{
  "model": "Hubbard",
  "lattice": "rectangular-4x16",
  "n_sites": 64,
  "boundary": "P",
  "params": {
    "Nf": 28,
    "U": 8
  },
  "instance_id": "Hubbard/rectangular-4x16_64_P_28_8",
  "rows": [
    {
      "energy": -48.192,
      "sigma": 0.003,
      "energy_variance": 0.44,
      "dof": 56,
      "einf": 98,
      "v_score": 0.0011529049089181764,
      "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 6). Soft mean-field constraint for lambda = 8 stripe order.",
      "bound_type": "variational",
      "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast",
      "reference": "[paper](https://www.pnas.org/doi/full/10.1073/pnas.2122059119)",
      "source": "varbench@2024-10-22",
      "provenance": "imported"
    },
    {
      "energy": -49.05408,
      "sigma": 0.00128,
      "energy_variance": null,
      "dof": 56,
      "einf": 98,
      "v_score": null,
      "method": "ACE (16 conv layers) trial state + fixed-node GFMC",
      "bound_type": "projected",
      "bound_type_reason": "assigned from the source text during verification",
      "reference": "Gu et al., arXiv:2604.25775 (2026-04-28)",
      "peer_reviewed": false,
      "source": "literature-2025-26",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-10",
        "method": "source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription",
        "reported_as": "-0.76647 (+/- 0.00002) per site",
        "note": "Sec. III.3.1: \"a GFMC calculation with the ACE trial wave-function gives an even lower energy of -0.76647(2)\". Fixed-node: upper bound but node-dependent.",
        "secondary_of": null
      }
    },
    {
      "energy": -49.03872,
      "sigma": 0.00064,
      "energy_variance": null,
      "dof": 56,
      "einf": 98,
      "v_score": null,
      "method": "ACE (16 conv layers) + full symmetry projection",
      "bound_type": "variational",
      "bound_type_reason": "assigned from the source text during verification",
      "reference": "Gu et al., arXiv:2604.25775 (2026-04-28)",
      "peer_reviewed": false,
      "source": "literature-2025-26",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-10",
        "method": "source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription",
        "reported_as": "-0.76623 (+/- 0.00001) per site",
        "note": "Sec. III.3.1: \"reach a new state-of-the-art energy of -0.76623(1)\".",
        "secondary_of": null
      }
    },
    {
      "energy": -48.9984,
      "sigma": 0.00064,
      "energy_variance": null,
      "dof": 56,
      "einf": 98,
      "v_score": null,
      "method": "NNBF, 32 determinants + free projection to the fully symmetric state",
      "bound_type": "variational",
      "bound_type_reason": "assigned from the source text during verification",
      "reference": "Loehr & Clark, arXiv:2510.26906 (2025-10-30)",
      "peer_reviewed": false,
      "source": "literature-2025-26",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-10",
        "method": "source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription",
        "reported_as": "-0.7656 (+/- 0.00001) per site",
        "note": "Sec. III: \"further projected at essentially no cost to the fully symmetric state to reach an energy of -0.76560(1)\". This is the best 4x16 number in arXiv:2510.26906v1.",
        "secondary_of": null
      }
    },
    {
      "energy": -48.95104,
      "sigma": 0.00128,
      "energy_variance": null,
      "dof": 56,
      "einf": 98,
      "v_score": null,
      "method": "NNBF, symmetry optimization with 32 determinants",
      "bound_type": "variational",
      "bound_type_reason": "assigned from the source text during verification",
      "reference": "Loehr & Clark, arXiv:2510.26906 (2025-10-30)",
      "peer_reviewed": false,
      "source": "literature-2025-26",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-10",
        "method": "source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription",
        "reported_as": "-0.76486 (+/- 0.00002) per site",
        "note": "Same section, before the free projection.",
        "secondary_of": null
      }
    },
    {
      "energy": -48.93696,
      "sigma": 0.00064,
      "energy_variance": null,
      "dof": 56,
      "einf": 98,
      "v_score": null,
      "method": "ACE (16 conv layers), no explicit symmetry",
      "bound_type": "variational",
      "bound_type_reason": "assigned from the source text during verification",
      "reference": "Gu et al., arXiv:2604.25775 (2026-04-28)",
      "peer_reviewed": false,
      "source": "literature-2025-26",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-10",
        "method": "source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription",
        "reported_as": "-0.76464 (+/- 0.00001) per site",
        "note": "Sec. III.3.1: \"achieves a ground-state energy of -0.76464(1)\".",
        "secondary_of": null
      }
    },
    {
      "energy": -48.83072,
      "sigma": null,
      "energy_variance": null,
      "dof": 56,
      "einf": 98,
      "v_score": null,
      "method": "Transformer backflow + MARCH optimizer",
      "bound_type": "variational",
      "bound_type_reason": "assigned from the source text during verification",
      "reference": "Gu et al., Nat. Commun. (2026), arXiv:2507.02644",
      "peer_reviewed": true,
      "source": "literature-2025-26",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-09-10",
        "method": "source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription",
        "reported_as": "-0.76298 per site",
        "note": "Supp. Sec. 4.1: \"our NQS yields a ground state energy of -0.76298\" on 16x4 PBC. Instance identity confirmed: same passage quotes HFDS -0.753(2), matching the VarBench row -0.753000(47). No error bar given.",
        "secondary_of": null
      }
    },
    {
      "energy": -48.90432,
      "sigma": 0.00192,
      "energy_variance": 2.002112,
      "dof": 56,
      "einf": 98,
      "v_score": 0.0051952596917330335,
      "method": "HFPS (hidden-fermion Pfaffian state)",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)",
      "reference": "Chen, Wan, Sengupta & Georges, Neural network-augmented Pfaffian wave-functions for scalable simulations of interacting fermions, Proc. Natl. Acad. Sci. U.S.A. 123, e2535288123 (2026), arXiv:2507.10705",
      "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": "-0.76413 (+/- 0.00003) per site",
        "note": "Sec. III: \"The energy we obtain in the 16x4 lattice is E/N = -0.76413, outperforming a recent result E/N = -0.76298 produced by Transformer-NNBF\". Instance identity confirmed by that comparison value; the number is independently quoted as the HFPS result by both arXiv:2604.25775 and arXiv:2510.26906. No error bar in Sec. III; Table IV of the same paper (raw data of Fig. 7, L x 4 PBC lattices, 1/8 doping, U = 8) gives -0.76413(3) and sigma^2/N_site = 0.031283, added 2026-09-15. Published as PNAS 123, e2535288123 (2026); read from the arXiv version, the journal version not compared.",
        "secondary_of": null
      }
    }
  ],
  "url": "https://qmbl.org/i/Hubbard/rectangular-4x16_64_P_28_8/",
  "per_site_divisor": 64,
  "per_site_label": "E/site",
  "record": {
    "energy": -49.03872,
    "sigma": 0.00064,
    "method": "ACE (16 conv layers) + full symmetry projection",
    "reference": "Gu et al., arXiv:2604.25775 (2026-04-28)",
    "energy_per_site": -0.76623
  },
  "no_record_reason": null
}