{
  "model": "Heisenberg",
  "lattice": "kagome-6x6",
  "n_sites": 108,
  "boundary": "P",
  "params": {},
  "instance_id": "Heisenberg/kagome-6x6_108_P",
  "rows": [
    {
      "energy": -192.72,
      "sigma": null,
      "energy_variance": null,
      "dof": 108,
      "einf": 0,
      "v_score": null,
      "method": "GCNN, spinon pair-density-wave state (chi0, P_Z2 = -1, q = (pi, pi/sqrt3))",
      "bound_type": "variational",
      "bound_type_reason": "assigned during source verification (RULES.md 4)",
      "reference": "Ðurić, Chung, Yang & Sengupta, Spin-1/2 Kagome Heisenberg Antiferromagnet: Machine Learning Discovery of the Spinon Pair-Density-Wave Ground State, Phys. Rev. X 15, 011047 (2025), arXiv:2401.02866",
      "peer_reviewed": true,
      "source": "sweep-pdf-2026-09-14",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-14",
        "method": "arXiv PDF text extracted locally with pypdf, no LLM transcription",
        "reported_as": "-0.4461111111111111 per site in S.S units",
        "note": "Sec. IV: \"The obtained ground state energy E0 = -48.18(0) is ~1.78% lower than the ground state energy obtained with DMRG calculations (E_DMRG = -47.33964)\". Total in S.S units; E/N = -0.4461111. The stated error (0) is not a usable error bar, so no sigma is recorded. DISPUTED - see the defect flag.",
        "secondary_of": null
      },
      "defect": {
        "flag": "sampling-nonergodic",
        "finding": "Published in Phys. Rev. X 15, 011047 (2025) as a ground state 1.78% below the DMRG benchmark. A Comment (arXiv:2605.28861) shows the single-spin-flip update used at this size does not conserve total magnetisation, so the Markov chains freeze and the low energy is a sampling artifact. Under ergodic sampling the same ansatz converges ~3.5% ABOVE DMRG.",
        "diagnosis": "The reported energy is an artifact of non-ergodic Monte Carlo sampling, not a property of the ansatz. The kagome Heisenberg Hamiltonian is SU(2) symmetric and its ground state lies in a fixed total-magnetisation sector, so a single-spin-flip update - which changes S^z_tot - is incompatible with the symmetry. As the network concentrates on the physical S^z_tot = 0 sector the acceptance rate collapses to exactly zero beyond 5000 iterations and the chains freeze, so the reported average is taken over a non-representative set of configurations. With the magnetisation-preserving exchange update the same architecture and optimizer converge stably to E ~ -45.6 against a DMRG value of E ~ -47.3; re-evaluating the spin-flip-optimised PARAMETERS under ergodic sampling raises the energy further, to E ~ -42.6. The published -48.18 lies below all of them.",
        "ruled_out": "NOT a variational-principle violation, and that is the point: DMRG at finite bond dimension is itself an upper bound, so an energy below it is not on its own evidence of error - it would ordinarily just be a better state. Nothing about the number alone identifies it as wrong. The refutation had to come from the sampler.",
        "evidence": "arXiv:2605.28861 (Kamal, Kufel, Vu, Laumann & Yao), Fig. 1(a) acceptance-rate collapse and Fig. 1(b) energy convergence; Ðurić et al., Phys. Rev. X 15, 011047 (2025), arXiv:2401.02866, Sec. IV."
      }
    },
    {
      "energy": -189.3456,
      "sigma": 0.1296,
      "energy_variance": null,
      "dof": 108,
      "einf": 0,
      "v_score": null,
      "method": "DMRG, truncation-error extrapolated (Torus 6)",
      "bound_type": "extrapolated",
      "bound_type_reason": "assigned during source verification (RULES.md 4)",
      "reference": "Depenbrock, McCulloch & Schollwoeck, Nature of the Spin Liquid Ground State of the S=1/2 Heisenberg Model on the Kagome Lattice, Phys. Rev. Lett. 109, 067201 (2012), arXiv:1205.4858",
      "peer_reviewed": true,
      "source": "sweep-pdf-2026-09-14",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-14",
        "method": "arXiv PDF text extracted locally with pypdf, no LLM transcription",
        "reported_as": "-0.4383 per site in S.S units",
        "note": "Table I row \"Torus 6 -0.4383(3)\", the 108-site torus (\"we also consider tori of up to 108 sites\"). E/N x 108 = -47.336, consistent with the -47.33964 that Ðurić et al. quote as the DMRG benchmark. Extrapolated in the truncation error, so not a strict bound (RULES.md 4).",
        "secondary_of": null
      }
    }
  ],
  "url": "https://qmbl.org/i/Heisenberg/kagome-6x6_108_P/",
  "per_site_divisor": 432,
  "per_site_label": "E/N (S.S)",
  "record": null,
  "no_record_reason": "every variational row is flagged"
}