "v_score": 0.000506572854979116,
      "method": "DMRG (MaxTruncError ~1.87E-6, MaxBondDim=10000, Extrapolated Energy = - 251.4628 +/- 0.0019)",
      "bound_type": "extrapolated",
      "bound_type_reason": "extrapolat",
      "reference": "[code](https://github.com/varbench/methods/blob/main/programs/dmrg_itensors_heisenberg/square_100_O.jl)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -251.45538,
      "sigma": 0.00044,
      "energy_variance": 0.19397,
      "dof": 100,
      "einf": 0,
      "v_score": 0.0003067698696765014,
      "method": "2D Gated RNN",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[paper](https://arxiv.org/abs/2207.14314) [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_100_O/vmc_2DgatedtensorizedRNN.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported"
    },
    {
      "energy": -251.4404,
      "sigma": null,
      "energy_variance": null,
      "dof": 100,
      "einf": 0,
      "v_score": null,
      "method": "PEPS (bond dimension = 10)",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.95.195154)",
      "source": "varbench@2024-10-22",
      "provenance": "imported"
    },
    {
      "energy": -250.346,
      "sigma": 0.0041,
      "energy_variance": 16.83691,
      "dof": 100,
      "einf": 0,
      "v_score": 0.02686464320955393,
      "method": "1D MPS-RNN (bond dimension = 40)",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[paper](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.L032001) [code](https://github.com/cqsl/mps-rnn)",
      "source": "varbench@2024-10-22",
      "provenance": "imported"
    },
    {
      "energy": -251.0788,
      "sigma": 0.0022,
      "energy_variance": 4.928276,
      "dof": 100,
      "einf": 0,
      "v_score": 0.00781762678217357,
      "method": "2D MPS-RNN (bond dimension = 40)",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[paper](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.L032001) [code](https://github.com/cqsl/mps-rnn)",
      "source": "varbench@2024-10-22",
      "provenance": "imported"
    },
    {
      "energy": -251.4112,
      "sigma": 0.0015,
      "energy_variance": 2.289447,
      "dof": 100,
      "einf": 0,
      "v_score": 0.003622107639597973,
      "method": "Tensor-RNN (bond dimension = 40)",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[paper](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.L032001) [code](https://github.com/cqsl/mps-rnn)",
      "source": "varbench@2024-10-22",
      "provenance": "imported"
    },
    {
      "energy": -249.1033,
      "sigma": 0.0051,
      "energy_variance": 26.1306,
      "dof": 100,
      "einf": 0,
      "v_score": 0.04211050214089497,
      "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_100_O/vmc_rbm.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -248.1073,
      "sigma": 0.0056,
      "energy_variance": 31.736,
      "dof": 100,
      "einf": 0,
      "v_score": 0.05155527434775247,
      "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_100_O/vmc_jastrow.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -251.46244,
      "sigma": 0.00008,
      "energy_variance": null,
      "dof": 100,
      "einf": 0,
      "v_score": null,
      "method": "QMC (stochastic series expansion)",
      "bound_type": "exact",
      "bound_type_reason": "assigned during source verification (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": 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": "-0.6286561 (+/- 2e-7) per site in S.S units",
        "note": "Read from Table 2 of arXiv:2605.13807 (2D Heisenberg, square lattice, OBC, E/N in S.S units), parsed from the arXiv HTML. Column \"QMC [49]\" = arXiv:2601.20189. The square-lattice Heisenberg AFM is bipartite and sign-problem-free, so SSE QMC is numerically exact here (RULES.md 4).",
        "secondary_of": "Parallel Scan Recurrent Neural Quantum States, arXiv:2605.13807",
        "found_via": null
      }
    },
    {
      "energy": -251.4624,
      "sigma": 0.0036,
      "energy_variance": null,
      "dof": 100,
      "einf": 0,
      "v_score": null,
      "method": "2D tensorized RNN (symmetry + annealing)",
      "bound_type": "variational",
      "bound_type_reason": "assigned during source verification (RULES.md 4)",
      "reference": "Hibat-Allah, Melko & Carrasquilla, Supplementing Recurrent Neural Network Wave Functions with Symmetry and Annealing to Improve Accuracy, arXiv:2207.14314 (2024)",
      "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": "-0.628656 (+/- 0.000009) per site in S.S units",
        "note": "Read from Table 2 of arXiv:2605.13807 (2D Heisenberg, square lattice, OBC, E/N in S.S units), parsed from the arXiv HTML. Column \"2D TRNN [40]\". The same value appears in arXiv:2502.17144 Table 5 as \"This work (best RNN)\", OBC 10x10. It sits 1e-7 below the QMC reference, well inside its own 9e-6 error bar.",
        "secondary_of": "Parallel Scan Recurrent Neural Quantum States, arXiv:2605.13807",
        "found_via": null
      }
    },
    {
      "energy": -251.4548,
      "sigma": 0.0016,
      "energy_variance": null,
      "dof": 100,
      "einf": 0,
      "v_score": null,
      "method": "2D minGRU (3 layers, c4v symmetry, parallel scan)",
      "bound_type": "variational",
      "bound_type_reason": "assigned during source verification (RULES.md 4)",
      "reference": "Parallel Scan Recurrent Neural Quantum States, arXiv:2605.13807",
      "peer_reviewed": false,
      "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": "-0.628637 (+/- 0.000004) per site in S.S units",
        "note": "Read from Table 2 of arXiv:2605.13807 (2D Heisenberg, square lattice, OBC, E/N in S.S units), parsed from the arXiv HTML. Column \"2D minGRU (ours)\".",
        "secondary_of": null,
        "found_via": null
      }
    },
    {
      "energy": -251.4508,
      "sigma": 0.0004,
      "energy_variance": null,
      "dof": 100,
      "einf": 0,
      "v_score": null,
      "method": "PixelCNN (deep autoregressive)",
      "bound_type": "variational",
      "bound_type_reason": "assigned during source verification (RULES.md 4)",
      "reference": "Sharir, Levine, Wies, Carleo & Shashua, Deep Autoregressive Models for the Efficient Variational Simulation of Many-Body Quantum Systems, Phys. Rev. Lett. 124, 020503 (2020)",
      "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": "-0.628627 (+/- 0.000001) per site in S.S units",
        "note": "Read from Table 2 of arXiv:2605.13807 (2D Heisenberg, square lattice, OBC, E/N in S.S units), parsed from the arXiv HTML. Column \"PixelCNN [52]\" = Phys. Rev. Lett. 124, 020503 (2020).",
        "secondary_of": "Parallel Scan Recurrent Neural Quantum States, arXiv:2605.13807",
        "found_via": null
      }
    },
    {
      "energy": -251.4404,
      "sigma": 0.0008,
      "energy_variance": null,
      "dof": 100,
      "einf": 0,
      "v_score": null,
      "method": "Finite PEPS, gradient optimization",
      "bound_type": "variational",
      "bound_type_reason": "assigned during source verification (RULES.md 4)",
      "reference": "Liu, Dong, Han, Guo & He, Gradient optimization of finite projected entangled pair states, Phys. Rev. B 95, 195154 (2017)",
      "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": "-0.628601 (+/- 0.000002) per site in S.S units",
        "note": "Read from Table 2 of arXiv:2605.13807 (2D Heisenberg, square lattice, OBC, E/N in S.S units), parsed from the arXiv HTML. Column \"PEPS [33]\" = Phys. Rev. B 95, 195154 (2017). Finite PEPS on the 10x10 lattice, not an infinite-lattice value.",
        "secondary_of": "Parallel Scan Recurrent Neural Quantum States, arXiv:2605.13807",
        "found_via": null
      }
    }
  ],
  "url": "https://qmbl.org/i/Heisenberg/square_100_O/",
  "per_site_divisor": 400,
  "per_site_label": "E/N (S.S)",
  "record": {
    "energy": -251.4624,
    "sigma": 0.0036,
    "method": "2D tensorized RNN (symmetry + annealing)",
    "reference": "Hibat-Allah, Melko & Carrasquilla, Supplementing Recurrent Neural Network Wave Functions with Symmetry and Annealing to Improve Accuracy, arXiv:2207.14314 (2024)",
    "energy_per_site": -0.628656
  },
  "no_record_reason": null
}