{
  "model": "J1J2",
  "lattice": "square",
  "n_sites": 64,
  "boundary": "P",
  "params": {
    "J2": 0.5
  },
  "instance_id": "J1J2/square_64_P_0.5",
  "rows": [
    {
      "energy": -127.73465,
      "sigma": 0.00048,
      "energy_variance": 0.25,
      "dof": 64,
      "einf": 0,
      "v_score": 0.0009806240449692198,
      "method": "RBM+PP with momentum (K=0), spin-parity (even S), and point-group (A1) projections, 16 hidden units",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[paper](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.031034)",
      "source": "varbench@2024-10-22",
      "provenance": "imported"
    },
    {
      "energy": -127.65838,
      "sigma": 0.00059,
      "energy_variance": 1.85,
      "dof": 64,
      "einf": 0,
      "v_score": 0.0072652915127413426,
      "method": "RBM with momentum (K=0), spin-parity (even S), and point-group (A1) projections, 96 hidden units",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[paper](https://iopscience.iop.org/article/10.1088/1361-648X/abe268)",
      "source": "varbench@2024-10-22",
      "provenance": "imported"
    },
    {
      "energy": -127.115,
      "sigma": 0.004,
      "energy_variance": 6.92,
      "dof": 64,
      "einf": 0,
      "v_score": 0.027408954179737793,
      "method": "VMC with projected BCS (Z2 spin liquid)",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/J1J2/square_64_P_0.5/vmc_gutzwiller.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -126.46793123817692,
      "sigma": null,
      "energy_variance": 8.65276397960406,
      "dof": 64,
      "einf": 0,
      "v_score": 0.03462374198896008,
      "method": "DMRG (bond dimension = 1024)",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/J1J2/square_64_P_0.5/dmrg.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -123.5715,
      "sigma": 0.0041,
      "energy_variance": 17.22527,
      "dof": 64,
      "einf": 0,
      "v_score": 0.07219537474821448,
      "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/J1J2/square_64_P_0.5/vmc_rbm.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -121.4943,
      "sigma": 0.0067,
      "energy_variance": 45.77505,
      "dof": 64,
      "einf": 0,
      "v_score": 0.19847097127429694,
      "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/J1J2/square_64_P_0.5/vmc_jastrow.sh)",
      "source": "varbench@2024-10-22",
      "provenance": "imported",
      "baseline": true
    },
    {
      "energy": -127.634,
      "sigma": 0.0128,
      "energy_variance": 1.208,
      "dof": 64,
      "einf": 0,
      "v_score": 0.0047458515362703855,
      "method": "ClebschTree",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.045123)",
      "source": "varbench@2024-10-22",
      "provenance": "imported"
    },
    {
      "energy": -128.0256,
      "sigma": 0.0256,
      "energy_variance": 1.4336,
      "dof": 64,
      "einf": 0,
      "v_score": 0.005597760671820845,
      "method": "Holographic Quantum Transformer (HQT)",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)",
      "reference": "Holographic Quantum Transformer, arXiv:2607.00398 (conference proceedings)",
      "peer_reviewed": true,
      "source": "sweep-2026-09-13",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-13",
        "method": "arXiv HTML parsed locally, no LLM transcription",
        "reported_as": "-0.5001 (+/- 0.0001) per site in S.S units",
        "note": "Abstract: \"HQT reaches a ground-state energy per site of -0.5001(1)\" on 8x8 at J2=0.5. That is 1.1e-3 below the best known variational energy for this instance (RBM+PP, -0.4989635), on a well-studied lattice. See the defect flag.",
        "secondary_of": null
      },
      "defect": {
        "flag": "energy-variance-inconsistent",
        "finding": "Claims E/N = -0.5001, below RBM+PP's -0.4989635, while reporting a variance that gives a V-score of 5.6e-3 against RBM+PP's 9.81e-4 - 5.7x worse. Taken at face value a state further from an eigenstate should not be lower in energy: by the V-score calibration (rel. err ~ V/63) this energy would sit ~1.2e-3 ABOVE where it is reported. That is grounds for objection, not proof of error. The inference assumes the lower-variance competitor sits near the ground state; a state pinned near a competing, excited configuration can have a small variance and a large energy error (RULES.md 9, the HFDS stripe case). Nothing suggests RBM+PP is such a state here, but the variance alone cannot rule it out.",
        "diagnosis": "The reported energy is inconsistent with the paper's OWN reported variance. At 8x8 the claimed E/N = -0.5001 beats RBM+PP's -0.4989635 while the reported variance (sigma^2 = 1.4e-3 per site, S.S units) gives a V-score of 5.6e-3 against RBM+PP's 9.81e-4 - 5.7x worse. Energy and variance move together - a state further from an eigenstate cannot be lower in energy - so by the V-score calibration this energy should be ~1.2e-3 higher than claimed.",
        "ruled_out": "NOT 'below the exact ground state'. There is no exact reference at 8x8 or 10x10: ED for this model reaches about 6x6, and the paper correctly uses 6x6 ED (-0.5038) as its only exact anchor. Chen & Heyl's -0.497715(9) is a zero-variance extrapolation, not a bound, so a lower variational energy would only mean the extrapolation carries systematic error. The sigma^2 is per site, so the V-score is 5.6e-3 and not a range. Separately, the paper is internally inconsistent about it: Table 1 gives sigma^2 = 1.4e-3 for the 8x8 run while the J2 scan lists 0.0034 at J2 = 0.50, which would make the V-score 1.4e-2 and the contradiction larger. No variance is reported at 10x10 at all, so that row's flag rests only on it being an unclaimed record.",
        "evidence": "arXiv:2607.00398 Tables 1-3; V-scores recomputed from this instance's own rows."
      }
    }
  ],
  "url": "https://qmbl.org/i/J1J2/square_64_P_0.5/",
  "per_site_divisor": 256,
  "per_site_label": "E/N (S.S)",
  "record": {
    "energy": -127.73465,
    "sigma": 0.00048,
    "method": "RBM+PP with momentum (K=0), spin-parity (even S), and point-group (A1) projections, 16 hidden units",
    "reference": "[paper](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.031034)",
    "energy_per_site": -0.4989634765625
  },
  "no_record_reason": null
}