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B 83, 212401 (2011), arXiv:1103.1159", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "Total E = -5.444875216 (S.S), cluster 12", "note": "Table I, \"Cluster studied in this work\", row N = 12: basis vectors a = (2,0), b = (0,2) in units of the kagome lattice vectors a1, a2 (each of length 2a), periodic (torus). E/N = -0.453740 as printed in the table's own E/N column; Pauli total = 4 E. 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Rev. B 83, 212401 (2011), arXiv:1103.1159", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "Total E = -13.154318948 (S.S), cluster 30", "note": "Table I, \"Cluster studied in this work\", row N = 30: basis vectors a = (2,1), b = (-2,4) in units of the kagome lattice vectors a1, a2 (each of length 2a), periodic (torus). E/N = -0.438477 as printed in the table's own E/N column; Pauli total = 4 E.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/kagome-30_30_P/", "per_site_divisor": 120, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "kagome-36a", "n_sites": 36, "boundary": "P", "params": {}, "instance_id": "Heisenberg/kagome-36a_36_P", "rows": [ { "energy": -63.151499388, "sigma": null, "energy_variance": null, "dof": 36, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Lauchli, Sudan & Sorensen, Ground-state energy and spin gap of spin-1/2 kagome Heisenberg antiferromagnetic clusters: large-scale exact diagonalization results, Phys. Rev. B 83, 212401 (2011), arXiv:1103.1159", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "Total E = -15.787874847 (S.S), cluster 36a", "note": "Table I, \"Cluster studied in this work\", row N = 36a: basis vectors a = (-2,3), b = (4,0) in units of the kagome lattice vectors a1, a2 (each of length 2a), periodic (torus). E/N = -0.438552 as printed in the table's own E/N column; Pauli total = 4 E.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/kagome-36a_36_P/", "per_site_divisor": 144, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "kagome-36b", "n_sites": 36, "boundary": "P", "params": {}, "instance_id": "Heisenberg/kagome-36b_36_P", "rows": [ { "energy": -63.227711024, "sigma": null, "energy_variance": null, "dof": 36, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Lauchli, Sudan & Sorensen, Ground-state energy and spin gap of spin-1/2 kagome Heisenberg antiferromagnetic clusters: large-scale exact diagonalization results, Phys. Rev. B 83, 212401 (2011), arXiv:1103.1159", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "Total E = -15.806927756 (S.S), cluster 36b", "note": "Table I, \"Cluster studied in this work\", row N = 36b: basis vectors a = (3,0), b = (-3,4) in units of the kagome lattice vectors a1, a2 (each of length 2a), periodic (torus). E/N = -0.439081 as printed in the table's own E/N column; Pauli total = 4 E.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/kagome-36b_36_P/", "per_site_divisor": 144, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "kagome-36c", "n_sites": 36, "boundary": "P", "params": {}, "instance_id": "Heisenberg/kagome-36c_36_P", "rows": [ { "energy": -63.257336008, "sigma": null, "energy_variance": null, "dof": 36, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Lauchli, Sudan & Sorensen, Ground-state energy and spin gap of spin-1/2 kagome Heisenberg antiferromagnetic clusters: large-scale exact diagonalization results, Phys. Rev. B 83, 212401 (2011), arXiv:1103.1159", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "Total E = -15.814334002 (S.S), cluster 36c", "note": "Table I, \"Cluster studied in this work\", row N = 36c: basis vectors a = (3,0), b = (-1,4) in units of the kagome lattice vectors a1, a2 (each of length 2a), periodic (torus). E/N = -0.439287 as printed in the table's own E/N column; Pauli total = 4 E.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/kagome-36c_36_P/", "per_site_divisor": 144, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "kagome-36d", "n_sites": 36, "boundary": "P", "params": {}, "instance_id": "Heisenberg/kagome-36d_36_P", "rows": [ { "energy": -63.126220472, "sigma": null, "energy_variance": null, "dof": 36, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Lauchli, Sudan & Sorensen, Ground-state energy and spin gap of spin-1/2 kagome Heisenberg antiferromagnetic clusters: large-scale exact diagonalization results, Phys. Rev. B 83, 212401 (2011), arXiv:1103.1159", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "Total E = -15.781555118 (S.S), cluster 36d", "note": "Table I, \"Cluster studied in this work\", row N = 36d: basis vectors a = (4,-2), b = (-2,4) in units of the kagome lattice vectors a1, a2 (each of length 2a), periodic (torus). E/N = -0.438377 as printed in the table's own E/N column; Pauli total = 4 E. The 36d cluster (|a| = |b| = d = sqrt 12) is the standard 36-site kagome cluster with the full symmetry of the plane.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/kagome-36d_36_P/", "per_site_divisor": 144, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "kagome-42a", "n_sites": 42, "boundary": "P", "params": {}, "instance_id": "Heisenberg/kagome-42a_42_P", "rows": [ { "energy": -73.583839936, "sigma": null, "energy_variance": null, "dof": 42, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Lauchli, Sudan & Sorensen, Ground-state energy and spin gap of spin-1/2 kagome Heisenberg antiferromagnetic clusters: large-scale exact diagonalization results, Phys. Rev. B 83, 212401 (2011), arXiv:1103.1159", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "Total E = -18.395959984 (S.S), cluster 42a", "note": "Table I, \"Cluster studied in this work\", row N = 42a: basis vectors a = (-1,3), b = (5,-1) in units of the kagome lattice vectors a1, a2 (each of length 2a), periodic (torus). E/N = -0.437999 as printed in the table's own E/N column; Pauli total = 4 E.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/kagome-42a_42_P/", "per_site_divisor": 168, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "kagome-42b", "n_sites": 42, "boundary": "P", "params": {}, "instance_id": "Heisenberg/kagome-42b_42_P", "rows": [ { "energy": -73.607955684, "sigma": null, "energy_variance": null, "dof": 42, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Lauchli, Sudan & Sorensen, Ground-state energy and spin gap of spin-1/2 kagome Heisenberg antiferromagnetic clusters: large-scale exact diagonalization results, Phys. Rev. B 83, 212401 (2011), arXiv:1103.1159", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "Total E = -18.401988921 (S.S), cluster 42b", "note": "Table I, \"Cluster studied in this work\", row N = 42b: basis vectors a = (-2,4), b = (4,-1) in units of the kagome lattice vectors a1, a2 (each of length 2a), periodic (torus). E/N = -0.438143 as printed in the table's own E/N column; Pauli total = 4 E.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/kagome-42b_42_P/", "per_site_divisor": 168, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "kagome-4x4", "n_sites": 48, "boundary": "P", "params": {}, "instance_id": "Heisenberg/kagome-4x4_48_P", "rows": [ { "energy": -84.231148254, "sigma": 4e-10, "energy_variance": null, "dof": 48, "einf": 0, "v_score": null, "method": "Exact Diagonalization", "bound_type": "exact", "bound_type_reason": "exact diagonalization|ex", "reference": "[paper](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.98.033309)", "source": "varbench@2024-10-22", "provenance": "imported" }, { "energy": -82.643, "sigma": 0.005, "energy_variance": 10.6, "dof": 48, "einf": 0, "v_score": 0.07449634312700791, "method": "VMC with Dirac spin liquid + Jastrow", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/kagome-4x4_48_P/vmc_gutzwiller.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -84, "sigma": 0.04, "energy_variance": null, "dof": 48, "einf": 0, "v_score": null, "method": "GCNN (6 layers, 6 feature maps), symmetric ansatz", "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": "-0.4375 per site in S.S units", "note": "Sec. III: \"R = (E_GCNN - E_ED)/E_ED ~ 0.3% with E_ED = -21.057787063 and E_GCNN = -21.00 +/- 0.01\" on the 48-site (4x4x3) PBC cluster, chi0 sector. Total energy in S.S units; E/N = -0.4375. The instance's existing exact row is -0.4387039, i.e. -21.057787063/48 to 8 digits, which confirms the cluster and the convention. Sampling here is the ergodic spin-exchange update, so the objection in arXiv:2605.28861 does not apply to this size.", "secondary_of": null } }, { "energy": -84.1536, "sigma": 0.0384, "energy_variance": null, "dof": 48, "einf": 0, "v_score": null, "method": "DMRG, truncation-error extrapolated (Torus 4)", "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 4 -0.4383(2)\". The paper states energies are \"extrapolated in the truncation error of single-site DMRG\", so this is not a strict variational bound (RULES.md 4). Sits above the exact -0.4387039 on this instance.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/kagome-4x4_48_P/", "per_site_divisor": 192, "per_site_label": "E/N (S.S)", "record": { "energy": -84, "sigma": 0.04, "method": "GCNN (6 layers, 6 feature maps), symmetric ansatz", "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", "energy_per_site": -0.4375 }, "no_record_reason": null }, { "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" }, { "model": "Heisenberg", "lattice": "kagome-8x8", "n_sites": 192, "boundary": "P", "params": {}, "instance_id": "Heisenberg/kagome-8x8_192_P", "rows": [ { "energy": -330.143, "sigma": 0.011, "energy_variance": 42.3, "dof": 192, "einf": 0, "v_score": 0.07451391967425162, "method": "VMC with Dirac spin liquid + Jastrow", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/kagome-8x8_192_P/vmc_gutzwiller.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true } ], "url": "https://qmbl.org/i/Heisenberg/kagome-8x8_192_P/", "per_site_divisor": 768, "per_site_label": "E/N (S.S)", "record": { "energy": -330.143, "sigma": 0.011, "method": "VMC with Dirac spin liquid + Jastrow", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/kagome-8x8_192_P/vmc_gutzwiller.sh)", "energy_per_site": -0.42987369791666663 }, "no_record_reason": null }, { "model": "Heisenberg", "lattice": "pyrochlore-2x2x2", "n_sites": 128, "boundary": "P", "params": {}, "instance_id": "Heisenberg/pyrochlore-2x2x2_128_P", "rows": [ { "energy": -252.007, "sigma": 0.021, "energy_variance": 9.93, "dof": 128, "einf": 0, "v_score": 0.020014005253350654, "method": "mVMC (PP + RBM + 1st step Lanczos, spin-parity projection, Number of RBM neurons: 128)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://arxiv.org/abs/2311.11561) [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/pyrochlore-2x2x2_128_P/mVMC.sh)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Heisenberg/pyrochlore-2x2x2_128_P/", "per_site_divisor": 512, "per_site_label": "E/N (S.S)", "record": { "energy": -252.007, "sigma": 0.021, "method": "mVMC (PP + RBM + 1st step Lanczos, spin-parity projection, Number of RBM neurons: 128)", "reference": "[paper](https://arxiv.org/abs/2311.11561) [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/pyrochlore-2x2x2_128_P/mVMC.sh)", "energy_per_site": -0.492201171875 }, "no_record_reason": null }, { "model": "Heisenberg", "lattice": "pyrochlore-2x2x2", "n_sites": 32, "boundary": "P", "params": {}, "instance_id": "Heisenberg/pyrochlore-2x2x2_32_P", "rows": [ { "energy": -66.1514, "sigma": 0, "energy_variance": 0, "dof": 32, "einf": 0, "v_score": 0, "method": "Exact diagonalization", "bound_type": "exact", "bound_type_reason": "exact diagonalization|ex", "reference": "[paper](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041021)", "source": "varbench@2024-10-22", "provenance": "imported" }, { "energy": -65.949, "sigma": 0.003, "energy_variance": 6.08, "dof": 32, "einf": 0, "v_score": 0.044733937675725685, "method": "RBM with symmetry projections", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041021)", "source": "varbench@2024-10-22", "provenance": "imported" }, { "energy": -66.082, "sigma": 0.003, "energy_variance": 4.36, "dof": 32, "einf": 0, "v_score": 0.03194994466655219, "method": "mVMC with SU(2) and symmetry projections", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041021)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Heisenberg/pyrochlore-2x2x2_32_P/", "per_site_divisor": 128, "per_site_label": "E/N (S.S)", "record": { "energy": -66.082, "sigma": 0.003, "method": "mVMC with SU(2) and symmetry projections", "reference": "[paper](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041021)", "energy_per_site": -0.516265625 }, "no_record_reason": null }, { "model": "Heisenberg", "lattice": "pyrochlore-3x3x3", "n_sites": 108, "boundary": "P", "params": {}, "instance_id": "Heisenberg/pyrochlore-3x3x3_108_P", "rows": [ { "energy": -210.43, "sigma": 0.04, "energy_variance": 39.88, "dof": 108, "einf": 0, "v_score": 0.0972665685517241, "method": "mVMC with SU(2) and symmetry projections", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041021)", "source": "varbench@2024-10-22", "provenance": "imported" }, { "energy": -208.48, "sigma": 0.04, "energy_variance": 18.06, "dof": 108, "einf": 0, "v_score": 0.04487585175953243, "method": "CNN NQS with symmetry projections", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041021)", "source": "varbench@2024-10-22", "provenance": "imported" }, { "energy": -209.5632, "sigma": 0, "energy_variance": null, "dof": 108, "einf": 0, "v_score": null, "method": "DMRG", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.117204)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Heisenberg/pyrochlore-3x3x3_108_P/", "per_site_divisor": 432, "per_site_label": "E/N (S.S)", "record": { "energy": -210.43, "sigma": 0.04, "method": "mVMC with SU(2) and symmetry projections", "reference": "[paper](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041021)", "energy_per_site": -0.4871064814814815 }, "no_record_reason": null }, { "model": "Heisenberg", "lattice": "pyrochlore-3x3x3", "n_sites": 432, "boundary": "P", "params": {}, "instance_id": "Heisenberg/pyrochlore-3x3x3_432_P", "rows": [ { "energy": -844.084, "sigma": 0.045, "energy_variance": 60.76, "dof": 432, "einf": 0, "v_score": 0.036840895301969837, "method": "mVMC (PP + 1st step Lanczos, spin-parity projection, C3 point-group projection)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://arxiv.org/abs/2311.11561) [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/pyrochlore-3x3x3_432_P/mVMC.sh)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Heisenberg/pyrochlore-3x3x3_432_P/", "per_site_divisor": 1728, "per_site_label": "E/N (S.S)", "record": { "energy": -844.084, "sigma": 0.045, "method": "mVMC (PP + 1st step Lanczos, spin-parity projection, C3 point-group projection)", "reference": "[paper](https://arxiv.org/abs/2311.11561) [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/pyrochlore-3x3x3_432_P/mVMC.sh)", "energy_per_site": -0.488474537037037 }, "no_record_reason": null }, { "model": "Heisenberg", "lattice": "pyrochlore-4x4x4", "n_sites": 1024, "boundary": "P", "params": {}, "instance_id": "Heisenberg/pyrochlore-4x4x4_1024_P", "rows": [ { "energy": -1998.857, "sigma": 0.048, "energy_variance": 185.53, "dof": 1024, "einf": 0, "v_score": 0.047550014135740046, "method": "mVMC (PP + 1st step Lanczos, spin-parity projection, C3 point-group projection)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://arxiv.org/abs/2311.11561) [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/pyrochlore-4x4x4_1024_P/mVMC.sh)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Heisenberg/pyrochlore-4x4x4_1024_P/", "per_site_divisor": 4096, "per_site_label": "E/N (S.S)", "record": { "energy": -1998.857, "sigma": 0.048, "method": "mVMC (PP + 1st step Lanczos, spin-parity projection, C3 point-group projection)", "reference": "[paper](https://arxiv.org/abs/2311.11561) [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/pyrochlore-4x4x4_1024_P/mVMC.sh)", "energy_per_site": -0.488002197265625 }, "no_record_reason": null }, { "model": "Heisenberg", "lattice": "pyrochlore-4x4x4", "n_sites": 256, "boundary": "P", "params": {}, "instance_id": "Heisenberg/pyrochlore-4x4x4_256_P", "rows": [ { "energy": -494.69, "sigma": 0.076, "energy_variance": 109.41, "dof": 256, "einf": 0, "v_score": 0.11445393291700551, "method": "mVMC with SU(2) and symmetry projections", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041021)", "source": "varbench@2024-10-22", "provenance": "imported" }, { "energy": -497.152, "sigma": null, "energy_variance": null, "dof": 256, "einf": 0, "v_score": null, "method": "Generalized RVB ansatz, unrestricted VMC optimization (no quantum-number projection)", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "Closely competing valence bond crystal orders in the ground state of the spin-1/2 antiferromagnetic Heisenberg model on the pyrochlore lattice: a large scale unrestricted variational study, arXiv:2509.13746", "peer_reviewed": false, "source": "sweep-worklist-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF text extracted locally with pypdf in layout mode, no LLM transcription", "reported_as": "E0 ~ -0.4855 J/site, approximate, no error bar", "note": "Sec. A: \"The obtained ground state energy is E0 ~ -0.5118 J/site for the L = 2 cluster and E0 ~ -0.4855 J/site for the L = 4 cluster\", periodic boundary conditions on both; the L = 4 cluster carries Nv = 4N^2 = 262144 parameters, so N = 256. Units: the same passage quotes the mVMC energy -0.5162 J/site for L = 2, which is the pyrochlore-2x2x2_32_P record -0.516266 in the S.S per-site convention. Approximate and without an error bar, so it is eligible for nothing (RULES.md 6). It is nonetheless 0.5% below the VarBench mVMC record -0.4830957, although the ansatz has no symmetry projection and sits above mVMC at L = 2 - which says more about that record than about this row.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/pyrochlore-4x4x4_256_P/", "per_site_divisor": 1024, "per_site_label": "E/N (S.S)", "record": { "energy": -494.69, "sigma": 0.076, "method": "mVMC with SU(2) and symmetry projections", "reference": "[paper](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041021)", "energy_per_site": -0.483095703125 }, "no_record_reason": null }, { "model": "Heisenberg", "lattice": "rectangular-10x20", "n_sites": 200, "boundary": "PO", "params": {}, "instance_id": "Heisenberg/rectangular-10x20_200_PO", "rows": [ { "energy": -257.21212, "sigma": 0.000152, "energy_variance": null, "dof": 200, "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/N_b = 0.3384370(2) (magnitude, per bond, N_b = 190), L x 2L = 10 x 20, cylinder", "note": "Table 3, \"SSE data for L x 2L lattices with cylindrical boundary conditions (periodic in the shorter direction and open in the longer direction). The ground state energy is normalized by the number of interaction bonds N_b = 2 L^2 - L\", row L = 10. Boundary PO (VarBench lattice.md): periodic along the 10-site direction, open along the 20-site one. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.32151515 in S.S units, Pauli total = 4 N E/N.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/rectangular-10x20_200_PO/", "per_site_divisor": 800, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "rectangular-12x24", "n_sites": 288, "boundary": "PO", "params": {}, "instance_id": "Heisenberg/rectangular-12x24_288_PO", "rows": [ { "energy": -372.7789584, "sigma": 0.000221, "energy_variance": null, "dof": 288, "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/N_b = 0.3376621(2) (magnitude, per bond, N_b = 276), L x 2L = 12 x 24, cylinder", "note": "Table 3, \"SSE data for L x 2L lattices with cylindrical boundary conditions (periodic in the shorter direction and open in the longer direction). The ground state energy is normalized by the number of interaction bonds N_b = 2 L^2 - L\", row L = 12. Boundary PO (VarBench lattice.md): periodic along the 12-site direction, open along the 24-site one. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.32359285 in S.S units, Pauli total = 4 N E/N.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/rectangular-12x24_288_PO/", "per_site_divisor": 1152, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "rectangular-14x28", "n_sites": 392, "boundary": "PO", "params": {}, "instance_id": "Heisenberg/rectangular-14x28_392_PO", "rows": [ { "energy": -509.7877344, "sigma": 0.000302, "energy_variance": null, "dof": 392, "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/N_b = 0.3371612(2) (magnitude, per bond, N_b = 378), L x 2L = 14 x 28, cylinder", "note": "Table 3, \"SSE data for L x 2L lattices with cylindrical boundary conditions (periodic in the shorter direction and open in the longer direction). The ground state energy is normalized by the number of interaction bonds N_b = 2 L^2 - L\", row L = 14. Boundary PO (VarBench lattice.md): periodic along the 14-site direction, open along the 28-site one. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.32511973 in S.S units, Pauli total = 4 N E/N.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/rectangular-14x28_392_PO/", "per_site_divisor": 1568, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "rectangular-16x32", "n_sites": 512, "boundary": "PO", "params": {}, "instance_id": "Heisenberg/rectangular-16x32_512_PO", "rows": [ { "energy": -668.2302464, "sigma": 0.000397, "energy_variance": null, "dof": 512, "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/N_b = 0.3368096(2) (magnitude, per bond, N_b = 496), L x 2L = 16 x 32, cylinder", "note": "Table 3, \"SSE data for L x 2L lattices with cylindrical boundary conditions (periodic in the shorter direction and open in the longer direction). The ground state energy is normalized by the number of interaction bonds N_b = 2 L^2 - L\", row L = 16. Boundary PO (VarBench lattice.md): periodic along the 16-site direction, open along the 32-site one. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.32628430 in S.S units, Pauli total = 4 N E/N.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/rectangular-16x32_512_PO/", "per_site_divisor": 2048, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "rectangular-18x36", "n_sites": 648, "boundary": "PO", "params": {}, "instance_id": "Heisenberg/rectangular-18x36_648_PO", "rows": [ { "energy": -848.103732, "sigma": 0.000504, "energy_variance": null, "dof": 648, "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/N_b = 0.3365491(2) (magnitude, per bond, N_b = 630), L x 2L = 18 x 36, cylinder", "note": "Table 3, \"SSE data for L x 2L lattices with cylindrical boundary conditions (periodic in the shorter direction and open in the longer direction). The ground state energy is normalized by the number of interaction bonds N_b = 2 L^2 - L\", row L = 18. Boundary PO (VarBench lattice.md): periodic along the 18-site direction, open along the 36-site one. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.32720051 in S.S units, Pauli total = 4 N E/N.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/rectangular-18x36_648_PO/", "per_site_divisor": 2592, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "rectangular-20x40", "n_sites": 800, "boundary": "PO", "params": {}, "instance_id": "Heisenberg/rectangular-20x40_800_PO", "rows": [ { "energy": -1049.404512, "sigma": 0.000624, "energy_variance": null, "dof": 800, "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/N_b = 0.3363476(2) (magnitude, per bond, N_b = 780), L x 2L = 20 x 40, cylinder", "note": "Table 3, \"SSE data for L x 2L lattices with cylindrical boundary conditions (periodic in the shorter direction and open in the longer direction). The ground state energy is normalized by the number of interaction bonds N_b = 2 L^2 - L\", row L = 20. Boundary PO (VarBench lattice.md): periodic along the 20-site direction, open along the 40-site one. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.32793891 in S.S units, Pauli total = 4 N E/N.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/rectangular-20x40_800_PO/", "per_site_divisor": 3200, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "rectangular-24x48", "n_sites": 1152, "boundary": "PO", "params": {}, "instance_id": "Heisenberg/rectangular-24x48_1152_PO", "rows": [ { "energy": -1516.282416, "sigma": 0.00135, "energy_variance": null, "dof": 1152, "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/N_b = 0.3360555(3) (magnitude, per bond, N_b = 1128), L x 2L = 24 x 48, cylinder", "note": "Table 3, \"SSE data for L x 2L lattices with cylindrical boundary conditions (periodic in the shorter direction and open in the longer direction). The ground state energy is normalized by the number of interaction bonds N_b = 2 L^2 - L\", row L = 24. Boundary PO (VarBench lattice.md): periodic along the 24-site direction, open along the 48-site one. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.32905434 in S.S units, Pauli total = 4 N E/N.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/rectangular-24x48_1152_PO/", "per_site_divisor": 4608, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "rectangular-32x64", "n_sites": 2048, "boundary": "PO", "params": {}, "instance_id": "Heisenberg/rectangular-32x64_2048_PO", "rows": [ { "energy": -2707.1339904, "sigma": 0.00242, "energy_variance": null, "dof": 2048, "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/N_b = 0.3357061(3) (magnitude, per bond, N_b = 2016), L x 2L = 32 x 64, cylinder", "note": "Table 3, \"SSE data for L x 2L lattices with cylindrical boundary conditions (periodic in the shorter direction and open in the longer direction). The ground state energy is normalized by the number of interaction bonds N_b = 2 L^2 - L\", row L = 32. Boundary PO (VarBench lattice.md): periodic along the 32-site direction, open along the 64-site one. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.33046069 in S.S units, Pauli total = 4 N E/N.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/rectangular-32x64_2048_PO/", "per_site_divisor": 8192, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "rectangular-48x96", "n_sites": 4608, "boundary": "PO", "params": {}, "instance_id": "Heisenberg/rectangular-48x96_4608_PO", "rows": [ { "energy": -6117.145152, "sigma": 0.00547, "energy_variance": null, "dof": 4608, "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/N_b = 0.3353698(3) (magnitude, per bond, N_b = 4560), L x 2L = 48 x 96, cylinder", "note": "Table 3, \"SSE data for L x 2L lattices with cylindrical boundary conditions (periodic in the shorter direction and open in the longer direction). The ground state energy is normalized by the number of interaction bonds N_b = 2 L^2 - L\", row L = 48. Boundary PO (VarBench lattice.md): periodic along the 48-site direction, open along the 96-site one. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.33187636 in S.S units, Pauli total = 4 N E/N.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/rectangular-48x96_4608_PO/", "per_site_divisor": 18432, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "rectangular-4x8", "n_sites": 32, "boundary": "PO", "params": {}, "instance_id": "Heisenberg/rectangular-4x8_32_PO", "rows": [ { "energy": -39.5624656, "sigma": 0.0000224, "energy_variance": null, "dof": 32, "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/N_b = 0.3532363(2) (magnitude, per bond, N_b = 28), L x 2L = 4 x 8, cylinder", "note": "Table 3, \"SSE data for L x 2L lattices with cylindrical boundary conditions (periodic in the shorter direction and open in the longer direction). The ground state energy is normalized by the number of interaction bonds N_b = 2 L^2 - L\", row L = 4. Boundary PO (VarBench lattice.md): periodic along the 4-site direction, open along the 8-site one. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.30908176 in S.S units, Pauli total = 4 N E/N.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/rectangular-4x8_32_PO/", "per_site_divisor": 128, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "rectangular-64x128", "n_sites": 8192, "boundary": "PO", "params": {}, "instance_id": "Heisenberg/rectangular-64x128_8192_PO", "rows": [ { "energy": -10898.1947136, "sigma": 0.0065, "energy_variance": null, "dof": 8192, "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/N_b = 0.3352053(2) (magnitude, per bond, N_b = 8128), L x 2L = 64 x 128, cylinder", "note": "Table 3, \"SSE data for L x 2L lattices with cylindrical boundary conditions (periodic in the shorter direction and open in the longer direction). The ground state energy is normalized by the number of interaction bonds N_b = 2 L^2 - L\", row L = 64. Boundary PO (VarBench lattice.md): periodic along the 64-site direction, open along the 128-site one. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.33258651 in S.S units, Pauli total = 4 N E/N.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/rectangular-64x128_8192_PO/", "per_site_divisor": 32768, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "rectangular-6x12", "n_sites": 72, "boundary": "PO", "params": {}, "instance_id": "Heisenberg/rectangular-6x12_72_PO", "rows": [ { "energy": -90.4960056, "sigma": 0.0000528, "energy_variance": null, "dof": 72, "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/N_b = 0.3427879(2) (magnitude, per bond, N_b = 66), L x 2L = 6 x 12, cylinder", "note": "Table 3, \"SSE data for L x 2L lattices with cylindrical boundary conditions (periodic in the shorter direction and open in the longer direction). The ground state energy is normalized by the number of interaction bonds N_b = 2 L^2 - L\", row L = 6. Boundary PO (VarBench lattice.md): periodic along the 6-site direction, open along the 12-site one. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.31422224 in S.S units, Pauli total = 4 N E/N.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/rectangular-6x12_72_PO/", "per_site_divisor": 288, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "rectangular-6x8", "n_sites": 48, "boundary": "P", "params": {}, "instance_id": "Heisenberg/rectangular-6x8_48_P", "rows": [ { "energy": -129.7894394912, "sigma": 4e-10, "energy_variance": null, "dof": 48, "einf": 0, "v_score": null, "method": "Exact Diagonalization", "bound_type": "exact", "bound_type_reason": "exact diagonalization|ex", "reference": "[paper](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.98.033309)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Heisenberg/rectangular-6x8_48_P/", "per_site_divisor": 192, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "rectangular-8x16", "n_sites": 128, "boundary": "PO", "params": {}, "instance_id": "Heisenberg/rectangular-8x16_128_PO", "rows": [ { "energy": -163.103184, "sigma": 0.000096, "energy_variance": null, "dof": 128, "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/N_b = 0.3397983(2) (magnitude, per bond, N_b = 120), L x 2L = 8 x 16, cylinder", "note": "Table 3, \"SSE data for L x 2L lattices with cylindrical boundary conditions (periodic in the shorter direction and open in the longer direction). The ground state energy is normalized by the number of interaction bonds N_b = 2 L^2 - L\", row L = 8. Boundary PO (VarBench lattice.md): periodic along the 8-site direction, open along the 16-site one. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.31856091 in S.S units, Pauli total = 4 N E/N.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/rectangular-8x16_128_PO/", "per_site_divisor": 512, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "shuriken", "n_sites": 216, "boundary": "P", "params": {}, "instance_id": "Heisenberg/shuriken_216_P", "rows": [ { "energy": -378.086, "sigma": 0.086, "energy_variance": 17.491, "dof": 216, "einf": 0, "v_score": 0.026429393571538857, "method": "mVMC with SU(2) and point group projection", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.L220408)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Heisenberg/shuriken_216_P/", "per_site_divisor": 864, "per_site_label": "E/N (S.S)", "record": { "energy": -378.086, "sigma": 0.086, "method": "mVMC with SU(2) and point group projection", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.L220408)", "energy_per_site": -0.4375995370370371 }, "no_record_reason": null }, { "model": "Heisenberg", "lattice": "shuriken", "n_sites": 24, "boundary": "P", "params": {}, "instance_id": "Heisenberg/shuriken_24_P", "rows": [ { "energy": -43.039584, "sigma": 0, "energy_variance": 0, "dof": 24, "einf": 0, "v_score": 0, "method": "Exact diagonalization", "bound_type": "exact", "bound_type_reason": "exact diagonalization|ex", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.L220408)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Heisenberg/shuriken_24_P/", "per_site_divisor": 96, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "shuriken", "n_sites": 384, "boundary": "P", "params": {}, "instance_id": "Heisenberg/shuriken_384_P", "rows": [ { "energy": -671.43, "sigma": 0.1536, "energy_variance": 46.96, "dof": 384, "einf": 0, "v_score": 0.039999800815514866, "method": "mVMC with SU(2) and point group projection", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.L220408)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Heisenberg/shuriken_384_P/", "per_site_divisor": 1536, "per_site_label": "E/N (S.S)", "record": { "energy": -671.43, "sigma": 0.1536, "method": "mVMC with SU(2) and point group projection", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.L220408)", "energy_per_site": -0.43712890624999995 }, "no_record_reason": null }, { "model": "Heisenberg", "lattice": "shuriken", "n_sites": 96, "boundary": "P", "params": {}, "instance_id": "Heisenberg/shuriken_96_P", "rows": [ { "energy": -168.2918, "sigma": 0.0048, "energy_variance": 7.563, "dof": 96, "einf": 0, "v_score": 0.025635360100123884, "method": "mVMC with SU(2) and point group projection", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.L220408)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Heisenberg/shuriken_96_P/", "per_site_divisor": 384, "per_site_label": "E/N (S.S)", "record": { "energy": -168.2918, "sigma": 0.0048, "method": "mVMC with SU(2) and point group projection", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.L220408)", "energy_per_site": -0.4382598958333333 }, "no_record_reason": null }, { "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 }, { "model": "Heisenberg", "lattice": "square", "n_sites": 100, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_100_P", "rows": [ { "energy": -267.741, "sigma": 0.004, "energy_variance": 12.2, "dof": 100, "einf": 0, "v_score": 0.017018844431084285, "method": "VMC with fermions (flux+neel+Jastrow)", "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_P/vmc_gutzwiller.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -268.4538, "sigma": 0.0005, "energy_variance": 0.5409, "dof": 100, "einf": 0, "v_score": 0.0007505469591770483, "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_P/vmc_rnn.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -268.56, "sigma": 0.005, "energy_variance": 0.092, "dof": 100, "einf": 0, "v_score": 0.00012755725679223536, "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_P/vmc_rnn.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -265.9424657801722, "sigma": null, "energy_variance": 42.496814964441, "dof": 100, "einf": 0, "v_score": 0.06008706618210692, "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/Heisenberg/square_100_P/dmrg.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -265.7726, "sigma": 0.006, "energy_variance": 37.11238, "dof": 100, "einf": 0, "v_score": 0.05254100749080764, "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_P/vmc_rbm.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -265.6846, "sigma": 0.0056, "energy_variance": 31.98062, "dof": 100, "einf": 0, "v_score": 0.045305832517367715, "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_P/vmc_jastrow.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -268.62104, "sigma": 0.000012, "energy_variance": null, "dof": 100, "einf": 0, "v_score": null, "method": "CNN + MinSR", "bound_type": "variational", "bound_type_reason": "assigned during source verification (RULES.md 4)", "reference": "Chen & Heyl, Empowering deep neural quantum states through efficient optimization, Nat. Phys. 20, 1476 (2024), arXiv:2302.01941", "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": "-0.6715526 (+/- 3e-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 \"VMC - CNN+MinSR [12]\" = Chen & Heyl, Nat. Phys. 20, 1476 (2024).", "secondary_of": "Moss, Wiersema, Hibat-Allah, Carrasquilla & Melko, Phys. Rev. B (2025), arXiv:2502.17144", "found_via": null } }, { "energy": -268.6076, "sigma": 0.0016, "energy_variance": null, "dof": 100, "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": "-0.671519 (+/- 0.000004) 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: \"10x10 0.671549(4) 5.3124(3) -0.671519(4) 5.38(6)\" - Energy(QMC) beside Energy(RBM).", "secondary_of": null, "found_via": "Moss, Wiersema, Hibat-Allah, Carrasquilla & Melko, Phys. Rev. B (2025), arXiv:2502.17144" } }, { "energy": -268.604, "sigma": 0.008, "energy_variance": null, "dof": 100, "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": "-0.67151 (+/- 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": -268.54, "sigma": null, "energy_variance": null, "dof": 100, "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": "-0.67135 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). No error bar given.", "secondary_of": "Moss, Wiersema, Hibat-Allah, Carrasquilla & Melko, Phys. Rev. B (2025), arXiv:2502.17144", "found_via": null } }, { "energy": -268.638, "sigma": 0.0002, "energy_variance": null, "dof": 100, "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": "-0.671595 (+/- 5e-7) 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. Zero-variance extrapolation, not an achieved energy (RULES.md 4).", "secondary_of": null, "found_via": null } }, { "energy": -268.621064, "sigma": 0.000008, "energy_variance": null, "dof": 100, "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 = -0.67155266(2) per spin (S.S), L = 10, 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 = 10. 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_100_P/", "per_site_divisor": 400, "per_site_label": "E/N (S.S)", "record": { "energy": -268.62104, "sigma": 0.000012, "method": "CNN + MinSR", "reference": "Chen & Heyl, Empowering deep neural quantum states through efficient optimization, Nat. Phys. 20, 1476 (2024), arXiv:2302.01941", "energy_per_site": -0.6715525999999999 }, "no_record_reason": null }, { "model": "Heisenberg", "lattice": "square", "n_sites": 1024, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_1024_O", "rows": [ { "energy": -2688.3231744, "sigma": 0.00159, "energy_variance": null, "dof": 1024, "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/N_b = 0.3387504(2) (magnitude, per bond, N_b = 1984), L = 32, open", "note": "Table 2, \"SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)\", row L = 32. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.65632890 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_1024_O/", "per_site_divisor": 4096, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 1024, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_1024_P", "rows": [ { "energy": -2742.31840768, "sigma": 0.000123, "energy_variance": null, "dof": 1024, "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 = -0.66951133(3) per spin (S.S), L = 32, 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 = 32. 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_1024_P/", "per_site_divisor": 4096, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 1296, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_1296_P", "rows": [ { "energy": -3470.64087744, "sigma": 0.000156, "energy_variance": null, "dof": 1296, "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 = -0.66949091(3) per spin (S.S), L = 36, 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 = 36. 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_1296_P/", "per_site_divisor": 5184, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 144, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_144_O", "rows": [ { "energy": -365.8755936, "sigma": 0.000106, "energy_variance": null, "dof": 144, "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/N_b = 0.3464731(1) (magnitude, per bond, N_b = 264), L = 12, open", "note": "Table 2, \"SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)\", row L = 12. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.63520068 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_144_O/", "per_site_divisor": 576, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 144, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_144_P", "rows": [ { "energy": -386.31278592, "sigma": 0.0000115, "energy_variance": null, "dof": 144, "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 = -0.67068192(2) per spin (S.S), L = 12, 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 = 12. 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_144_P/", "per_site_divisor": 576, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 16, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_16_P", "rows": [ { "energy": -44.9139328, "sigma": null, "energy_variance": null, "dof": 16, "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_16_P/vqe.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -44.9116731, "sigma": null, "energy_variance": 0.1064, "dof": 16, "einf": 0, "v_score": 0.000844001351839292, "method": "VQE + symm. circuit (64 pars., exact grad, statevector)", "bound_type": "variational", "bound_type_reason": "\\bvqe\\b|\\bcircuit\\b|\\bpq", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_16_P/vqe.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -44.9104126, "sigma": null, "energy_variance": 0.1599, "dof": 16, "einf": 0, "v_score": 0.0012684529311267963, "method": "VQE + symm. circuit (64 pars., 2^14 samples/grad)", "bound_type": "variational", "bound_type_reason": "\\bvqe\\b|\\bcircuit\\b|\\bpq", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_16_P/vqe_noisy.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true } ], "url": "https://qmbl.org/i/Heisenberg/square_16_P/", "per_site_divisor": 64, "per_site_label": "E/N (S.S)", "record": { "energy": -44.9116731, "sigma": null, "method": "VQE + symm. circuit (64 pars., exact grad, statevector)", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_16_P/vqe.sh)", "energy_per_site": -0.7017448921875 }, "no_record_reason": null }, { "model": "Heisenberg", "lattice": "square", "n_sites": 1600, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_1600_P", "rows": [ { "energy": -4284.657664, "sigma": 0.000192, "energy_variance": null, "dof": 1600, "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 = -0.66947776(3) per spin (S.S), L = 40, 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 = 40. 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_1600_P/", "per_site_divisor": 6400, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 16384, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_16384_O", "rows": [ { "energy": -43656.0784179, "sigma": 0.0117, "energy_variance": null, "dof": 16384, "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/N_b = 0.33569204(9) (magnitude, per bond, N_b = 32512), L = 128, open", "note": "Table 2, \"SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)\", row L = 128. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.66613889 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_16384_O/", "per_site_divisor": 65536, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 1936, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_1936_P", "rows": [ { "energy": -5184.36708416, "sigma": 0.000232, "energy_variance": null, "dof": 1936, "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 = -0.66946889(3) per spin (S.S), L = 44, 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 = 44. 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_1936_P/", "per_site_divisor": 7744, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 196, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_196_O", "rows": [ { "energy": -501.7137216, "sigma": 0.000146, "energy_variance": null, "dof": 196, "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/N_b = 0.3445836(1) (magnitude, per bond, N_b = 364), L = 14, open", "note": "Table 2, \"SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)\", row L = 14. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.63994097 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_196_O/", "per_site_divisor": 784, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 196, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_196_P", "rows": [ { "energy": -523.983, "sigma": 0.006, "energy_variance": 22.07, "dof": 196, "einf": 0, "v_score": 0.0157552032593818, "method": "VMC with fermions (flux+neel+Jastrow)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_196_P/vmc_gutzwiller.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -516.3911695223306, "sigma": null, "energy_variance": 127.79517553787447, "dof": 100, "einf": 0, "v_score": 0.04792441769886959, "method": "DMRG (bond dimension = 512)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_196_P/dmrg.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true, "defect": { "flag": "dof-mismatch", "finding": "Row stores dof = 100 on a 196-site instance. Upstream typo; the V-score derived from it would be wrong by a factor of ~2." } }, { "energy": -520.4234, "sigma": 0.0079, "energy_variance": 63.70537, "dof": 196, "einf": 0, "v_score": 0.04610186370257413, "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_196_P/vmc_rbm.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -519.4475, "sigma": 0.0081, "energy_variance": 66.74661, "dof": 196, "einf": 0, "v_score": 0.04848439309211061, "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_196_P/vmc_jastrow.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -525.462084, "sigma": 0.0000157, "energy_variance": null, "dof": 196, "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 = -0.67023225(2) per spin (S.S), L = 14, 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 = 14. 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_196_P/", "per_site_divisor": 784, "per_site_label": "E/N (S.S)", "record": { "energy": -523.983, "sigma": 0.006, "method": "VMC with fermions (flux+neel+Jastrow)", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_196_P/vmc_gutzwiller.sh)", "energy_per_site": -0.6683456632653061 }, "no_record_reason": null }, { "model": "Heisenberg", "lattice": "square", "n_sites": 2304, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_2304_O", "rows": [ { "energy": -6088.7256192, "sigma": 0.00361, "energy_variance": null, "dof": 2304, "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/N_b = 0.3373629(2) (magnitude, per bond, N_b = 4512), L = 48, open", "note": "Table 2, \"SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)\", row L = 48. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.66066901 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_2304_O/", "per_site_divisor": 9216, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 2304, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_2304_P", "rows": [ { "energy": -6169.76861184, "sigma": 0.000276, "energy_variance": null, "dof": 2304, "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 = -0.66946274(3) per spin (S.S), L = 48, 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 = 48. 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_2304_P/", "per_site_divisor": 9216, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 256, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_256_O", "rows": [ { "energy": -658.9764608, "sigma": 0.0002048, "energy_variance": null, "dof": 256, "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": "-0.6435317 (+/- 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.", "secondary_of": "Parallel Scan Recurrent Neural Quantum States, arXiv:2605.13807", "found_via": null } }, { "energy": -658.948096, "sigma": 0.003072, "energy_variance": null, "dof": 256, "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": "-0.643504 (+/- 0.000003) 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. The citing paper's own result; 29 GPU-days on a single L40S.", "secondary_of": null, "found_via": null } }, { "energy": -658.890752, "sigma": 0.001024, "energy_variance": null, "dof": 256, "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": "-0.643448 (+/- 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]\".", "secondary_of": "Parallel Scan Recurrent Neural Quantum States, arXiv:2605.13807", "found_via": null } }, { "energy": -658.832384, "sigma": 0.003072, "energy_variance": null, "dof": 256, "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": "-0.643391 (+/- 0.000003) 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]\".", "secondary_of": "Parallel Scan Recurrent Neural Quantum States, arXiv:2605.13807", "found_via": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_256_O/", "per_site_divisor": 1024, "per_site_label": "E/N (S.S)", "record": { "energy": -658.948096, "sigma": 0.003072, "method": "2D minGRU (3 layers, c4v symmetry, parallel scan)", "reference": "Parallel Scan Recurrent Neural Quantum States, arXiv:2605.13807", "energy_per_site": -0.643504 }, "no_record_reason": null }, { "model": "Heisenberg", "lattice": "square", "n_sites": 256, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_256_P", "rows": [ { "energy": -686.0560384, "sigma": 0.0000205, "energy_variance": null, "dof": 256, "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 = -0.66997660(2) per spin (S.S), L = 16, 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 = 16. 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_256_P/", "per_site_divisor": 1024, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 2704, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_2704_P", "rows": [ { "energy": -7240.86129728, "sigma": 0.000324, "energy_variance": null, "dof": 2704, "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 = -0.66945833(3) per spin (S.S), L = 52, 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 = 52. 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_2704_P/", "per_site_divisor": 10816, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 3136, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_3136_P", "rows": [ { "energy": -8397.64377088, "sigma": 0.000376, "energy_variance": null, "dof": 3136, "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 = -0.66945502(3) per spin (S.S), L = 56, 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 = 56. 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_3136_P/", "per_site_divisor": 12544, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 324, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_324_O", "rows": [ { "energy": -837.6632496, "sigma": 0.00049, "energy_variance": null, "dof": 324, "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/N_b = 0.3421827(2) (magnitude, per bond, N_b = 612), L = 18, open", "note": "Table 2, \"SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)\", row L = 18. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.64634510 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_324_O/", "per_site_divisor": 1296, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 324, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_324_P", "rows": [ { "energy": -868.08727728, "sigma": 0.0000259, "energy_variance": null, "dof": 324, "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 = -0.66982043(2) per spin (S.S), L = 18, 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 = 18. 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_324_P/", "per_site_divisor": 1296, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 36, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_36_O", "rows": [ { "energy": -86.9071441699763, "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_O/ed_lattice_symmetries.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -86.90713263791655, "sigma": null, "energy_variance": 0.00045932030843687244, "dof": 36, "einf": 0, "v_score": 0.0000021893102276886876, "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_O/dmrg.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -85.4857, "sigma": 0.0035, "energy_variance": 12.2737, "dof": 36, "einf": 0, "v_score": 0.06046319048946153, "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_O/vmc_rbm.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -85.6567, "sigma": 0.0036, "energy_variance": 13.457, "dof": 36, "einf": 0, "v_score": 0.06602798980657479, "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_O/vmc_jastrow.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -86.907192, "sigma": 0.000024, "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/N_b = 0.3621133(1) (magnitude, per bond, N_b = 60), L = 6, open", "note": "Table 2, \"SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)\", row L = 6. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.60352217 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_36_O/", "per_site_divisor": 144, "per_site_label": "E/N (S.S)", "record": { "energy": -86.90713263791655, "sigma": null, "method": "DMRG (bond dimension = 2048)", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_36_O/dmrg.sh)", "energy_per_site": -0.603521754429976 }, "no_record_reason": null }, { "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 }, { "model": "Heisenberg", "lattice": "square", "n_sites": 3600, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_3600_P", "rows": [ { "energy": -9640.11744, "sigma": 0.000432, "energy_variance": null, "dof": 3600, "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 = -0.66945260(3) per spin (S.S), L = 60, 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 = 60. 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_3600_P/", "per_site_divisor": 14400, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 40, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_40_P", "rows": [ { "energy": -108.379401, "sigma": null, "energy_variance": null, "dof": 40, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Richter & Schulenburg, The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins, Eur. Phys. J. B 73, 117 (2010), arXiv:0909.3723", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "E_GS(S = 0) = -27.09485025 (total, S.S, J1 = 1), N = 40, J2 = 0", "note": "Table 1, \"Ground state energy E_GS(S = 0) ...\", row J2 = 0. The tilted 40-site square-lattice cluster of the paper's Fig. 1 with periodic boundaries (Sec. 2); E/N = -0.6773713 in S.S units, Pauli total = 4 E. J2 = 0 is the plain Heisenberg antiferromagnet on this cluster.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_40_P/", "per_site_divisor": 160, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 400, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_400_O", "rows": [ { "energy": -1037.774528, "sigma": 0.000608, "energy_variance": null, "dof": 400, "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/N_b = 0.3413732(2) (magnitude, per bond, N_b = 760), L = 20, open", "note": "Table 2, \"SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)\", row L = 20. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.64860908 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_400_O/", "per_site_divisor": 1600, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 400, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_400_P", "rows": [ { "energy": -1071.551472, "sigma": 0.000032, "energy_variance": null, "dof": 400, "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 = -0.66971967(2) per spin (S.S), L = 20, 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 = 20. 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_400_P/", "per_site_divisor": 1600, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 4096, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_4096_O", "rows": [ { "energy": -10860.159744, "sigma": 0.00645, "energy_variance": null, "dof": 4096, "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/N_b = 0.3366865(2) (magnitude, per bond, N_b = 8064), L = 64, open", "note": "Table 2, \"SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)\", row L = 64. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.66285155 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_4096_O/", "per_site_divisor": 16384, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 4096, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_4096_P", "rows": [ { "energy": -10968.2805965, "sigma": 0.000492, "energy_variance": null, "dof": 4096, "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 = -0.66945072(3) per spin (S.S), L = 64, 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 = 64. 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_4096_P/", "per_site_divisor": 16384, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 484, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_484_O", "rows": [ { "energy": -1259.3077728, "sigma": 0.000739, "energy_variance": null, "dof": 484, "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/N_b = 0.3407218(2) (magnitude, per bond, N_b = 924), L = 22, open", "note": "Table 2, \"SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)\", row L = 22. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.65046889 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_484_O/", "per_site_divisor": 1936, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 484, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_484_P", "rows": [ { "energy": -1296.44580736, "sigma": 0.0000387, "energy_variance": null, "dof": 484, "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 = -0.66965176(2) per spin (S.S), L = 22, 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 = 22. 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_484_P/", "per_site_divisor": 1936, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 50, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_50_P", "rows": [ { "energy": -135.020407726, "sigma": 4e-10, "energy_variance": null, "dof": 50, "einf": 0, "v_score": null, "method": "Exact Diagonalization", "bound_type": "exact", "bound_type_reason": "exact diagonalization|ex", "reference": "[paper](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.98.033309)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Heisenberg/square_50_P/", "per_site_divisor": 200, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 5184, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_5184_P", "rows": [ { "energy": -13881.6768384, "sigma": 0.000622, "energy_variance": null, "dof": 5184, "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 = -0.66944815(3) per spin (S.S), L = 72, 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 = 72. 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_5184_P/", "per_site_divisor": 20736, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 576, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_576_O", "rows": [ { "energy": -1502.265792, "sigma": 0.000883, "energy_variance": null, "dof": 576, "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/N_b = 0.3401870(2) (magnitude, per bond, N_b = 1104), L = 24, open", "note": "Table 2, \"SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)\", row L = 24. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.65202508 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_576_O/", "per_site_divisor": 2304, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 576, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_576_P", "rows": [ { "energy": -1542.76839936, "sigma": 0.0000461, "energy_variance": null, "dof": 576, "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 = -0.66960434(2) per spin (S.S), L = 24, 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 = 24. 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_576_P/", "per_site_divisor": 2304, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 64, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_64_O", "rows": [ { "energy": -158.473504, "sigma": 0.0000448, "energy_variance": null, "dof": 64, "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/N_b = 0.3537355(1) (magnitude, per bond, N_b = 112), L = 8, open", "note": "Table 2, \"SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)\", row L = 8. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.61903712 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_64_O/", "per_site_divisor": 256, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 64, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_64_P", "rows": [ { "energy": -172.4134528, "sigma": 0.00000512, "energy_variance": null, "dof": 64, "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 = -0.67349005(2) per spin (S.S), L = 8, 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 = 8. 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_64_P/", "per_site_divisor": 256, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 6400, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_6400_P", "rows": [ { "energy": -17137.82912, "sigma": 0.000768, "energy_variance": null, "dof": 6400, "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 = -0.66944645(3) per spin (S.S), L = 80, 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 = 80. 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_6400_P/", "per_site_divisor": 25600, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 676, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_676_O", "rows": [ { "energy": -1766.64644, "sigma": 0.00104, "energy_variance": null, "dof": 676, "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/N_b = 0.3397397(2) (magnitude, per bond, N_b = 1300), L = 26, open", "note": "Table 2, \"SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)\", row L = 26. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.65334558 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_676_O/", "per_site_divisor": 2704, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 676, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_676_P", "rows": [ { "energy": -1810.51773968, "sigma": 0.0000811, "energy_variance": null, "dof": 676, "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 = -0.66957017(3) per spin (S.S), L = 26, 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 = 26. 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_676_P/", "per_site_divisor": 2704, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 7744, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_7744_P", "rows": [ { "energy": -20736.7376128, "sigma": 0.000929, "energy_variance": null, "dof": 7744, "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 = -0.66944530(3) per spin (S.S), L = 88, 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 = 88. 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_7744_P/", "per_site_divisor": 30976, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 784, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_784_O", "rows": [ { "energy": -2052.4510944, "sigma": 0.00121, "energy_variance": null, "dof": 784, "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/N_b = 0.3393603(2) (magnitude, per bond, N_b = 1512), L = 28, open", "note": "Table 2, \"SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)\", row L = 28. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.65448058 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_784_O/", "per_site_divisor": 3136, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 784, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_784_P", "rows": [ { "energy": -2099.69299456, "sigma": 0.0000941, "energy_variance": null, "dof": 784, "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 = -0.66954496(3) per spin (S.S), L = 28, 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 = 28. 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_784_P/", "per_site_divisor": 3136, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 900, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_900_P", "rows": [ { "energy": -2410.29342, "sigma": 0.000108, "energy_variance": null, "dof": 900, "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 = -0.66952595(3) per spin (S.S), L = 30, 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 = 30. 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_900_P/", "per_site_divisor": 3600, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 9216, "boundary": "O", "params": {}, "instance_id": "Heisenberg/square_9216_O", "rows": [ { "energy": -24516.0892416, "sigma": 0.00584, "energy_variance": null, "dof": 9216, "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/N_b = 0.33602096(8) (magnitude, per bond, N_b = 18240), L = 96, open", "note": "Table 2, \"SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)\", row L = 96. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.66504148 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Heisenberg/square_9216_O/", "per_site_divisor": 36864, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "square", "n_sites": 9216, "boundary": "P", "params": {}, "instance_id": "Heisenberg/square_9216_P", "rows": [ { "energy": -24678.4035226, "sigma": 0.00111, "energy_variance": null, "dof": 9216, "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 = -0.66944454(3) per spin (S.S), L = 96, 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 = 96. 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_9216_P/", "per_site_divisor": 36864, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "Heisenberg", "lattice": "triangular", "n_sites": 100, "boundary": "O", "params": {}, "instance_id": "Heisenberg/triangular_100_O", "rows": [ { "energy": -200.276, "sigma": 0.004, "energy_variance": 5.032, "dof": 100, "einf": 0, "v_score": 0.01254535094003916, "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/triangular_100_O/vmc_rnn.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -205.90673107198964, "sigma": null, "energy_variance": 11.224306962726814, "dof": 100, "einf": 0, "v_score": 0.026473931907252366, "method": "DMRG (bond dimension = 4096)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/triangular_100_O/dmrg.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -195.8572, "sigma": 0.0089, "energy_variance": 80.60808, "dof": 100, "einf": 0, "v_score": 0.21013553201026944, "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": -203.2108, "sigma": 0.005, "energy_variance": 25.73127, "dof": 100, "einf": 0, "v_score": 0.06231142033639836, "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": -205.5452, "sigma": 0.0035, "energy_variance": 12.31392, "dof": 100, "einf": 0, "v_score": 0.029146180428250745, "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": -193.8193, "sigma": 0.0082, "energy_variance": 69.24535, "dof": 100, "einf": 0, "v_score": 0.1843302335200169, "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/triangular_100_O/vmc_rbm.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -190.6983, "sigma": 0.0093, "energy_variance": 87.62059, "dof": 100, "einf": 0, "v_score": 0.24094201066103857, "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/triangular_100_O/vmc_jastrow.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true } ], "url": "https://qmbl.org/i/Heisenberg/triangular_100_O/", "per_site_divisor": 400, "per_site_label": "E/N (S.S)", "record": { "energy": -205.90673107198964, "sigma": null, "method": "DMRG (bond dimension = 4096)", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/triangular_100_O/dmrg.sh)", "energy_per_site": -0.5147668276799741 }, "no_record_reason": null }, { "model": "Heisenberg", "lattice": "triangular", "n_sites": 108, "boundary": "P", "params": {}, "instance_id": "Heisenberg/triangular_108_P", "rows": [ { "energy": -238.9608, "sigma": 0.01296, "energy_variance": null, "dof": 108, "einf": 0, "v_score": null, "method": "GCNN (deep group-equivariant CNN)", "bound_type": "variational", "bound_type_reason": "assigned during source verification (RULES.md 4)", "reference": "Roth, Szabo & MacDonald, High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks, Phys. Rev. B 108, 054410 (2023), arXiv:2211.07749", "peer_reviewed": true, "source": "sweep-pdf-2026-09-14", "provenance": "primary", "verified": { "checked_on": "2026-09-14", "method": "arXiv PDF extracted locally with pypdf in layout mode, no LLM transcription", "reported_as": "-0.55315 (+/- 0.00003) per site in S.S units", "note": "Read from Table III of arXiv:2211.07749 (J1-J2 triangular lattice, periodic, energies in units of J1 per spin, i.e. the S.S per-site convention), extracted from the PDF with pypdf in layout mode so the column positions are unambiguous. Row J2/J1 = 0, N = 108.", "secondary_of": null } }, { "energy": -238.4208, "sigma": 0.1728, "energy_variance": null, "dof": 108, "einf": 0, "v_score": null, "method": "Graph neural network", "bound_type": "variational", "bound_type_reason": "assigned during source verification (RULES.md 4)", "reference": "Kochkov, Pfaff, Sanchez-Gonzalez, Battaglia & Clark, Learning ground states of quantum Hamiltonians with graph networks, arXiv:2110.06390 (2021)", "peer_reviewed": null, "source": "sweep-pdf-2026-09-14", "provenance": "secondary", "verified": { "checked_on": "2026-09-14", "method": "arXiv PDF extracted locally with pypdf in layout mode, no LLM transcription", "reported_as": "-0.5519 (+/- 0.0004) per site in S.S units", "note": "Read from Table III of arXiv:2211.07749 (J1-J2 triangular lattice, periodic, energies in units of J1 per spin, i.e. the S.S per-site convention), extracted from the PDF with pypdf in layout mode so the column positions are unambiguous. Column \"Graph NN [64]\" at J2/J1 = 0, N = 108 = arXiv:2110.06390.", "secondary_of": "Roth, Szabo & MacDonald, High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks, Phys. Rev. B 108, 054410 (2023), arXiv:2211.07749" } } ], "url": "https://qmbl.org/i/Heisenberg/triangular_108_P/", "per_site_divisor": 432, "per_site_label": "E/N (S.S)", "record": { "energy": -238.9608, "sigma": 0.01296, "method": "GCNN (deep group-equivariant CNN)", "reference": "Roth, Szabo & MacDonald, High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks, Phys. Rev. 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Same typo." } }, { "energy": -282.04227, "sigma": 0.013, "energy_variance": 168.431, "dof": 144, "einf": 0, "v_score": 0.3048990817330671, "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/triangular_144_P/vmc_rbm.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -279.66142, "sigma": 0.012, "energy_variance": 154.749, "dof": 144, "einf": 0, "v_score": 0.28492150283393475, "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/triangular_144_P/vmc_jastrow.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true } ], "url": "https://qmbl.org/i/Heisenberg/triangular_144_P/", "per_site_divisor": 576, "per_site_label": "E/N (S.S)", 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Row \"Group Convolutional Neural Network [32]\" = arXiv:2211.07749. Sits 6.0e-5 ABOVE the exact -0.5603734 already on this instance, as a variational bound must. CONFIRMED IN PRIMARY SOURCE - arXiv:2211.07749, results table: \"J2/J1 N GCNN ... 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Row \"Lattice Convolutional Network [49]\" = arXiv:2206.07370. 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Row \"Group Convolutional Neural Network [104]\" = arXiv:2104.05085. No error bar is given in the table, so under RULES.md 6 this sampled row cannot hold a record. CONFIRMED IN PRIMARY SOURCE - arXiv:2104.05085, results table: \"J2 = 0 ... ED [28] -0.5603734 ... 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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. 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-0.514421875 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "rectangular-4x8", "n_sites": 32, "boundary": "P", "params": { "Nf": 14, "U": 8 }, "instance_id": "Hubbard/rectangular-4x8_32_P_14_8", "rows": [ { "energy": -24.426, "sigma": 0.002, "energy_variance": 0.021, "dof": 28, "einf": 49, "v_score": 0.00010906303678744678, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 14). 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": -24.5408, "sigma": 0.00032, "energy_variance": 0.52224, "dof": 28, "einf": 49, "v_score": 0.0027037807032315774, "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-worklist-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF text extracted locally with pypdf in layout mode, no LLM transcription", "reported_as": "-0.76690(1) per site, sigma^2/N_site = 0.016320", "note": "Table IV of arXiv:2507.10705, \"Raw data of Fig. 7 in the L x 4 lattice with 1/8 doping and U = 8\", row 8 x 4; Sec. III states Fig. 7 shows \"L x 4 PBC lattices\". 32 sites, 28 electrons, U = 8, periodic: this instance. The same table's 16 x 4 value -0.76413(3) is the HFPS row on rectangular-4x16_64_P_28_8, and its 4 x 4 value -0.74177(5) sits 3e-5 above the ED energy -0.7418 quoted by arXiv:2606.00924. New record, 0.47% below the VarBench HFDS row -0.7633125 and below the RHB -0.7641(3) of arXiv:2606.00924 on the same lattice. Variance: sigma^2/N_site stored as a total (x 32). Its V-score, 2.7e-3, is ~25x the VarBench HFDS row's 1.1e-4 at a higher energy; the HFDS variance is the likelier outlier, since on 4x16 the same VarBench ansatz has ten times the per-site variance and this paper puts HFDS 3e-4 above exact already at 4x4. Read from the arXiv version; the PNAS version was not compared (pnas.org refuses this sandbox).", "secondary_of": null } }, { "energy": -24.4512, "sigma": 0.0096, "energy_variance": null, "dof": 28, "einf": 49, "v_score": null, "method": "RHB (hierarchical backflow K = 2 + nonlocal FNN factor, Ndet = 5, Nneuron = 100)", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "Zhou, Zhou & Liu, Locality-Induced Hierarchical Backflow Wavefunctions for Correlated Fermions, arXiv:2606.00924", "peer_reviewed": false, "source": "sweep-worklist-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF text extracted locally with pypdf in layout mode, no LLM transcription", "reported_as": "-0.7641 per site, table-wide sampling error ~0.0003", "note": "Table IV of arXiv:2606.00924, \"Energies per site for the Hubbard model at nh = 0.125, U = 8 for different systems under periodic boundary conditions. The energy sampling errors are around 0.0003\", row 4 x 8, column RHB (Ndet = 5, Nneuron = 100, the setting adopted for all large-scale runs after Table III). Instance identity: the same row's HFDS column, -0.7633, is the VarBench HFDS row -0.7633125. sigma is the caption's table-wide \"around 0.0003\". Layout mode shifts this row one column left in sweep-tables.tsv; read here from the printed table. Internal check: the same ansatz gives -0.7410 on 4 x 4 against ED -0.7418 (Table III), above exact as it must be.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/rectangular-4x8_32_P_14_8/", "per_site_divisor": 32, "per_site_label": "E/site", "record": { "energy": -24.5408, "sigma": 0.00032, "method": "HFPS (hidden-fermion Pfaffian state)", "reference": "Chen, Wan, Sengupta & Georges, Neural network-augmented Pfaffian wave-functions for scalable simulations of interacting fermions, Proc. Natl. Acad. Sci. 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Single hidden layer fully connected net with alpha = 14). 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": -23.46944, "sigma": 0.00256, "energy_variance": 0.72, "dof": 28, "einf": 49, "v_score": 0.0038386694558110793, "method": "Hidden-fermion determinant state (HFDS), 8x4 cylinder", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. B 113, 245104 (2026), arXiv:2510.11710", "peer_reviewed": true, "source": "sweep-tables-2026-09-14", "provenance": "primary", "verified": { "checked_on": "2026-09-14", "method": "arXiv HTML parsed locally, no LLM transcription", "reported_as": "E_0/M = -0.73342 (+/- 0.00008), sigma^2/M = 0.0225", "note": "Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Row \"HFDS (8x4)\".", "secondary_of": null } }, { "energy": -23.4624, "sigma": 0.0192, "energy_variance": 0.752, "dof": 28, "einf": 49, "v_score": 0.004010056058287358, "method": "Jastrow-backflow (JBf), 8x4 cylinder", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. B 113, 245104 (2026), arXiv:2510.11710", "peer_reviewed": true, "source": "sweep-tables-2026-09-14", "provenance": "primary", "verified": { "checked_on": "2026-09-14", "method": "arXiv HTML parsed locally, no LLM transcription", "reported_as": "E_0/M = -0.7332 (+/- 0.0006), sigma^2/M = 0.0235", "note": "Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Row \"JBf (8x4)\".", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/rectangular-4x8_32_PO_14_8/", "per_site_divisor": 32, "per_site_label": "E/site", "record": { "energy": -23.5645, "sigma": null, "method": "DMRG (MaxLinkDim = 12000, MaxTruncErr~6.6E-06, extrap energy -23.5658 +/- 0.0003)", "reference": "[code](https://github.com/varbench/methods/blob/main/programs/dmrg_itensors_hubbard/rectangular-4x8_32_PO_14_8.jl)", "energy_per_site": -0.736390625 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "rectangular-6x48", "n_sites": 288, "boundary": "P", "params": { "Nf": 120, "U": 8 }, "instance_id": "Hubbard/rectangular-6x48_288_P_120_8", "rows": [ { "energy": -236.376, "sigma": 0.009, "energy_variance": 17.3, "dof": 240, "einf": 400, "v_score": 0.01025249957247314, "method": "VMC with stripe of wavelength 6 (+Jastrow and backflow)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": 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Release; no strict upper bound property", "bound_type": "projected", "bound_type_reason": "no strict upper bound|co", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.88.125132)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_100_P_40_4/", "per_site_divisor": 100, "per_site_label": "E/site", "record": null, "no_record_reason": "no variational row" }, { "model": "Hubbard", "lattice": "square", "n_sites": 100, "boundary": "P", "params": { "Nf": 50, "U": 2 }, "instance_id": "Hubbard/square_100_P_50_2", "rows": [ { "energy": -116.908, "sigma": 0.004, "energy_variance": null, "dof": 100, "einf": 50, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -116.908(4) (total), 10 x 10, U = 2, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 10 x 10, column U = 2 PBC. Half filling: N_up = N_dn = 50. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_100_P_50_2/", "per_site_divisor": 100, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 100, "boundary": "P", "params": { "Nf": 50, "U": 4 }, "instance_id": "Hubbard/square_100_P_50_4", "rows": [ { "energy": -86.12, "sigma": 0.04, "energy_variance": null, "dof": 100, "einf": 100, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -86.12(4) (total), 10 x 10, U = 4, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 10 x 10, column U = 4 PBC. Half filling: N_up = N_dn = 50. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_100_P_50_4/", "per_site_divisor": 100, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 100, "boundary": "P", "params": { "Nf": 50, "U": 6 }, "instance_id": "Hubbard/square_100_P_50_6", "rows": [ { "energy": -65.8, "sigma": 0.02, "energy_variance": null, "dof": 100, "einf": 150, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -65.80(2) (total), 10 x 10, U = 6, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 10 x 10, column U = 6 PBC. Half filling: N_up = N_dn = 50. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_100_P_50_6/", "per_site_divisor": 100, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 100, "boundary": "P", "params": { "Nf": 50, "U": 8 }, "instance_id": "Hubbard/square_100_P_50_8", "rows": [ { "energy": -52.54, "sigma": 0.03, "energy_variance": null, "dof": 100, "einf": 200, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -52.54(3) (total), 10 x 10, U = 8, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 10 x 10, column U = 8 PBC. Half filling: N_up = N_dn = 50. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_100_P_50_8/", "per_site_divisor": 100, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 100, "boundary": "PA", "params": { "Nf": 50, "U": 2 }, "instance_id": "Hubbard/square_100_PA_50_2", "rows": [ { "energy": -118.505, "sigma": 0.004, "energy_variance": null, "dof": 100, "einf": 50, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -118.505(4) (total), 10 x 10, U = 2, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 10 x 10, column U = 2 PBC-APBC. Half filling: N_up = N_dn = 50. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_100_PA_50_2/", "per_site_divisor": 100, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 100, "boundary": "PA", "params": { "Nf": 50, "U": 4 }, "instance_id": "Hubbard/square_100_PA_50_4", "rows": [ { "energy": -86.2, "sigma": 0.02, "energy_variance": null, "dof": 100, "einf": 100, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -86.20(2) (total), 10 x 10, U = 4, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 10 x 10, column U = 4 PBC-APBC. Half filling: N_up = N_dn = 50. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_100_PA_50_4/", "per_site_divisor": 100, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 100, "boundary": "PA", "params": { "Nf": 50, "U": 6 }, "instance_id": "Hubbard/square_100_PA_50_6", "rows": [ { "energy": -65.76, "sigma": 0.02, "energy_variance": null, "dof": 100, "einf": 150, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -65.76(2) (total), 10 x 10, U = 6, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 10 x 10, column U = 6 PBC-APBC. Half filling: N_up = N_dn = 50. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_100_PA_50_6/", "per_site_divisor": 100, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 100, "boundary": "PA", "params": { "Nf": 50, "U": 8 }, "instance_id": "Hubbard/square_100_PA_50_8", "rows": [ { "energy": -52.49, "sigma": 0.02, "energy_variance": null, "dof": 100, "einf": 200, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -52.49(2) (total), 10 x 10, U = 8, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 10 x 10, column U = 8 PBC-APBC. Half filling: N_up = N_dn = 50. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_100_PA_50_8/", "per_site_divisor": 100, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 144, "boundary": "P", "params": { "Nf": 58, "U": 4 }, "instance_id": "Hubbard/square_144_P_58_4", "rows": [ { "energy": -159.6816, "sigma": 0.0288, "energy_variance": null, "dof": 116, "einf": 93.44444444444444, "v_score": null, "method": "CP AFQMC with Constraint Release; no strict upper bound property", "bound_type": "projected", "bound_type_reason": "no strict upper bound|co", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.88.125132)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_144_P_58_4/", "per_site_divisor": 144, "per_site_label": "E/site", "record": null, "no_record_reason": "no variational row" }, { "model": "Hubbard", "lattice": "square", "n_sites": 144, "boundary": "P", "params": { "Nf": 72, "U": 2 }, "instance_id": "Hubbard/square_144_P_72_2", "rows": [ { "energy": -168.749, "sigma": 0.007, "energy_variance": null, "dof": 144, "einf": 72, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -168.749(7) (total), 12 x 12, U = 2, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 12 x 12, column U = 2 PBC. Half filling: N_up = N_dn = 72. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_144_P_72_2/", "per_site_divisor": 144, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 144, "boundary": "P", "params": { "Nf": 72, "U": 4 }, "instance_id": "Hubbard/square_144_P_72_4", "rows": [ { "energy": -123.95, "sigma": 0.02, "energy_variance": null, "dof": 144, "einf": 144, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -123.95(2) (total), 12 x 12, U = 4, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 12 x 12, column U = 4 PBC. Half filling: N_up = N_dn = 72. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_144_P_72_4/", "per_site_divisor": 144, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 144, "boundary": "P", "params": { "Nf": 72, "U": 6 }, "instance_id": "Hubbard/square_144_P_72_6", "rows": [ { "energy": -94.66, "sigma": 0.02, "energy_variance": null, "dof": 144, "einf": 216, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -94.66(2) (total), 12 x 12, U = 6, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 12 x 12, column U = 6 PBC. Half filling: N_up = N_dn = 72. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_144_P_72_6/", "per_site_divisor": 144, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 144, "boundary": "P", "params": { "Nf": 72, "U": 8 }, "instance_id": "Hubbard/square_144_P_72_8", "rows": [ { "energy": -75.54, "sigma": 0.02, "energy_variance": null, "dof": 144, "einf": 288, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -75.54(2) (total), 12 x 12, U = 8, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 12 x 12, column U = 8 PBC. Half filling: N_up = N_dn = 72. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_144_P_72_8/", "per_site_divisor": 144, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 144, "boundary": "PA", "params": { "Nf": 72, "U": 2 }, "instance_id": "Hubbard/square_144_PA_72_2", "rows": [ { "energy": -170.112, "sigma": 0.003, "energy_variance": null, "dof": 144, "einf": 72, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -170.112(3) (total), 12 x 12, U = 2, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 12 x 12, column U = 2 PBC-APBC. Half filling: N_up = N_dn = 72. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_144_PA_72_2/", "per_site_divisor": 144, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 144, "boundary": "PA", "params": { "Nf": 72, "U": 4 }, "instance_id": "Hubbard/square_144_PA_72_4", "rows": [ { "energy": -123.99, "sigma": 0.03, "energy_variance": null, "dof": 144, "einf": 144, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -123.99(3) (total), 12 x 12, U = 4, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 12 x 12, column U = 4 PBC-APBC. Half filling: N_up = N_dn = 72. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_144_PA_72_4/", "per_site_divisor": 144, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 144, "boundary": "PA", "params": { "Nf": 72, "U": 6 }, "instance_id": "Hubbard/square_144_PA_72_6", "rows": [ { "energy": -94.67, "sigma": 0.02, "energy_variance": null, "dof": 144, "einf": 216, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -94.67(2) (total), 12 x 12, U = 6, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 12 x 12, column U = 6 PBC-APBC. Half filling: N_up = N_dn = 72. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_144_PA_72_6/", "per_site_divisor": 144, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 144, "boundary": "PA", "params": { "Nf": 72, "U": 8 }, "instance_id": "Hubbard/square_144_PA_72_8", "rows": [ { "energy": -75.58, "sigma": 0.03, "energy_variance": null, "dof": 144, "einf": 288, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -75.58(3) (total), 12 x 12, U = 8, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 12 x 12, column U = 8 PBC-APBC. Half filling: N_up = N_dn = 72. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_144_PA_72_8/", "per_site_divisor": 144, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 16, "boundary": "P", "params": { "Nf": 4, "U": 10 }, "instance_id": "Hubbard/square_16_P_4_10", "rows": [ { "energy": -16.143208170202413, "sigma": null, "energy_variance": null, "dof": 8, "einf": 10, "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/Hubbard/square_16_P_4_10/ed_lattice_symmetries.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -16.1383, "sigma": 0.0001, "energy_variance": 0.078, "dof": 8, "einf": 10, "v_score": 0.0009133346059135673, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 8. Single hidden layer fully connected net with alpha = 32)", "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": -16.14320807, "sigma": null, "energy_variance": 1.81e-7, "dof": 8, "einf": 10, "v_score": 2.118608952138064e-9, "method": "DMRG (MaxBondDim = 3200)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/programs/dmrg_itensors_hubbard/square_16_P_4_10.jl)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true } ], "url": "https://qmbl.org/i/Hubbard/square_16_P_4_10/", "per_site_divisor": 16, "per_site_label": "E/site", "record": { "energy": -16.14320807, "sigma": null, "method": "DMRG (MaxBondDim = 3200)", "reference": "[code](https://github.com/varbench/methods/blob/main/programs/dmrg_itensors_hubbard/square_16_P_4_10.jl)", "energy_per_site": -1.008950504375 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 16, "boundary": "P", "params": { "Nf": 4, "U": 2 }, "instance_id": "Hubbard/square_16_P_4_2", "rows": [ { "energy": -18.495436619460737, "sigma": null, "energy_variance": null, "dof": 8, "einf": 2, "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/Hubbard/square_16_P_4_2/ed_lattice_symmetries.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -18.495436027, "sigma": null, "energy_variance": 2.067e-7, "dof": 8, "einf": 2, "v_score": 3.936553330755232e-9, "method": "DMRG (MaxLinkDim ~ 3200)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/programs/dmrg_itensors_hubbard/square_16_P_4_2.jl)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true } ], "url": "https://qmbl.org/i/Hubbard/square_16_P_4_2/", "per_site_divisor": 16, "per_site_label": "E/site", "record": { "energy": -18.495436027, "sigma": null, "method": "DMRG (MaxLinkDim ~ 3200)", "reference": "[code](https://github.com/varbench/methods/blob/main/programs/dmrg_itensors_hubbard/square_16_P_4_2.jl)", "energy_per_site": -1.1559647516875 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 16, "boundary": "P", "params": { "Nf": 4, "U": 3.5981 }, "instance_id": "Hubbard/square_16_P_4_3.5981", "rows": [ { "energy": -17.698030018702067, "sigma": null, "energy_variance": null, "dof": 8, "einf": 3.5981, "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/Hubbard/square_16_P_4_3.5981/ed_lattice_symmetries.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -17.69357, "sigma": 0.00004, "energy_variance": 0.033, "dof": 8, "einf": 3.5981, "v_score": 0.0005823505286323852, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 8. 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Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_16_P_8_4/", "per_site_divisor": 16, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 16, "boundary": "P", "params": { "Nf": 8, "U": 6 }, "instance_id": "Hubbard/square_16_P_8_6", "rows": [ { "energy": -10.541, "sigma": 0.004, "energy_variance": null, "dof": 16, "einf": 24, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -10.541(4) (total), 4 x 4, U = 6, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 4 x 4, column U = 6 PBC. Half filling: N_up = N_dn = 8. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_16_P_8_6/", "per_site_divisor": 16, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 16, "boundary": "P", "params": { "Nf": 8, "U": 8 }, "instance_id": "Hubbard/square_16_P_8_8", "rows": [ { "energy": -8.476, "sigma": 0.009, "energy_variance": null, "dof": 16, "einf": 32, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -8.476(9) (total), 4 x 4, U = 8, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 4 x 4, column U = 8 PBC. Half filling: N_up = N_dn = 8. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_16_P_8_8/", "per_site_divisor": 16, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 16, "boundary": "PA", "params": { "Nf": 8, "U": 2 }, "instance_id": "Hubbard/square_16_PA_8_2", "rows": [ { "energy": -20.114, "sigma": 0.002, "energy_variance": null, "dof": 16, "einf": 8, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -20.114(2) (total), 4 x 4, U = 2, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 4 x 4, column U = 2 PBC-APBC. Half filling: N_up = N_dn = 8. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_16_PA_8_2/", "per_site_divisor": 16, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 16, "boundary": "PA", "params": { "Nf": 8, "U": 4 }, "instance_id": "Hubbard/square_16_PA_8_4", "rows": [ { "energy": -14.594, "sigma": 0.003, "energy_variance": null, "dof": 16, "einf": 16, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -14.594(3) (total), 4 x 4, U = 4, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 4 x 4, column U = 4 PBC-APBC. Half filling: N_up = N_dn = 8. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_16_PA_8_4/", "per_site_divisor": 16, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 16, "boundary": "PA", "params": { "Nf": 8, "U": 6 }, "instance_id": "Hubbard/square_16_PA_8_6", "rows": [ { "energy": -10.902, "sigma": 0.007, "energy_variance": null, "dof": 16, "einf": 24, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -10.902(7) (total), 4 x 4, U = 6, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 4 x 4, column U = 6 PBC-APBC. Half filling: N_up = N_dn = 8. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_16_PA_8_6/", "per_site_divisor": 16, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 16, "boundary": "PA", "params": { "Nf": 8, "U": 8 }, "instance_id": "Hubbard/square_16_PA_8_8", "rows": [ { "energy": -8.646, "sigma": 0.008, "energy_variance": null, "dof": 16, "einf": 32, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -8.646(8) (total), 4 x 4, U = 8, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 4 x 4, column U = 8 PBC-APBC. Half filling: N_up = N_dn = 8. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_16_PA_8_8/", "per_site_divisor": 16, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 196, "boundary": "P", "params": { "Nf": 98, "U": 2 }, "instance_id": "Hubbard/square_196_P_98_2", "rows": [ { "energy": -229.981, "sigma": 0.006, "energy_variance": null, "dof": 196, "einf": 98, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -229.981(6) (total), 14 x 14, U = 2, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 14 x 14, column U = 2 PBC. Half filling: N_up = N_dn = 98. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_196_P_98_2/", "per_site_divisor": 196, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 196, "boundary": "P", "params": { "Nf": 98, "U": 4 }, "instance_id": "Hubbard/square_196_P_98_4", "rows": [ { "energy": -168.67, "sigma": 0.02, "energy_variance": null, "dof": 196, "einf": 196, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -168.67(2) (total), 14 x 14, U = 4, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 14 x 14, column U = 4 PBC. Half filling: N_up = N_dn = 98. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_196_P_98_4/", "per_site_divisor": 196, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 196, "boundary": "P", "params": { "Nf": 98, "U": 6 }, "instance_id": "Hubbard/square_196_P_98_6", "rows": [ { "energy": -128.76, "sigma": 0.02, "energy_variance": null, "dof": 196, "einf": 294, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -128.76(2) (total), 14 x 14, U = 6, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 14 x 14, column U = 6 PBC. Half filling: N_up = N_dn = 98. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_196_P_98_6/", "per_site_divisor": 196, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 196, "boundary": "P", "params": { "Nf": 98, "U": 8 }, "instance_id": "Hubbard/square_196_P_98_8", "rows": [ { "energy": -102.85, "sigma": 0.03, "energy_variance": null, "dof": 196, "einf": 392, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -102.85(3) (total), 14 x 14, U = 8, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 14 x 14, column U = 8 PBC. Half filling: N_up = N_dn = 98. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_196_P_98_8/", "per_site_divisor": 196, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 196, "boundary": "PA", "params": { "Nf": 98, "U": 2 }, "instance_id": "Hubbard/square_196_PA_98_2", "rows": [ { "energy": -231.134, "sigma": 0.004, "energy_variance": null, "dof": 196, "einf": 98, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -231.134(4) (total), 14 x 14, U = 2, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 14 x 14, column U = 2 PBC-APBC. Half filling: N_up = N_dn = 98. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_196_PA_98_2/", "per_site_divisor": 196, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 196, "boundary": "PA", "params": { "Nf": 98, "U": 4 }, "instance_id": "Hubbard/square_196_PA_98_4", "rows": [ { "energy": -168.69, "sigma": 0.03, "energy_variance": null, "dof": 196, "einf": 196, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -168.69(3) (total), 14 x 14, U = 4, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 14 x 14, column U = 4 PBC-APBC. Half filling: N_up = N_dn = 98. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_196_PA_98_4/", "per_site_divisor": 196, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 196, "boundary": "PA", "params": { "Nf": 98, "U": 6 }, "instance_id": "Hubbard/square_196_PA_98_6", "rows": [ { "energy": -128.78, "sigma": 0.03, "energy_variance": null, "dof": 196, "einf": 294, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -128.78(3) (total), 14 x 14, U = 6, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 14 x 14, column U = 6 PBC-APBC. Half filling: N_up = N_dn = 98. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_196_PA_98_6/", "per_site_divisor": 196, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 196, "boundary": "PA", "params": { "Nf": 98, "U": 8 }, "instance_id": "Hubbard/square_196_PA_98_8", "rows": [ { "energy": -102.83, "sigma": 0.04, "energy_variance": null, "dof": 196, "einf": 392, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -102.83(4) (total), 14 x 14, U = 8, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 14 x 14, column U = 8 PBC-APBC. Half filling: N_up = N_dn = 98. 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"[paper](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.92.033603) [code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_P_112_-4/AFQMC/)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_256_P_112_-4/", "per_site_divisor": 256, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "P", "params": { "Nf": 112, "U": -8 }, "instance_id": "Hubbard/square_256_P_112_-8", "rows": [ { "energy": -1028.169, "sigma": 0.014, "energy_variance": null, "dof": 224, "einf": -392, "v_score": null, "method": "AFQMC (Metropolis), numerically exact", "bound_type": "exact", "bound_type_reason": "exact diagonalization|ex", "reference": "[paper](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.92.033603) [code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_P_112_-8/AFQMC/)", "source": 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"source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -190.451, "sigma": 0.008, "energy_variance": 26.6, "dof": 224, "einf": 392, "v_score": 0.017563491889007465, "method": "VMC with uniform BCS pairing (+Jastrow and backflow)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_P_112_8/VMC-uniform/vmc_hubbard.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -191.977, "sigma": 0.007, "energy_variance": null, "dof": 224, "einf": 392, "v_score": null, "method": "FN on the state above", "bound_type": "projected", "bound_type_reason": "fixed[- ]node|^fn\\b|\\bfn", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_P_112_8/FN-uniform/fn_hubbard.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -194.215, "sigma": 0.007, "energy_variance": 1.5, "dof": 224, "einf": 392, "v_score": 0.0009777445943484262, "method": "VAFQMC stripe length 8", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "source": "varbench@2024-10-22", "provenance": "imported" }, { "energy": -194.1248, "sigma": null, "energy_variance": null, "dof": 224, "einf": 392, "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": "-0.7583 per site", "note": "Read directly from the Table 1 HTML of arXiv:2604.25775, column t'=0 PBC (16x16, U=8, delta=1/8). ACE Table 1 energies are WITHOUT symmetry projection (Sec. III.C.2), so they are not the ansatz floor.", "secondary_of": null } }, { "energy": -193.8688, "sigma": null, "energy_variance": null, "dof": 224, "einf": 392, "v_score": null, "method": "ACE (16 conv layers), no 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": "-0.7573 per site", "note": "Read directly from the Table 1 HTML of arXiv:2604.25775, column t'=0 PBC (16x16, U=8, delta=1/8). ACE Table 1 energies are WITHOUT symmetry projection (Sec. 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III.C.2), so they are not the ansatz floor.", "secondary_of": null } }, { "energy": -193.3312, "sigma": null, "energy_variance": null, "dof": 224, "einf": 392, "v_score": null, "method": "Tensor-Backflow + Lanczos", "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": "secondary", "verified": { "checked_on": "2026-09-10", "method": "source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription", "reported_as": "-0.7552 per site", "note": "Read directly from the Table 1 HTML of arXiv:2604.25775, column t'=0 PBC (16x16, U=8, delta=1/8). ACE Table 1 energies are WITHOUT symmetry projection (Sec. III.C.2), so they are not the ansatz floor.", "secondary_of": "ACE Table 1 row 'Tensor-Backflow+Lanczos [21]'" } }, { "energy": -192.7424, "sigma": null, "energy_variance": null, "dof": 224, "einf": 392, "v_score": null, "method": "SCALE (1 conv layer), no 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": "-0.7529 per site", "note": "Read directly from the Table 1 HTML of arXiv:2604.25775, column t'=0 PBC (16x16, U=8, delta=1/8). ACE Table 1 energies are WITHOUT symmetry projection (Sec. 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III.C.2), so they are not the ansatz floor.", "secondary_of": "ACE Table 1 row 'HFPS+sym [31]'" } }, { "energy": -192.2304, "sigma": null, "energy_variance": null, "dof": 224, "einf": 392, "v_score": null, "method": "Tensor-Backflow", "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": "secondary", "verified": { "checked_on": "2026-09-10", "method": "source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription", "reported_as": "-0.7509 per site", "note": "Read directly from the Table 1 HTML of arXiv:2604.25775, column t'=0 PBC (16x16, U=8, delta=1/8). ACE Table 1 energies are WITHOUT symmetry projection (Sec. 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"energy_variance": 0.84, "dof": 228, "einf": 304.59375, "v_score": 0.0007088783371593957, "method": "VAFQMC stripe length=8", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "source": "varbench@2024-10-22", "provenance": "imported" }, { "energy": -212.665, "sigma": 0.004, "energy_variance": 11.85, "dof": 228, "einf": 304.59375, "v_score": 0.010098049736270393, "method": "VMC stripe length=8 (+Jastrow and backflow)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_P_114_6/VMC-uniform/vmc_hubbard.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true } ], "url": "https://qmbl.org/i/Hubbard/square_256_P_114_6/", "per_site_divisor": 256, "per_site_label": "E/site", "record": { "energy": -215.188224, "sigma": 0.005632, "method": "VAFQMC stripe length=8", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "energy_per_site": -0.840579 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "P", "params": { "Nf": 114, "U": 8 }, "instance_id": "Hubbard/square_256_P_114_8", "rows": [ { "energy": -186.978304, "sigma": 0.012032, "energy_variance": 2.04, "dof": 228, "einf": 406.125, "v_score": 0.001322221843925223, "method": "VAFQMC stripe length=8", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "source": "varbench@2024-10-22", "provenance": "imported" }, { "energy": -184.398, "sigma": 0.007, "energy_variance": 15.83, "dof": 228, "einf": 406.125, "v_score": 0.010350042355139942, "method": "VMC stripe length=8 (+Jastrow and backflow)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_P_114_8/VMC-uniform/vmc_hubbard.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true } ], "url": "https://qmbl.org/i/Hubbard/square_256_P_114_8/", "per_site_divisor": 256, "per_site_label": "E/site", "record": { "energy": -186.978304, "sigma": 0.012032, "method": "VAFQMC stripe length=8", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "energy_per_site": -0.730384 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "P", "params": { "Nf": 115, "U": 8 }, "instance_id": "Hubbard/square_256_P_115_8", "rows": [ { "energy": -183.254272, "sigma": 0.011776, "energy_variance": 1.06, "dof": 230, "einf": 413.28125, "v_score": 0.0006851112227477799, "method": "VAFQMC stripe length=8", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_256_P_115_8/", "per_site_divisor": 256, "per_site_label": "E/site", "record": { "energy": -183.254272, "sigma": 0.011776, "method": "VAFQMC stripe length=8", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "energy_per_site": -0.715837 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "P", "params": { "Nf": 116, "U": 8 }, "instance_id": "Hubbard/square_256_P_116_8", "rows": [ { "energy": -179.546368, "sigma": 0.008192, "energy_variance": 1.63, "dof": 232, "einf": 420.5, "v_score": 0.0010502821065695599, "method": "VAFQMC stripe length=8", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "source": "varbench@2024-10-22", "provenance": "imported" }, { "energy": -176.978, "sigma": 0.006, "energy_variance": 15.46, "dof": 232, "einf": 420.5, "v_score": 0.010047398725511254, "method": "VMC stripe length=8 (+Jastrow and backflow)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_P_116_8/VMC-uniform/vmc_hubbard.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true } ], "url": "https://qmbl.org/i/Hubbard/square_256_P_116_8/", "per_site_divisor": 256, "per_site_label": "E/site", "record": { "energy": -179.546368, "sigma": 0.008192, "method": "VAFQMC stripe length=8", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "energy_per_site": -0.701353 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "P", "params": { "Nf": 117, "U": 8 }, "instance_id": "Hubbard/square_256_P_117_8", "rows": [ { "energy": -175.81568, "sigma": 0.013312, "energy_variance": 1.86, "dof": 234, "einf": 427.78125, "v_score": 0.0011946336873431052, "method": "VAFQMC stripe length=8", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_256_P_117_8/", "per_site_divisor": 256, "per_site_label": "E/site", "record": { "energy": -175.81568, "sigma": 0.013312, "method": "VAFQMC stripe length=8", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "energy_per_site": -0.68678 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "P", "params": { "Nf": 118, "U": 8 }, "instance_id": "Hubbard/square_256_P_118_8", "rows": [ { "energy": -172.069888, "sigma": 0.013312, "energy_variance": 1.75, "dof": 236, "einf": 435.125, "v_score": 0.0011201955382548377, "method": "VAFQMC stripe length=8", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_256_P_118_8/", "per_site_divisor": 256, "per_site_label": "E/site", "record": { "energy": -172.069888, "sigma": 0.013312, "method": "VAFQMC stripe length=8", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "energy_per_site": -0.672148 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "P", "params": { "Nf": 120, "U": 8 }, "instance_id": "Hubbard/square_256_P_120_8", "rows": [ { "energy": -163.975424, "sigma": 0.012544, "energy_variance": 1.86, "dof": 240, "einf": 450, "v_score": 0.0011841922392035297, "method": "VAFQMC stripe length=8", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_256_P_120_8/", "per_site_divisor": 256, "per_site_label": "E/site", "record": { "energy": -163.975424, "sigma": 0.012544, "method": "VAFQMC stripe length=8", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "energy_per_site": -0.640529 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "P", "params": { "Nf": 128, "U": -4 }, "instance_id": "Hubbard/square_256_P_128_-4", "rows": [ { "energy": -732.289, "sigma": 0.028, "energy_variance": null, "dof": 256, "einf": -256, "v_score": null, "method": "AFQMC (Metropolis), numerically exact", "bound_type": "exact", "bound_type_reason": "exact diagonalization|ex", "reference": "[paper](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.92.033603) [code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_P_128_-4/AFQMC/)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_256_P_128_-4/", "per_site_divisor": 256, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "P", "params": { "Nf": 128, "U": -8 }, "instance_id": "Hubbard/square_256_P_128_-8", "rows": [ { "energy": -1158.261, "sigma": 0.015, "energy_variance": null, "dof": 256, "einf": -512, "v_score": null, "method": "AFQMC (Metropolis), numerically exact", "bound_type": "exact", "bound_type_reason": "exact diagonalization|ex", "reference": "[paper](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.92.033603) [code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_P_128_-8/AFQMC/)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_256_P_128_-8/", "per_site_divisor": 256, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "P", "params": { "Nf": 128, "U": 2 }, "instance_id": "Hubbard/square_256_P_128_2", "rows": [ { "energy": -300.596, "sigma": 0.006, "energy_variance": null, "dof": 256, "einf": 128, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -300.596(6) (total), 16 x 16, U = 2, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 16 x 16, column U = 2 PBC. Half filling: N_up = N_dn = 128. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_256_P_128_2/", "per_site_divisor": 256, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "P", "params": { "Nf": 128, "U": 4 }, "instance_id": "Hubbard/square_256_P_128_4", "rows": [ { "energy": -220.29, "sigma": 0.04, "energy_variance": null, "dof": 256, "einf": 256, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -220.29(4) (total), 16 x 16, U = 4, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 16 x 16, column U = 4 PBC. Half filling: N_up = N_dn = 128. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_256_P_128_4/", "per_site_divisor": 256, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "P", "params": { "Nf": 128, "U": 6 }, "instance_id": "Hubbard/square_256_P_128_6", "rows": [ { "energy": -168.19, "sigma": 0.03, "energy_variance": null, "dof": 256, "einf": 384, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -168.19(3) (total), 16 x 16, U = 6, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 16 x 16, column U = 6 PBC. Half filling: N_up = N_dn = 128. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_256_P_128_6/", "per_site_divisor": 256, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "P", "params": { "Nf": 128, "U": 8 }, "instance_id": "Hubbard/square_256_P_128_8", "rows": [ { "energy": -134.23, "sigma": 0.03, "energy_variance": null, "dof": 256, "einf": 512, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -134.23(3) (total), 16 x 16, U = 8, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 16 x 16, column U = 8 PBC. Half filling: N_up = N_dn = 128. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_256_P_128_8/", "per_site_divisor": 256, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "PA", "params": { "Nf": 107, "U": 8 }, "instance_id": "Hubbard/square_256_PA_107_8", "rows": [ { "energy": -204.48, "sigma": 0.0175, "energy_variance": 49.7, "dof": 214, "einf": 357.78125, "v_score": 0.03364293329830187, "method": "mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos, (U=8), alpha = 1", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_PA_107_8/mVMC/mVMC.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -210.772224, "sigma": 0.00768, "energy_variance": 1.57, "dof": 214, "einf": 357.78125, "v_score": 0.001039371467933221, "method": "VAFQMC stripe length 8 APBC along the stripe", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_256_PA_107_8/", "per_site_divisor": 256, "per_site_label": "E/site", "record": { "energy": -210.772224, "sigma": 0.00768, "method": "VAFQMC stripe length 8 APBC along the stripe", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "energy_per_site": -0.823329 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "PA", "params": { "Nf": 107, "U": 8 }, "instance_id": "Hubbard/square_256_PA_107_8_t12", "rows": [ { "energy": -204.64, "sigma": 0.0095, "energy_variance": 43.6, "dof": 214, "einf": 357.78125, "v_score": 0.02949693016724527, "method": "mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos (Ncond = 214), alpha = 1, with 4x4 RBM subspace", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_PA_107_8_t12/mVMC/mVMC.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true } ], "url": "https://qmbl.org/i/Hubbard/square_256_PA_107_8_t12/", "per_site_divisor": 256, "per_site_label": "E/site", "record": { "energy": -204.64, "sigma": 0.0095, "method": "mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos (Ncond = 214), alpha = 1, with 4x4 RBM subspace", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_PA_107_8_t12/mVMC/mVMC.sh)", "energy_per_site": -0.799375 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "PA", "params": { "Nf": 107, "U": 8 }, "instance_id": "Hubbard/square_256_PA_107_8_t12_UV1V2", "rows": [ { "energy": 322.35, "sigma": 0.00696, "energy_variance": 44.71, "dof": 214, "einf": 894.0400735294118, "v_score": 0.029275015945507306, "method": "mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos (Ne = 214), alpha = 2, with 1x1 RBM subpsace", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_PA_107_8_t12_UV1V2/mVMC/mVMC.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true } ], "url": "https://qmbl.org/i/Hubbard/square_256_PA_107_8_t12_UV1V2/", "per_site_divisor": 256, "per_site_label": "E/site", "record": { "energy": 322.35, "sigma": 0.00696, "method": "mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos (Ne = 214), alpha = 2, with 1x1 RBM subpsace", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_PA_107_8_t12_UV1V2/mVMC/mVMC.sh)", "energy_per_site": 1.2591796875 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "PA", "params": { "Nf": 128, "U": 2 }, "instance_id": "Hubbard/square_256_PA_128_2", "rows": [ { "energy": -301.562, "sigma": 0.005, "energy_variance": null, "dof": 256, "einf": 128, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -301.562(5) (total), 16 x 16, U = 2, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 16 x 16, column U = 2 PBC-APBC. Half filling: N_up = N_dn = 128. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_256_PA_128_2/", "per_site_divisor": 256, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "PA", "params": { "Nf": 128, "U": 4 }, "instance_id": "Hubbard/square_256_PA_128_4", "rows": [ { "energy": -220.3, "sigma": 0.04, "energy_variance": null, "dof": 256, "einf": 256, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -220.30(4) (total), 16 x 16, U = 4, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 16 x 16, column U = 4 PBC-APBC. Half filling: N_up = N_dn = 128. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_256_PA_128_4/", "per_site_divisor": 256, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "PA", "params": { "Nf": 128, "U": 6 }, "instance_id": "Hubbard/square_256_PA_128_6", "rows": [ { "energy": -168.21, "sigma": 0.05, "energy_variance": null, "dof": 256, "einf": 384, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -168.21(5) (total), 16 x 16, U = 6, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 16 x 16, column U = 6 PBC-APBC. Half filling: N_up = N_dn = 128. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_256_PA_128_6/", "per_site_divisor": 256, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 256, "boundary": "PA", "params": { "Nf": 128, "U": 8 }, "instance_id": "Hubbard/square_256_PA_128_8", "rows": [ { "energy": -132.86, "sigma": 0.0044, "energy_variance": 8.1, "dof": 256, "einf": 512, "v_score": 0.0049864802850897744, "method": "mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos, (U=8)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_256_PA_128_8/mVMC/mVMC.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -134.2125, "sigma": 0.00923, "energy_variance": 1, "dof": 256, "einf": 512, "v_score": 0.0006130406263201699, "method": "VAFQMC", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "source": "varbench@2024-10-22", "provenance": "imported" }, { "energy": -134.25, "sigma": 0.03, "energy_variance": null, "dof": 256, "einf": 512, "v_score": null, "method": "AFQMC", "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.94.085103)", "source": "varbench@2024-10-22", "provenance": "imported" }, { "energy": -134.25, "sigma": 0.03, "energy_variance": null, "dof": 256, "einf": 512, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -134.25(3) (total), 16 x 16, U = 8, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 16 x 16, column U = 8 PBC-APBC. Half filling: N_up = N_dn = 128. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_256_PA_128_8/", "per_site_divisor": 256, "per_site_label": "E/site", "record": { "energy": -134.2125, "sigma": 0.00923, "method": "VAFQMC", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "energy_per_site": -0.524267578125 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 36, "boundary": "P", "params": { "Nf": 12, "U": 4 }, "instance_id": "Hubbard/square_36_P_12_4", "rows": [ { "energy": -42.669, "sigma": 0.00144, "energy_variance": null, "dof": 24, "einf": 16, "v_score": null, "method": "CP AFQMC with Constraint Release; no strict upper bound property", "bound_type": "projected", "bound_type_reason": "no strict upper bound|co", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.88.125132)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_36_P_12_4/", "per_site_divisor": 36, "per_site_label": "E/site", "record": null, "no_record_reason": "no variational row" }, { "model": "Hubbard", "lattice": "square", "n_sites": 36, "boundary": "P", "params": { "Nf": 18, "U": 2 }, "instance_id": "Hubbard/square_36_P_18_2", "rows": [ { "energy": -41.457, "sigma": 0.005, "energy_variance": null, "dof": 36, "einf": 18, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -41.457(5) (total), 6 x 6, U = 2, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 6 x 6, column U = 2 PBC. Half filling: N_up = N_dn = 18. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_36_P_18_2/", "per_site_divisor": 36, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 36, "boundary": "P", "params": { "Nf": 18, "U": 4 }, "instance_id": "Hubbard/square_36_P_18_4", "rows": [ { "energy": -30.865, "sigma": 0.009, "energy_variance": null, "dof": 36, "einf": 36, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -30.865(9) (total), 6 x 6, U = 4, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 6 x 6, column U = 4 PBC. Half filling: N_up = N_dn = 18. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_36_P_18_4/", "per_site_divisor": 36, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 36, "boundary": "P", "params": { "Nf": 18, "U": 6 }, "instance_id": "Hubbard/square_36_P_18_6", "rows": [ { "energy": -23.74, "sigma": 0.01, "energy_variance": null, "dof": 36, "einf": 54, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -23.74(1) (total), 6 x 6, U = 6, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 6 x 6, column U = 6 PBC. Half filling: N_up = N_dn = 18. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_36_P_18_6/", "per_site_divisor": 36, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 36, "boundary": "P", "params": { "Nf": 18, "U": 8 }, "instance_id": "Hubbard/square_36_P_18_8", "rows": [ { "energy": -19, "sigma": 0.02, "energy_variance": null, "dof": 36, "einf": 72, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -19.00(2) (total), 6 x 6, U = 8, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 6 x 6, column U = 8 PBC. Half filling: N_up = N_dn = 18. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_36_P_18_8/", "per_site_divisor": 36, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 36, "boundary": "P", "params": { "Nf": 24, "U": 4 }, "instance_id": "Hubbard/square_36_P_24_4", "rows": [ { "energy": 5.33124, "sigma": 0.00144, "energy_variance": null, "dof": 48, "einf": 64, "v_score": null, "method": "CP AFQMC with Constraint Release; no strict upper bound property", "bound_type": "projected", "bound_type_reason": "no strict upper bound|co", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.88.125132)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_36_P_24_4/", "per_site_divisor": 36, "per_site_label": "E/site", "record": null, "no_record_reason": "no variational row" }, { "model": "Hubbard", "lattice": "square", "n_sites": 36, "boundary": "PA", "params": { "Nf": 18, "U": 2 }, "instance_id": "Hubbard/square_36_PA_18_2", "rows": [ { "energy": -76.162, "sigma": 0.006, "energy_variance": 0.14, "dof": 36, "einf": 18, "v_score": 0.0005684328757789593, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel 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", "defect": { "flag": "wrong-instance", "finding": "Energy, sigma and variance (-76.162, 0.006, 0.14(2)) are identical to the 8x8 row on square_64_PA_32_2. The HFDS first author uploaded them to the 6x6 file 67 minutes after the 8x8 file (VarBench commits c810bdcd22, then 21f00e54f6), and they never changed. -76.162 / 64 = -1.19003 is the paper's 8x8, U = 2 value, -1.1900(2) (PNAS SI Table 5, arXiv SI Table V). As a 36-site total it is -2.1156 per site: 16.45 below the non-interacting ground state of this lattice, -59.712813, which a U >= 0 Hamiltonian cannot go below, and 32.7 below the sign-free AFQMC total -43.499(2) (Qin, Shi & Zhang, PRB 94, 085103 (2016), Table IV). The paper's 6x6 value, -1.2079(1), is carried as a separate row." } }, { "energy": -43.4844, "sigma": 0.0036, "energy_variance": null, "dof": 36, "einf": 18, "v_score": null, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 8, fully parametrized hidden sub-matrix, hidden-unit density alpha = 78)", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "Robledo Moreno, Carleo, Georges & Stokes, Fermionic wave functions from neural-network constrained hidden states, Proc. Natl. Acad. Sci. U.S.A. 119, e2122059119 (2022), arXiv:2111.10420, doi:10.1073/pnas.2122059119", "peer_reviewed": true, "source": "sweep-worklist-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "PNAS supplementary PDF (via Europe PMC, PMC9371695) extracted locally, cross-read against the arXiv version; no LLM transcription", "reported_as": "-1.2079(1) per site, no variance", "note": "PNAS SI Table 5 (identical to arXiv SI Table V), \"Variational energy per site in the L x L Hubbard model at half filling with periodic boundary conditions along one of the sides of the square and anti-periodic boundary conditions along the other side\", row L = 6, column U = 2. SI Sec. 6: N_hidden = 8, alpha = 78 for 6x6; no projection or constraint stated, nor what the error bar is. Column alignment checked: the VarBench HFDS rows for 6x6 at U = 4, 6, 8 and 8x8 at U = 2, 4, 8 match the printed digits. As a total, -43.4844 sits 3.4e-4 (relative) above the sign-free AFQMC value -43.499(2) for this lattice (Qin, Shi & Zhang, PRB 94, 085103 (2016), Table IV, PBC-APBC) and above the non-interacting bound -59.712813.", "secondary_of": null } }, { "energy": -43.499, "sigma": 0.002, "energy_variance": null, "dof": 36, "einf": 18, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -43.499(2) (total), 6 x 6, U = 2, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 6 x 6, column U = 2 PBC-APBC. Half filling: N_up = N_dn = 18. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_36_PA_18_2/", "per_site_divisor": 36, "per_site_label": "E/site", "record": { "energy": -43.4844, "sigma": 0.0036, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 8, fully parametrized hidden sub-matrix, hidden-unit density alpha = 78)", "reference": "Robledo Moreno, Carleo, Georges & Stokes, Fermionic wave functions from neural-network constrained hidden states, Proc. Natl. Acad. Sci. U.S.A. 119, e2122059119 (2022), arXiv:2111.10420, doi:10.1073/pnas.2122059119", "energy_per_site": -1.2079 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 36, "boundary": "PA", "params": { "Nf": 18, "U": 4 }, "instance_id": "Hubbard/square_36_PA_18_4", "rows": [ { "energy": -31.3822, "sigma": 0.0008, "energy_variance": 0.036, "dof": 36, "einf": 36, "v_score": 0.00028543986593902417, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel 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": -31.43, "sigma": 0.02, "energy_variance": null, "dof": 36, "einf": 36, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -31.43(2) (total), 6 x 6, U = 4, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 6 x 6, column U = 4 PBC-APBC. Half filling: N_up = N_dn = 18. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_36_PA_18_4/", "per_site_divisor": 36, "per_site_label": "E/site", "record": { "energy": -31.3822, "sigma": 0.0008, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel order.", "reference": "[paper](https://www.pnas.org/doi/full/10.1073/pnas.2122059119)", "energy_per_site": -0.8717277777777778 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 36, "boundary": "PA", "params": { "Nf": 18, "U": 6 }, "instance_id": "Hubbard/square_36_PA_18_6", "rows": [ { "energy": -23.794, "sigma": 0.001, "energy_variance": 0.044, "dof": 36, "einf": 54, "v_score": 0.0002617357053931823, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel 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": -23.84, "sigma": 0.01, "energy_variance": null, "dof": 36, "einf": 54, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -23.84(1) (total), 6 x 6, U = 6, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 6 x 6, column U = 6 PBC-APBC. Half filling: N_up = N_dn = 18. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_36_PA_18_6/", "per_site_divisor": 36, "per_site_label": "E/site", "record": { "energy": -23.794, "sigma": 0.001, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel order.", "reference": "[paper](https://www.pnas.org/doi/full/10.1073/pnas.2122059119)", "energy_per_site": -0.6609444444444444 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 36, "boundary": "PA", "params": { "Nf": 18, "U": 8 }, "instance_id": "Hubbard/square_36_PA_18_8", "rows": [ { "energy": -18.9738, "sigma": 0.001, "energy_variance": 0.063, "dof": 36, "einf": 72, "v_score": 0.0002740377410281188, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel 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": -19.01, "sigma": 0.01, "energy_variance": null, "dof": 36, "einf": 72, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -19.01(1) (total), 6 x 6, U = 8, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 6 x 6, column U = 8 PBC-APBC. Half filling: N_up = N_dn = 18. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_36_PA_18_8/", "per_site_divisor": 36, "per_site_label": "E/site", "record": { "energy": -18.9738, "sigma": 0.001, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel order.", "reference": "[paper](https://www.pnas.org/doi/full/10.1073/pnas.2122059119)", "energy_per_site": -0.52705 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 50, "boundary": "P", "params": { "Nf": 21, "U": 8 }, "instance_id": "Hubbard/square_50_P_21_8", "rows": [ { "energy": -41.1638, "sigma": 0.0016, "energy_variance": 0.3, "dof": 42, "einf": 70.56, "v_score": 0.0010094368334838178, "method": "VAFQMC", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_50_P_21_8/", "per_site_divisor": 50, "per_site_label": "E/site", "record": { "energy": -41.1638, "sigma": 0.0016, "method": "VAFQMC", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115133)", "energy_per_site": -0.823276 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "P", "params": { "Nf": 14, "U": 4 }, "instance_id": "Hubbard/square_64_P_14_4", "rows": [ { "energy": -68.5952, "sigma": 0.00128, "energy_variance": null, "dof": 28, "einf": 12.25, "v_score": null, "method": "CP AFQMC with Constraint Release; no strict upper bound property", "bound_type": "projected", "bound_type_reason": "no strict upper bound|co", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.88.125132)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_64_P_14_4/", "per_site_divisor": 64, "per_site_label": "E/site", "record": null, "no_record_reason": "no variational row" }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "P", "params": { "Nf": 22, "U": 4 }, "instance_id": "Hubbard/square_64_P_22_4", "rows": [ { "energy": -75.8912, "sigma": 0.0128, "energy_variance": null, "dof": 44, "einf": 30.25, "v_score": null, "method": "CP AFQMC with Constraint Release; no strict upper bound property", "bound_type": "projected", "bound_type_reason": "no strict upper bound|co", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.88.125132)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_64_P_22_4/", "per_site_divisor": 64, "per_site_label": "E/site", "record": null, "no_record_reason": "no variational row" }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "P", "params": { "Nf": 25, "U": -4 }, "instance_id": "Hubbard/square_64_P_25_-4", "rows": [ { "energy": -153.363, "sigma": 0.006, "energy_variance": null, "dof": 50, "einf": -39.0625, "v_score": null, "method": "AFQMC (Metropolis, Trotter error extrapolated), numerically exact", "bound_type": "exact", "bound_type_reason": "exact diagonalization|ex", "reference": "[paper](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.92.033603) [code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_25_-4/AFQMC/)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_64_P_25_-4/", "per_site_divisor": 64, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "P", "params": { "Nf": 25, "U": -8 }, "instance_id": "Hubbard/square_64_P_25_-8", "rows": [ { "energy": -232.05, "sigma": 0.007, "energy_variance": null, "dof": 50, "einf": -78.125, "v_score": null, "method": "AFQMC (Metropolis, Trotter error extrapolated), numerically exact", "bound_type": "exact", "bound_type_reason": "exact diagonalization|ex", "reference": "[paper](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.92.033603) [code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_25_-8/AFQMC/)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_64_P_25_-8/", "per_site_divisor": 64, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "P", "params": { "Nf": 25, "U": 4 }, "instance_id": "Hubbard/square_64_P_25_4", "rows": [ { "energy": -72.344, "sigma": 0.000331, "energy_variance": 0.59, "dof": 50, "einf": 39.0625, "v_score": 0.002376845513757116, "method": "mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos, (U=4), alpha = 8 with 1x1 RBM subspace", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_25_4/mVMC/mVMC.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -72.11732, "sigma": 0.00089, "energy_variance": 1.43, "dof": 50, "einf": 39.0625, "v_score": 0.005784343942887791, "method": "VMC with uniform pairing (+Jastrow and backflow)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_25_4/VMC-uniform/vmc_hubbard.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -72.2174, "sigma": 0.0006, "energy_variance": null, "dof": 50, "einf": 39.0625, "v_score": null, "method": "FN on the state above", "bound_type": "projected", "bound_type_reason": "fixed[- ]node|^fn\\b|\\bfn", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_25_4/FN-uniform/fn_hubbard.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true } ], "url": "https://qmbl.org/i/Hubbard/square_64_P_25_4/", "per_site_divisor": 64, "per_site_label": "E/site", "record": { "energy": -72.344, "sigma": 0.000331, "method": "mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos, (U=4), alpha = 8 with 1x1 RBM subspace", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_25_4/mVMC/mVMC.sh)", "energy_per_site": -1.130375 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "P", "params": { "Nf": 25, "U": 8 }, "instance_id": "Hubbard/square_64_P_25_8", "rows": [ { "energy": -58.549, "sigma": 0.001031, "energy_variance": 2.58, "dof": 50, "einf": 78.125, "v_score": 0.00690586207558429, "method": "mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos, (U=8), alpha = 8 with 1x1 RBM subspace", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": 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[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_28_-8/AFQMC/)", "source": "varbench@2024-10-22", "provenance": "imported" } ], "url": "https://qmbl.org/i/Hubbard/square_64_P_28_-8/", "per_site_divisor": 64, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "P", "params": { "Nf": 28, "U": 4 }, "instance_id": "Hubbard/square_64_P_28_4", "rows": [ { "energy": -64.371, "sigma": 0.002, "energy_variance": 2.79, "dof": 56, "einf": 49, "v_score": 0.012155932378347296, "method": "VMC with uniform BCS pairing (+Jastrow and backflow)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_28_4/VMC-uniform/vmc_hubbard.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -64.553, "sigma": 0.001, "energy_variance": null, "dof": 56, "einf": 49, "v_score": null, "method": "FN on the state above", "bound_type": "projected", "bound_type_reason": "fixed[- ]node|^fn\\b|\\bfn", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_28_4/FN-uniform/fn_hubbard.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true } ], "url": "https://qmbl.org/i/Hubbard/square_64_P_28_4/", "per_site_divisor": 64, "per_site_label": "E/site", "record": { "energy": -64.371, "sigma": 0.002, "method": "VMC with uniform BCS pairing (+Jastrow and backflow)", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_28_4/VMC-uniform/vmc_hubbard.sh)", "energy_per_site": -1.005796875 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "P", "params": { "Nf": 28, "U": 8 }, "instance_id": "Hubbard/square_64_P_28_8", "rows": [ { "energy": -47.501, "sigma": 0.002, "energy_variance": 6.62, "dof": 56, "einf": 98, "v_score": 0.017511125482456442, "method": "VMC with uniform BCS pairing (+Jastrow and backflow)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_28_8/VMC-uniform/vmc_hubbard.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -47.954, "sigma": 0.001, "energy_variance": null, "dof": 56, "einf": 98, "v_score": null, "method": "FN on the state above", "bound_type": "projected", "bound_type_reason": "fixed[- ]node|^fn\\b|\\bfn", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_28_8/FN-uniform/fn_hubbard.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -47.7312, "sigma": 0.0384, "energy_variance": 3.36, "dof": 56, "einf": 98, "v_score": 0.008859765401290735, "method": "Jastrow-backflow (JBf), 8x8 torus", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. B 113, 245104 (2026), arXiv:2510.11710", "peer_reviewed": true, "source": "sweep-tables-2026-09-14", "provenance": "primary", "verified": { "checked_on": "2026-09-14", "method": "arXiv HTML parsed locally, no LLM transcription", "reported_as": "E_0/M = -0.7458 (+/- 0.0006), sigma^2/M = 0.0525", "note": "Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Row \"JBf (8x8 Torus)\".", "secondary_of": null } }, { "energy": -47.7056, "sigma": 0.0576, "energy_variance": 3.6032, "dof": 56, "einf": 98, "v_score": 0.009504382557134799, "method": "Hidden-fermion determinant state (HFDS), 8x8 torus", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. B 113, 245104 (2026), arXiv:2510.11710", "peer_reviewed": true, "source": "sweep-tables-2026-09-14", "provenance": "primary", "verified": { "checked_on": "2026-09-14", "method": "arXiv HTML parsed locally, no LLM transcription", "reported_as": "E_0/M = -0.7454 (+/- 0.0009), sigma^2/M = 0.0563", "note": "Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Row \"HFDS (8x8 Torus)\".", "secondary_of": null } }, { "energy": -47.5008, "sigma": 0.0192, "energy_variance": 4.4672, "dof": 56, "einf": 98, "v_score": 0.011816603445266605, "method": "JBf-ViT (vision transformer backflow), 8x8 torus", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. B 113, 245104 (2026), arXiv:2510.11710", "peer_reviewed": true, "source": "sweep-tables-2026-09-14", "provenance": "primary", "verified": { "checked_on": "2026-09-14", "method": "arXiv HTML parsed locally, no LLM transcription", "reported_as": "E_0/M = -0.7422 (+/- 0.0003), sigma^2/M = 0.0698", "note": "Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Right-hand table, row \"JBf-ViT\"; same lattice and parameters per the caption.", "secondary_of": null } }, { "energy": -47.104, "sigma": 0.064, "energy_variance": 6.72, "dof": 56, "einf": 98, "v_score": 0.017873044264928566, "method": "HFDS-ViT, 8x8 torus", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. B 113, 245104 (2026), arXiv:2510.11710", "peer_reviewed": true, "source": "sweep-tables-2026-09-14", "provenance": "primary", "verified": { "checked_on": "2026-09-14", "method": "arXiv HTML parsed locally, no LLM transcription", "reported_as": "E_0/M = -0.736 (+/- 0.001), sigma^2/M = 0.105", "note": "Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Right-hand table, row \"HFDS-ViT\".", "secondary_of": null } }, { "energy": -46.656, "sigma": 0.256, "energy_variance": 8.896, "dof": 56, "einf": 98, "v_score": 0.02380728645124025, "method": "HFDS Symm-ViT-Ti, 8x8 torus", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. B 113, 245104 (2026), arXiv:2510.11710", "peer_reviewed": true, "source": "sweep-tables-2026-09-14", "provenance": "primary", "verified": { "checked_on": "2026-09-14", "method": "arXiv HTML parsed locally, no LLM transcription", "reported_as": "E_0/M = -0.729 (+/- 0.004), sigma^2/M = 0.139", "note": "Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Right-hand table, row \"HFDS Symm-ViT-Ti\".", "secondary_of": null } }, { "energy": -46.464, "sigma": 0.576, "energy_variance": 10.752, "dof": 56, "einf": 98, "v_score": 0.02885081004475789, "method": "JBf Symm-ViT-Ti, 8x8 torus", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. B 113, 245104 (2026), arXiv:2510.11710", "peer_reviewed": true, "source": "sweep-tables-2026-09-14", "provenance": "primary", "verified": { "checked_on": "2026-09-14", "method": "arXiv HTML parsed locally, no LLM transcription", "reported_as": "E_0/M = -0.726 (+/- 0.009), sigma^2/M = 0.168", "note": "Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Right-hand table, row \"JBf Symm-ViT-Ti\".", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_64_P_28_8/", "per_site_divisor": 64, "per_site_label": "E/site", "record": { "energy": -47.7312, "sigma": 0.0384, "method": "Jastrow-backflow (JBf), 8x8 torus", "reference": "Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model, Phys. Rev. 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Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -74.470(5) (total), 8 x 8, U = 2, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 8 x 8, column U = 2 PBC. Half filling: N_up = N_dn = 32. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_64_P_32_2/", "per_site_divisor": 64, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "P", "params": { "Nf": 32, "U": 4 }, "instance_id": "Hubbard/square_64_P_32_4", "rows": [ { "energy": -54.986, "sigma": 0.00049, "energy_variance": 0.47, "dof": 64, "einf": 64, "v_score": 0.002124643661906032, "method": "mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos, (U=4), alpha = 4", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_32_4/mVMC/mVMC.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -54.81638, "sigma": 0.00072, "energy_variance": 0.863, "dof": 64, "einf": 64, "v_score": 0.003912353934049426, "method": "VMC with Neel AF (+Jastrow and backflow)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_32_4/VMC-uniform/vmc_hubbard.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -54.8898, "sigma": 0.0006, "energy_variance": null, "dof": 64, "einf": 64, "v_score": null, "method": "FN on the state above", "bound_type": "projected", "bound_type_reason": "fixed[- ]node|^fn\\b|\\bfn", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_32_4/FN-uniform/fn_hubbard.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -55.063, "sigma": 0.004, "energy_variance": null, "dof": 64, "einf": 64, "v_score": null, "method": "AFQMC (Metropolis, Trotter error extrapolated), numerically exact", "bound_type": "exact", "bound_type_reason": "exact diagonalization|ex", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.94.085103)", "source": "varbench@2024-10-22", "provenance": "imported" }, { "energy": -55.05, "sigma": 0.01, "energy_variance": null, "dof": 64, "einf": 64, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -55.05(1) (total), 8 x 8, U = 4, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 8 x 8, column U = 4 PBC. Half filling: N_up = N_dn = 32. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_64_P_32_4/", "per_site_divisor": 64, "per_site_label": "E/site", "record": { "energy": -54.986, "sigma": 0.00049, "method": "mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos, (U=4), alpha = 4", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_32_4/mVMC/mVMC.sh)", "energy_per_site": -0.85915625 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "P", "params": { "Nf": 32, "U": 6 }, "instance_id": "Hubbard/square_64_P_32_6", "rows": [ { "energy": -42.16, "sigma": 0.02, "energy_variance": null, "dof": 64, "einf": 96, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -42.16(2) (total), 8 x 8, U = 6, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 8 x 8, column U = 6 PBC. Half filling: N_up = N_dn = 32. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_64_P_32_6/", "per_site_divisor": 64, "per_site_label": "E/site", "record": null, "no_record_reason": "solved exactly" }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "P", "params": { "Nf": 32, "U": 8 }, "instance_id": "Hubbard/square_64_P_32_8", "rows": [ { "energy": -33.574, "sigma": 0.00061, "energy_variance": 0.62, "dof": 64, "einf": 128, "v_score": 0.0015199479293909394, "method": "mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos, (U=8) (Ne = 64), alpha = 8 with 1x1 RBM-subspace", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_32_8/mVMC/mVMC.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -33.642, "sigma": 0.005, "energy_variance": null, "dof": 64, "einf": 128, "v_score": null, "method": "AFQMC (Metropolis, Trotter error extrapolated), numerically exact", "bound_type": "exact", "bound_type_reason": "exact diagonalization|ex", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.94.085103)", "source": "varbench@2024-10-22", "provenance": "imported" }, { "energy": -33.65248, "sigma": null, "energy_variance": null, "dof": 64, "einf": 128, "v_score": null, "method": "Transformer backflow + MARCH optimizer", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "Gu et al., Nat. Commun. (2026), arXiv:2507.02644", "peer_reviewed": true, "source": "sweep-worklist-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF text extracted locally with pypdf in layout mode, no LLM transcription", "reported_as": "-0.52582 per site, no error bar", "note": "Supplementary Table S1 of arXiv:2507.02644, \"Benchmark energy in pure Hubbard model at half-filling with PBC\", column 8 x 8, row \"NQS\"; the AFQMC reference in the same table is -0.5262(5). Same method string as this paper's 16 x 4 row. No error bar, so it is eligible for nothing (RULES.md 6). It sits 1.6e-4 per site below the stored AFQMC exact row -0.5256563(78) - see the defect flag. Read from the arXiv supplement; the Nat. Commun. version was not compared.", "secondary_of": null }, "defect": { "flag": "below-exact-suspected", "finding": "Reported without an error bar at E/N = -0.52582, 1.6e-4 per site below the stored AFQMC exact row -0.5256563(78): 2.1 sigma of the AFQMC error alone. Not established - the paper's own AFQMC reference for this lattice is -0.5262(5), lower still, and the NQS error is unknown - but a variational energy below a numerically exact one is grounds for objection (RULES.md 10). It holds nothing either way: a sampled energy with no sigma is eligible for nothing (RULES.md 6)." } }, { "energy": -33.68, "sigma": 0.03, "energy_variance": null, "dof": 64, "einf": 128, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -33.68(3) (total), 8 x 8, U = 8, PBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 8 x 8, column U = 8 PBC. Half filling: N_up = N_dn = 32. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_64_P_32_8/", "per_site_divisor": 64, "per_site_label": "E/site", "record": { "energy": -33.574, "sigma": 0.00061, "method": "mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos, (U=8) (Ne = 64), alpha = 8 with 1x1 RBM-subspace", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/Hubbard/square_64_P_32_8/mVMC/mVMC.sh)", "energy_per_site": -0.52459375 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "PA", "params": { "Nf": 32, "U": 2 }, "instance_id": "Hubbard/square_64_PA_32_2", "rows": [ { "energy": -76.162, "sigma": 0.006, "energy_variance": 0.14, "dof": 64, "einf": 32, "v_score": 0.000765876231075651, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel 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": -76.308, "sigma": 0.003, "energy_variance": null, "dof": 64, "einf": 32, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -76.308(3) (total), 8 x 8, U = 2, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 8 x 8, column U = 2 PBC-APBC. Half filling: N_up = N_dn = 32. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_64_PA_32_2/", "per_site_divisor": 64, "per_site_label": "E/site", "record": { "energy": -76.162, "sigma": 0.006, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel order.", "reference": "[paper](https://www.pnas.org/doi/full/10.1073/pnas.2122059119)", "energy_per_site": -1.19003125 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "PA", "params": { "Nf": 32, "U": 4 }, "instance_id": "Hubbard/square_64_PA_32_4", "rows": [ { "energy": -55.18, "sigma": 0.01, "energy_variance": 0.54, "dof": 64, "einf": 64, "v_score": 0.0024331392895362816, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel 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": -55.31, "sigma": 0.01, "energy_variance": null, "dof": 64, "einf": 64, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -55.31(1) (total), 8 x 8, U = 4, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 8 x 8, column U = 4 PBC-APBC. Half filling: N_up = N_dn = 32. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_64_PA_32_4/", "per_site_divisor": 64, "per_site_label": "E/site", "record": { "energy": -55.18, "sigma": 0.01, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel order.", "reference": "[paper](https://www.pnas.org/doi/full/10.1073/pnas.2122059119)", "energy_per_site": -0.8621875 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "PA", "params": { "Nf": 32, "U": 6 }, "instance_id": "Hubbard/square_64_PA_32_6", "rows": [ { "energy": -42.676, "sigma": 0.007, "energy_variance": 0.82, "dof": 64, "einf": 96, "v_score": 0.0027289225735036777, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel 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", "defect": { "flag": "below-exact", "finding": "VarBench stores -42.676 (E/N = -0.666813) and cites the HFDS paper, Moreno et al., PNAS 119, e2122059119 (2022). The paper gives -0.6574(2) for this lattice (8x8, half filling, U = 6, periodic along one side and antiperiodic along the other) in both the PNAS supplement (Table 5, read via Europe PMC PMC9371695) and the arXiv version (SI Table V). The stored number also sits 0.5 below the sign-free AFQMC total for this lattice, -42.17(2) (Qin, Shi & Zhang, PRB 94, 085103 (2016), Table IV) - 1.2%, about 25 sigma - which no variational energy can do. It breaks the paper's own size trend at U = 6 as well (-0.68135, -0.6609, -0.6574 for L = 4, 6, 8).", "diagnosis": "The published value is a minimum of the optimization trace, not a converged measurement. Three seeds each, netket 3.22.4, alpha=1 complex RBM, 2000 SR steps. Evaluating the trained parameters by FULL SUMMATION over all 1024 basis states (zero Monte-Carlo error) puts every converged energy ABOVE the exact value, as the variational principle requires, while every training trace dips 1.5e-3 to 3.5e-3 BELOW it. Each published value lies between the two.", "ruled_out": "Not an autocorrelation-underestimated error bar: tau_corr <= 0.05 and R_hat = 1.0000 across all six runs.", "evidence": "checks/results-tfising-rbm.json, checks/tfising_rbm_check.py" } }, { "energy": -42.0736, "sigma": 0.0128, "energy_variance": null, "dof": 64, "einf": 96, "v_score": null, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16, fully parametrized hidden sub-matrix, hidden-unit density alpha = 1)", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "Robledo Moreno, Carleo, Georges & Stokes, Fermionic wave functions from neural-network constrained hidden states, Proc. Natl. Acad. Sci. U.S.A. 119, e2122059119 (2022), arXiv:2111.10420, doi:10.1073/pnas.2122059119", "peer_reviewed": true, "source": "sweep-worklist-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "PNAS supplementary PDF (via Europe PMC, PMC9371695) extracted locally, cross-read against the arXiv version; no LLM transcription", "reported_as": "-0.6574(2) per site, no variance", "note": "PNAS SI Table 5 (identical to arXiv SI Table V), \"Variational energy per site in the L x L Hubbard model at half filling with periodic boundary conditions along one of the sides of the square and anti-periodic boundary conditions along the other side\", row L = 8, column U = 6. SI Sec. 6: N_hidden = 16, alpha = 1 for 8x8, the settings in the VarBench method string. Column alignment checked as for square_36_PA_18_2. As a total, -42.0736 sits 2.3e-3 (relative) above the sign-free AFQMC value -42.17(2) for this lattice (Qin, Shi & Zhang, PRB 94, 085103 (2016), Table IV, PBC-APBC), as a variational energy must, and fits the paper's size trend at U = 6 (-0.68135, -0.6609, -0.6574 for L = 4, 6, 8).", "secondary_of": null } }, { "energy": -42.17, "sigma": 0.02, "energy_variance": null, "dof": 64, "einf": 96, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -42.17(2) (total), 8 x 8, U = 6, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 8 x 8, column U = 6 PBC-APBC. Half filling: N_up = N_dn = 32. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_64_PA_32_6/", "per_site_divisor": 64, "per_site_label": "E/site", "record": { "energy": -42.0736, "sigma": 0.0128, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16, fully parametrized hidden sub-matrix, hidden-unit density alpha = 1)", "reference": "Robledo Moreno, Carleo, Georges & Stokes, Fermionic wave functions from neural-network constrained hidden states, Proc. Natl. Acad. Sci. U.S.A. 119, e2122059119 (2022), arXiv:2111.10420, doi:10.1073/pnas.2122059119", "energy_per_site": -0.6574 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "PA", "params": { "Nf": 32, "U": 8 }, "instance_id": "Hubbard/square_64_PA_32_8", "rows": [ { "energy": -33.57, "sigma": 0.01, "energy_variance": 0.8, "dof": 64, "einf": 128, "v_score": 0.001961320244181766, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel 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": -33.66, "sigma": 0.02, "energy_variance": null, "dof": 64, "einf": 128, "v_score": null, "method": "AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error)", "bound_type": "exact", "bound_type_reason": "sign-problem-free AFQMC at half filling on a bipartite lattice; the paper states the results are numerically exact (RULES.md 4)", "reference": "Qin, Shi & Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016), arXiv:1605.09421", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription", "reported_as": "E = -33.66(2) (total), 8 x 8, U = 8, PBC-APBC", "note": "Table IV, \"Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit.\" Row 8 x 8, column U = 8 PBC-APBC. Half filling: N_up = N_dn = 32. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.", "secondary_of": null } } ], "url": "https://qmbl.org/i/Hubbard/square_64_PA_32_8/", "per_site_divisor": 64, "per_site_label": "E/site", "record": { "energy": -33.57, "sigma": 0.01, "method": "VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel order.", "reference": "[paper](https://www.pnas.org/doi/full/10.1073/pnas.2122059119)", "energy_per_site": -0.52453125 }, "no_record_reason": null }, { "model": "Hubbard", "lattice": "square", "n_sites": 64, "boundary": "PO", "params": { "Nf": 32, "U": 8 }, "instance_id": "Hubbard/square_64_PO_32_8", "rows": [ { "energy": -31.8998, "sigma": null, "energy_variance": 1.107, "dof": 64, "einf": 128, "v_score": 0.0027709695526207064, "method": "DMRG (MaxLinkDim=10000, MaxTruncErr ~ 3.4E-5, Extrap Energy -32.0027 +/- 0.0206)", "bound_type": "variational", "bound_type_reason": "vmc|\\brbm\\b|\\brnn\\b|jast", "reference": "[code](https://github.com/varbench/methods/blob/main/programs/dmrg_itensors_hubbard/square_64_PO_32_8.jl)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -31.964, "sigma": 0.0023, "energy_variance": 0.03, "dof": 64, "einf": 128, "v_score": 0.00007503376139404315, "method": "VAFQMC", 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Phys. 20, 1476 (2024), arXiv:2302.01941", "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": "-0.4976921 (+/- 4e-7) per site in S.S units", "note": "PDF text: \"to attain the best variational energy E/N = -0.4976921(4)\"", "secondary_of": null } }, { "energy": -199.086, "sigma": 0.0036, "energy_variance": null, "dof": 100, "einf": 0, "v_score": null, "method": "ResNet2 MinSR, zero-variance extrapolation", "bound_type": "extrapolated", "bound_type_reason": "assigned from the source text during verification", "reference": "Chen & Heyl, Nat. Phys. 20, 1476 (2024), arXiv:2302.01941", "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": "-0.497715 (+/- 0.000009) per site in S.S units", "note": "PDF text: \"estimate the ground-state energy E_GS/N = -0.497715(9) by zero-variance extrapolation\". NOT a variational bound.", "secondary_of": null } }, { "energy": -199.0536, "sigma": 0.0004, "energy_variance": null, "dof": 100, "einf": 0, "v_score": null, "method": "fViT (vision transformer, 2.7e5 params)", "bound_type": "variational", "bound_type_reason": "assigned from the source text during verification", "reference": "Nutakki, Shokry & Vicentini, Phys. Rev. Research 7, 043099 (2025), arXiv:2505.03466", "peer_reviewed": true, "source": "literature-2025-26", "provenance": "secondary", "verified": { "checked_on": "2026-09-10", "method": "source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription", "reported_as": "-0.497634 (+/- 0.000001) per site in S.S units", "note": "Read from Table 1 of the arXiv HTML of 2505.03466; primary source not checked.", "secondary_of": "Rende et al., cited as ref [16] of arXiv:2505.03466" } }, { "energy": -199.0332, "sigma": 0.0024, "energy_variance": null, "dof": 100, "einf": 0, "v_score": null, "method": "ConvNext (6,3,3)[2,2], 2.6e5 params", "bound_type": "variational", "bound_type_reason": "assigned from the source text during verification", "reference": "Nutakki, Shokry & Vicentini, Phys. Rev. Research 7, 043099 (2025), arXiv:2505.03466", "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": "-0.497583 (+/- 0.000006) per site in S.S units", "note": "Table 1 of arXiv:2505.03466, their own result. Paper states L=10, PBC, S.S per site.", "secondary_of": null } }, { "energy": -199.07756, "sigma": 0.00008, "energy_variance": null, "dof": 100, "einf": 0, "v_score": null, "method": "CNN-MPS (h,D,l)=(32,20,20), Marshall sign transformation", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "arXiv:2603.14425, Disentangling Tensor Network States with Deep Neural Networks", "peer_reviewed": false, "source": "sweep-2026-09-13", "provenance": "primary", "verified": { "checked_on": "2026-09-13", "method": "arXiv HTML parsed locally, no LLM transcription", "reported_as": "-0.4976939 (+/- 2e-7) per site in S.S units", "note": "Read from Table 1 of arXiv:2603.14425 (square-lattice J1-J2 at J2/J1=0.5, PBC, E per site in S.S units), parsed from the arXiv HTML. Claim: \"the best energy obtained by CNN-MPS is -0.4976939(2) ... which is lower than the best previously reported result\". Supersedes Chen & Heyl -0.4976921(4) as the record for this instance.", "secondary_of": null } }, { "energy": -199.07692, "sigma": 0.00008, "energy_variance": null, "dof": 100, "einf": 0, "v_score": null, "method": "T-MPS", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "arXiv:2603.14425, Disentangling Tensor Network States with Deep Neural Networks", "peer_reviewed": false, "source": "sweep-2026-09-13", "provenance": "primary", "verified": { "checked_on": "2026-09-13", "method": "arXiv HTML parsed locally, no LLM transcription", "reported_as": "-0.4976923 (+/- 2e-7) per site in S.S units", "note": "Read from Table 1 of arXiv:2603.14425 (square-lattice J1-J2 at J2/J1=0.5, PBC, E per site in S.S units), parsed from the arXiv HTML.", "secondary_of": null } }, { "energy": -199.07056, "sigma": 0.00028, "energy_variance": null, "dof": 100, "einf": 0, "v_score": null, "method": "Convolutional transformer wave function (CTWF)", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "Chen, Naik & Heyl, Convolutional transformer wave functions, arXiv:2503.10462", "peer_reviewed": false, "source": "sweep-2026-09-13", "provenance": "primary", "verified": { "checked_on": "2026-09-13", "method": "arXiv HTML parsed locally, no LLM transcription", "reported_as": "-0.4976764 (+/- 7e-7) per site in S.S units", "note": "Read from Table 1 of arXiv:2603.14425 (square-lattice J1-J2 at J2/J1=0.5, PBC, E per site in S.S units), parsed from the arXiv HTML. Cross-checked against the CTWF paper's own text.", "secondary_of": null } }, { "energy": -199.128, "sigma": 0.012, "energy_variance": null, "dof": 100, "einf": 0, "v_score": null, "method": "Holographic Quantum Transformer (HQT), zero-shot 8x8->10x10 transfer", "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": "-0.49782 (+/- 0.00003) per site in S.S units", "note": "Abstract: \"This zero-shot protocol yields an energy of E/N = -0.49782(3), statistically consistent with the variational state of the art\". It is not consistent: it is 1.3e-4 BELOW the best variational energy (CNN-MPS -0.4976939(2)) and 1.05e-4 below the zero-variance extrapolated ground state -0.497715(9), i.e. below the ground state itself, which no variational energy can be. See the defect flag.", "secondary_of": null }, "defect": { "flag": "energy-variance-inconsistent", "finding": "Claimed 1.3e-4 below the best variational energy (CNN-MPS) while the paper itself calls the number 'statistically consistent with the variational state of the art' - it is lower by ~4x its own stated error bar, so if real it is an unclaimed record. There is NO exact reference at 10x10 (ED reaches ~6x6 for this model), so this is a contested record claim, not a proven error.", "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_100_P_0.5/", "per_site_divisor": 400, "per_site_label": "E/N (S.S)", "record": { "energy": -199.07756, "sigma": 0.00008, "method": "CNN-MPS (h,D,l)=(32,20,20), Marshall sign transformation", "reference": "arXiv:2603.14425, Disentangling Tensor Network States with Deep Neural Networks", "energy_per_site": -0.4976939 }, "no_record_reason": null }, { "model": "J1J2", "lattice": "square", "n_sites": 100, "boundary": "P", "params": { "J2": 0.6 }, "instance_id": "J1J2/square_100_P_0.6", "rows": [ { "energy": -190.98442540818522, "sigma": null, "energy_variance": 9.306874823283579, "dof": 100, "einf": 0, "v_score": 0.025515728236972162, "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_100_P_0.6/dmrg.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -181.3329, "sigma": 0.0074, "energy_variance": 56.47486, "dof": 100, "einf": 0, "v_score": 0.17175205762671678, "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_100_P_0.6/vmc_rbm.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -177.868, "sigma": 0.01, "energy_variance": 103.986, "dof": 100, "einf": 0, "v_score": 0.3286845036989973, "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_100_P_0.6/vmc_jastrow.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -190.648, "sigma": 0.012, "energy_variance": null, "dof": 100, "einf": 0, "v_score": null, "method": "RBM wave function", "bound_type": "variational", "bound_type_reason": "variational ansatz, or a Lanczos improvement of one (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": false, "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": "-0.47662 (+/- 0.00003) per site in S.S units", "note": "Read from Table 5 of arXiv:2206.14307 (J1-J2 on 6x6 and 10x10 square lattices, PBC, E/N in S.S units), extracted from the PDF with pypdf.", "secondary_of": null } }, { "energy": -190.416, "sigma": 0.004, "energy_variance": null, "dof": 100, "einf": 0, "v_score": null, "method": "CNN", "bound_type": "variational", "bound_type_reason": "variational ansatz, or a Lanczos improvement of one (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": true, "source": "sweep-pdf-2026-09-14", "provenance": "secondary", "verified": { "checked_on": "2026-09-14", "method": "arXiv PDF text extracted locally with pypdf, no LLM transcription", "reported_as": "-0.47604 (+/- 0.00001) per site in S.S units", "note": "Read from Table 5 of arXiv:2206.14307 (J1-J2 on 6x6 and 10x10 square lattices, PBC, E/N in S.S units), extracted from the PDF with pypdf.", "secondary_of": "Chen, Hendry, Weinberg & Feiguin, Systematic improvement of neural network quantum states using a Lanczos recursion, arXiv:2206.14307 (2022)" } } ], "url": "https://qmbl.org/i/J1J2/square_100_P_0.6/", "per_site_divisor": 400, "per_site_label": "E/N (S.S)", "record": { "energy": -190.98442540818522, "sigma": null, "method": "DMRG (bond dimension = 1024)", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/J1J2/square_100_P_0.6/dmrg.sh)", "energy_per_site": -0.477461063520463 }, "no_record_reason": null }, { "model": "J1J2", "lattice": "square", "n_sites": 100, "boundary": "P", "params": { "J2": 0.7 }, "instance_id": "J1J2/square_100_P_0.7", "rows": [ { "energy": -206.57227523794538, "sigma": null, "energy_variance": 13.807249386465855, "dof": 100, "einf": 0, "v_score": 0.03235661662296521, "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_100_P_0.7/dmrg.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -175.2267, "sigma": 0.0035, "energy_variance": 12.74215, "dof": 100, "einf": 0, "v_score": 0.041499431667546514, "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_100_P_0.7/vmc_rbm.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -189.3, "sigma": 0.016, "energy_variance": 269.96, "dof": 100, "einf": 0, "v_score": 0.7533524266705063, "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_100_P_0.7/vmc_jastrow.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -207.556, "sigma": 0.008, "energy_variance": null, "dof": 100, "einf": 0, "v_score": null, "method": "RBM wave function", "bound_type": "variational", "bound_type_reason": "variational ansatz, or a Lanczos improvement of one (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": false, "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": "-0.51889 (+/- 0.00002) per site in S.S units", "note": "Read from Table 5 of arXiv:2206.14307 (J1-J2 on 6x6 and 10x10 square lattices, PBC, E/N in S.S units), extracted from the PDF with pypdf.", "secondary_of": null } } ], "url": "https://qmbl.org/i/J1J2/square_100_P_0.7/", "per_site_divisor": 400, "per_site_label": "E/N (S.S)", "record": { "energy": -207.556, "sigma": 0.008, "method": "RBM wave function", "reference": "Chen, Hendry, Weinberg & Feiguin, Systematic improvement of neural network quantum states using a Lanczos recursion, arXiv:2206.14307 (2022)", "energy_per_site": -0.5188900000000001 }, "no_record_reason": null }, { "model": "J1J2", "lattice": "square", "n_sites": 100, "boundary": "P", "params": { "J2": 0.8 }, "instance_id": "J1J2/square_100_P_0.8", "rows": [ { "energy": -227.7026939728324, "sigma": null, "energy_variance": 28.508300812252628, "dof": 100, "einf": 0, "v_score": 0.05498383087574174, "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_100_P_0.8/dmrg.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -217.131, "sigma": 0.012, "energy_variance": 147.364, "dof": 100, "einf": 0, "v_score": 0.3125703192476002, "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_100_P_0.8/vmc_rbm.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -220.1402, "sigma": 0.0084, "energy_variance": 71.75249, "dof": 100, "einf": 0, "v_score": 0.1480601767260613, "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_100_P_0.8/vmc_jastrow.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -229.616, "sigma": 0.008, "energy_variance": null, "dof": 100, "einf": 0, "v_score": null, "method": "RBM wave function", "bound_type": "variational", "bound_type_reason": "variational ansatz, or a Lanczos improvement of one (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": false, "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": "-0.57404 (+/- 0.00002) per site in S.S units", "note": "Read from Table 5 of arXiv:2206.14307 (J1-J2 on 6x6 and 10x10 square lattices, PBC, E/N in S.S units), extracted from the PDF with 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"provenance": "imported", "baseline": true }, { "energy": -92.3972, "sigma": 0.0047, "energy_variance": 22.1831, "dof": 36, "einf": 0, "v_score": 0.0935421002336649, "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_36_P_0.9/vmc_rbm.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -91.7305, "sigma": 0.0046, "energy_variance": 21.5075, "dof": 36, "einf": 0, "v_score": 0.09201633065161303, "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_36_P_0.9/vmc_jastrow.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true } ], "url": "https://qmbl.org/i/J1J2/square_36_P_0.9/", "per_site_divisor": 144, "per_site_label": "E/N (S.S)", "record": { "energy": -93.29335086463179, "sigma": null, "method": "DMRG (bond dimension = 2048)", "reference": "[code](https://github.com/varbench/methods/blob/main/scripts/J1J2/square_36_P_0.9/dmrg.sh)", "energy_per_site": -0.6478704921154985 }, "no_record_reason": null }, { "model": "J1J2", "lattice": "square", "n_sites": 36, "boundary": "P", "params": { "J2": 1 }, "instance_id": "J1J2/square_36_P_1", "rows": [ { "energy": -102.867902314985, "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/J1J2/square_36_P_1/ed_lattice_symmetries.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -102.67869884652434, "sigma": null, "energy_variance": 7.164533866758574, "dof": 36, "einf": 0, "v_score": 0.02446412727299761, "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/J1J2/square_36_P_1/dmrg.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -102.12601, "sigma": 0.0045, "energy_variance": 20.4796, "dof": 36, "einf": 0, "v_score": 0.07068890290265885, "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_36_P_1/vmc_rbm.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -100.82252, "sigma": 0.005, "energy_variance": 25.3075, "dof": 36, "einf": 0, "v_score": 0.08962654391538695, "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_36_P_1/vmc_jastrow.sh)", "source": "varbench@2024-10-22", "provenance": "imported", "baseline": true }, { "energy": -102.85776, "sigma": 0.00144, "energy_variance": null, "dof": 36, "einf": 0, "v_score": null, "method": "RBM wave function", "bound_type": "variational", "bound_type_reason": "variational ansatz, or a Lanczos improvement of one (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": false, "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": "-0.71429 (+/- 0.00001) per site in S.S units", "note": "Read from Table 5 of arXiv:2206.14307 (J1-J2 on 6x6 and 10x10 square lattices, PBC, E/N in S.S units), extracted from the PDF with pypdf.", "secondary_of": null } }, { "energy": -102.74544, "sigma": 0.00144, "energy_variance": null, "dof": 36, "einf": 0, "v_score": null, "method": "CNN", "bound_type": "variational", "bound_type_reason": "variational ansatz, or a Lanczos improvement of one (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": true, "source": "sweep-pdf-2026-09-14", "provenance": "secondary", "verified": { "checked_on": "2026-09-14", "method": "arXiv PDF text extracted locally with pypdf, no LLM transcription", "reported_as": "-0.71351 (+/- 0.00001) per site in S.S units", "note": "Read from Table 5 of arXiv:2206.14307 (J1-J2 on 6x6 and 10x10 square lattices, PBC, E/N in S.S units), extracted from the PDF with pypdf.", "secondary_of": "Chen, Hendry, Weinberg & Feiguin, Systematic improvement of neural network quantum states using a Lanczos recursion, arXiv:2206.14307 (2022)" } } ], "url": "https://qmbl.org/i/J1J2/square_36_P_1/", "per_site_divisor": 144, "per_site_label": "E/N (S.S)", "record": { "energy": -102.85776, "sigma": 0.00144, "method": "RBM wave function", "reference": "Chen, Hendry, Weinberg & Feiguin, Systematic improvement of neural network quantum states using a Lanczos recursion, arXiv:2206.14307 (2022)", "energy_per_site": -0.71429 }, "no_record_reason": null }, { "model": "J1J2", "lattice": "square", "n_sites": 40, "boundary": "P", "params": { "J2": 0.1 }, "instance_id": "J1J2/square_40_P_0.1", "rows": [ { "energy": -101.8584104, "sigma": null, "energy_variance": null, "dof": 40, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Richter & Schulenburg, The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins, Eur. Phys. J. B 73, 117 (2010), arXiv:0909.3723", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "E_GS(S = 0) = -25.46460260 (total, S.S, J1 = 1), N = 40, J2 = 0.1", "note": "Table 1, \"Ground state energy E_GS(S = 0) ...\", row J2 = 0.1. The tilted 40-site square-lattice cluster of the paper's Fig. 1 with periodic boundaries (Sec. 2); E/N = -0.6366151 in S.S units, Pauli total = 4 E. The paper's J2 grid is 0, 0.1, ..., 0.5, 0.55, 0.6, 0.65, 0.7, 0.8, 0.9, 1.0.", "secondary_of": null } } ], "url": "https://qmbl.org/i/J1J2/square_40_P_0.1/", "per_site_divisor": 160, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "J1J2", "lattice": "square", "n_sites": 40, "boundary": "P", "params": { "J2": 0.2 }, "instance_id": "J1J2/square_40_P_0.2", "rows": [ { "energy": -95.60187672, "sigma": null, "energy_variance": null, "dof": 40, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Richter & Schulenburg, The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins, Eur. Phys. J. B 73, 117 (2010), arXiv:0909.3723", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "E_GS(S = 0) = -23.90046918 (total, S.S, J1 = 1), N = 40, J2 = 0.2", "note": "Table 1, \"Ground state energy E_GS(S = 0) ...\", row J2 = 0.2. The tilted 40-site square-lattice cluster of the paper's Fig. 1 with periodic boundaries (Sec. 2); E/N = -0.5975117 in S.S units, Pauli total = 4 E. The paper's J2 grid is 0, 0.1, ..., 0.5, 0.55, 0.6, 0.65, 0.7, 0.8, 0.9, 1.0.", "secondary_of": null } } ], "url": "https://qmbl.org/i/J1J2/square_40_P_0.2/", "per_site_divisor": 160, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "J1J2", "lattice": "square", "n_sites": 40, "boundary": "P", "params": { "J2": 0.3 }, "instance_id": "J1J2/square_40_P_0.3", "rows": [ { "energy": -89.70914572, "sigma": null, "energy_variance": null, "dof": 40, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Richter & Schulenburg, The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins, Eur. Phys. J. B 73, 117 (2010), arXiv:0909.3723", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "E_GS(S = 0) = -22.42728643 (total, S.S, J1 = 1), N = 40, J2 = 0.3", "note": "Table 1, \"Ground state energy E_GS(S = 0) ...\", row J2 = 0.3. The tilted 40-site square-lattice cluster of the paper's Fig. 1 with periodic boundaries (Sec. 2); E/N = -0.5606822 in S.S units, Pauli total = 4 E. The paper's J2 grid is 0, 0.1, ..., 0.5, 0.55, 0.6, 0.65, 0.7, 0.8, 0.9, 1.0.", "secondary_of": null } } ], "url": "https://qmbl.org/i/J1J2/square_40_P_0.3/", "per_site_divisor": 160, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "J1J2", "lattice": "square", "n_sites": 40, "boundary": "P", "params": { "J2": 0.4 }, "instance_id": "J1J2/square_40_P_0.4", "rows": [ { "energy": -84.3534668, "sigma": null, "energy_variance": null, "dof": 40, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Richter & Schulenburg, The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins, Eur. Phys. J. B 73, 117 (2010), arXiv:0909.3723", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "E_GS(S = 0) = -21.08836670 (total, S.S, J1 = 1), N = 40, J2 = 0.4", "note": "Table 1, \"Ground state energy E_GS(S = 0) ...\", row J2 = 0.4. The tilted 40-site square-lattice cluster of the paper's Fig. 1 with periodic boundaries (Sec. 2); E/N = -0.5272092 in S.S units, Pauli total = 4 E. The paper's J2 grid is 0, 0.1, ..., 0.5, 0.55, 0.6, 0.65, 0.7, 0.8, 0.9, 1.0.", "secondary_of": null } } ], "url": "https://qmbl.org/i/J1J2/square_40_P_0.4/", "per_site_divisor": 160, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "J1J2", "lattice": "square", "n_sites": 40, "boundary": "P", "params": { "J2": 0.5 }, "instance_id": "J1J2/square_40_P_0.5", "rows": [ { "energy": -79.85219356, "sigma": null, "energy_variance": null, "dof": 40, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Richter & Schulenburg, The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins, Eur. Phys. J. B 73, 117 (2010), arXiv:0909.3723", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "E_GS(S = 0) = -19.96304839 (total, S.S, J1 = 1), N = 40, J2 = 0.5", "note": "Table 1, \"Ground state energy E_GS(S = 0) ...\", row J2 = 0.5. The tilted 40-site square-lattice cluster of the paper's Fig. 1 with periodic boundaries (Sec. 2); E/N = -0.4990762 in S.S units, Pauli total = 4 E. The paper's J2 grid is 0, 0.1, ..., 0.5, 0.55, 0.6, 0.65, 0.7, 0.8, 0.9, 1.0.", "secondary_of": null } } ], "url": "https://qmbl.org/i/J1J2/square_40_P_0.5/", "per_site_divisor": 160, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "J1J2", "lattice": "square", "n_sites": 40, "boundary": "P", "params": { "J2": 0.55 }, "instance_id": "J1J2/square_40_P_0.55", "rows": [ { "energy": -78.07166104, "sigma": null, "energy_variance": null, "dof": 40, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Richter & Schulenburg, The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins, Eur. Phys. J. B 73, 117 (2010), arXiv:0909.3723", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "E_GS(S = 0) = -19.51791526 (total, S.S, J1 = 1), N = 40, J2 = 0.55", "note": "Table 1, \"Ground state energy E_GS(S = 0) ...\", row J2 = 0.55. The tilted 40-site square-lattice cluster of the paper's Fig. 1 with periodic boundaries (Sec. 2); E/N = -0.4879479 in S.S units, Pauli total = 4 E. The paper's J2 grid is 0, 0.1, ..., 0.5, 0.55, 0.6, 0.65, 0.7, 0.8, 0.9, 1.0.", "secondary_of": null } } ], "url": "https://qmbl.org/i/J1J2/square_40_P_0.55/", "per_site_divisor": 160, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "J1J2", "lattice": "square", "n_sites": 40, "boundary": "P", "params": { "J2": 0.6 }, "instance_id": "J1J2/square_40_P_0.6", "rows": [ { "energy": -76.73472152, "sigma": null, "energy_variance": null, "dof": 40, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Richter & Schulenburg, The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins, Eur. Phys. J. B 73, 117 (2010), arXiv:0909.3723", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "E_GS(S = 0) = -19.18368038 (total, S.S, J1 = 1), N = 40, J2 = 0.6", "note": "Table 1, \"Ground state energy E_GS(S = 0) ...\", row J2 = 0.6. The tilted 40-site square-lattice cluster of the paper's Fig. 1 with periodic boundaries (Sec. 2); E/N = -0.4795920 in S.S units, Pauli total = 4 E. The paper's J2 grid is 0, 0.1, ..., 0.5, 0.55, 0.6, 0.65, 0.7, 0.8, 0.9, 1.0.", "secondary_of": null } } ], "url": "https://qmbl.org/i/J1J2/square_40_P_0.6/", "per_site_divisor": 160, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "J1J2", "lattice": "square", "n_sites": 40, "boundary": "P", "params": { "J2": 0.65 }, "instance_id": "J1J2/square_40_P_0.65", "rows": [ { "energy": -80.1841302, "sigma": null, "energy_variance": null, "dof": 40, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Richter & Schulenburg, The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins, Eur. Phys. J. B 73, 117 (2010), arXiv:0909.3723", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "E_GS(S = 0) = -20.04603255 (total, S.S, J1 = 1), N = 40, J2 = 0.65", "note": "Table 1, \"Ground state energy E_GS(S = 0) ...\", row J2 = 0.65. The tilted 40-site square-lattice cluster of the paper's Fig. 1 with periodic boundaries (Sec. 2); E/N = -0.5011508 in S.S units, Pauli total = 4 E. The paper's J2 grid is 0, 0.1, ..., 0.5, 0.55, 0.6, 0.65, 0.7, 0.8, 0.9, 1.0.", "secondary_of": null } } ], "url": "https://qmbl.org/i/J1J2/square_40_P_0.65/", "per_site_divisor": 160, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "J1J2", "lattice": "square", "n_sites": 40, "boundary": "P", "params": { "J2": 0.7 }, "instance_id": "J1J2/square_40_P_0.7", "rows": [ { "energy": -84.22120956, "sigma": null, "energy_variance": null, "dof": 40, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Richter & Schulenburg, The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins, Eur. Phys. J. B 73, 117 (2010), arXiv:0909.3723", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "E_GS(S = 0) = -21.05530239 (total, S.S, J1 = 1), N = 40, J2 = 0.7", "note": "Table 1, \"Ground state energy E_GS(S = 0) ...\", row J2 = 0.7. The tilted 40-site square-lattice cluster of the paper's Fig. 1 with periodic boundaries (Sec. 2); E/N = -0.5263826 in S.S units, Pauli total = 4 E. The paper's J2 grid is 0, 0.1, ..., 0.5, 0.55, 0.6, 0.65, 0.7, 0.8, 0.9, 1.0.", "secondary_of": null } } ], "url": "https://qmbl.org/i/J1J2/square_40_P_0.7/", "per_site_divisor": 160, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "J1J2", "lattice": "square", "n_sites": 40, "boundary": "P", "params": { "J2": 0.8 }, "instance_id": "J1J2/square_40_P_0.8", "rows": [ { "energy": -93.36081708, "sigma": null, "energy_variance": null, "dof": 40, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Richter & Schulenburg, The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins, Eur. Phys. J. B 73, 117 (2010), arXiv:0909.3723", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "E_GS(S = 0) = -23.34020427 (total, S.S, J1 = 1), N = 40, J2 = 0.8", "note": "Table 1, \"Ground state energy E_GS(S = 0) ...\", row J2 = 0.8. The tilted 40-site square-lattice cluster of the paper's Fig. 1 with periodic boundaries (Sec. 2); E/N = -0.5835051 in S.S units, Pauli total = 4 E. The paper's J2 grid is 0, 0.1, ..., 0.5, 0.55, 0.6, 0.65, 0.7, 0.8, 0.9, 1.0.", "secondary_of": null } } ], "url": "https://qmbl.org/i/J1J2/square_40_P_0.8/", "per_site_divisor": 160, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "J1J2", "lattice": "square", "n_sites": 40, "boundary": "P", "params": { "J2": 0.9 }, "instance_id": "J1J2/square_40_P_0.9", "rows": [ { "energy": -103.34765148, "sigma": null, "energy_variance": null, "dof": 40, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Richter & Schulenburg, The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins, Eur. Phys. J. B 73, 117 (2010), arXiv:0909.3723", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "E_GS(S = 0) = -25.83691287 (total, S.S, J1 = 1), N = 40, J2 = 0.9", "note": "Table 1, \"Ground state energy E_GS(S = 0) ...\", row J2 = 0.9. The tilted 40-site square-lattice cluster of the paper's Fig. 1 with periodic boundaries (Sec. 2); E/N = -0.6459228 in S.S units, Pauli total = 4 E. The paper's J2 grid is 0, 0.1, ..., 0.5, 0.55, 0.6, 0.65, 0.7, 0.8, 0.9, 1.0.", "secondary_of": null } } ], "url": "https://qmbl.org/i/J1J2/square_40_P_0.9/", "per_site_divisor": 160, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "J1J2", "lattice": "square", "n_sites": 40, "boundary": "P", "params": { "J2": 1 }, "instance_id": "J1J2/square_40_P_1", "rows": [ { "energy": -113.75523568, "sigma": null, "energy_variance": null, "dof": 40, "einf": 0, "v_score": null, "method": "Exact diagonalization (Lanczos)", "bound_type": "exact", "bound_type_reason": "exact diagonalization of the full cluster; deterministic, exact to the printed digits (RULES.md 4)", "reference": "Richter & Schulenburg, The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins, Eur. Phys. J. B 73, 117 (2010), arXiv:0909.3723", "peer_reviewed": true, "source": "exact-2026-09-15", "provenance": "primary", "verified": { "checked_on": "2026-09-15", "method": "arXiv PDF extracted locally with pypdf in layout mode, parsed by column position and checked against the paper's own per-site column; no LLM transcription", "reported_as": "E_GS(S = 0) = -28.43880892 (total, S.S, J1 = 1), N = 40, J2 = 1", "note": "Table 1, \"Ground state energy E_GS(S = 0) ...\", row J2 = 1. The tilted 40-site square-lattice cluster of the paper's Fig. 1 with periodic boundaries (Sec. 2); E/N = -0.7109702 in S.S units, Pauli total = 4 E. The paper's J2 grid is 0, 0.1, ..., 0.5, 0.55, 0.6, 0.65, 0.7, 0.8, 0.9, 1.0.", "secondary_of": null } } ], "url": "https://qmbl.org/i/J1J2/square_40_P_1/", "per_site_divisor": 160, "per_site_label": "E/N (S.S)", "record": null, "no_record_reason": "solved exactly" }, { "model": "J1J2", "lattice": "square", "n_sites": 400, "boundary": "P", "params": { "J2": 0.5 }, "instance_id": "J1J2/square_400_P_0.5", "rows": [ { "energy": -794.87792, "sigma": 0.00096, "energy_variance": null, "dof": 400, "einf": 0, "v_score": null, "method": "CNN-MPS (h,D,l)=(32,15,20)", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "arXiv:2603.14425, Disentangling Tensor Network States with Deep Neural Networks", "peer_reviewed": false, "source": "sweep-2026-09-13", "provenance": "primary", "verified": { "checked_on": "2026-09-13", "method": "arXiv HTML parsed locally, no LLM transcription", "reported_as": "-0.4967987 (+/- 6e-7) per site in S.S units", "note": "Read from Table 1 of arXiv:2603.14425 (square-lattice J1-J2 at J2/J1=0.5, PBC, E per site in S.S units), parsed from the arXiv HTML. 20x20; no VarBench instance existed for this size.", "secondary_of": null } }, { "energy": -794.7712, "sigma": 0.0016, "energy_variance": null, "dof": 400, "einf": 0, "v_score": null, "method": "ViT with symmetry restoration", "bound_type": "variational", "bound_type_reason": "variational ansatz; energy is a strict upper bound (assigned during source verification)", "reference": "Approaching the Thermodynamic Limit with Neural-Network Quantum States, arXiv:2602.02665", "peer_reviewed": false, "source": "sweep-2026-09-13", "provenance": "primary", "verified": { "checked_on": "2026-09-13", "method": "arXiv HTML parsed locally, no LLM transcription", "reported_as": "-0.496732 (+/- 0.000001) per site in S.S units", "note": "Read from Table 1 of arXiv:2603.14425 (square-lattice J1-J2 at J2/J1=0.5, PBC, E per site in S.S units), parsed from the arXiv HTML. Listed in Table 1 as the L=20 ViT 2026 value; matches the source paper's own symmetry-restoration sequence.", "secondary_of": null } } ], "url": "https://qmbl.org/i/J1J2/square_400_P_0.5/", "per_site_divisor": 1600, "per_site_label": "E/N (S.S)", "record": { "energy": -794.87792, "sigma": 0.00096, "method": "CNN-MPS (h,D,l)=(32,15,20)", "reference": "arXiv:2603.14425, Disentangling Tensor Network States with Deep Neural Networks", "energy_per_site": -0.49679870000000004 }, "no_record_reason": null }, { "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 }, { "model": "J1J2", "lattice": "square", "n_sites": 64, "boundary": "P", "params": { "J2": 0.55 }, "instance_id": "J1J2/square_64_P_0.55", "rows": [ { "energy": -125.0176, "sigma": 0.0204, "energy_variance": 3.0368, "dof": 64, "einf": 0, "v_score": 0.012435230792484863, "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" } ], "url": "https://qmbl.org/i/J1J2/square_64_P_0.55/", "per_site_divisor": 256, "per_site_label": "E/N (S.S)", "record": { "energy": -125.0176, "sigma": 0.0204, "method": "ClebschTree", "reference": "[paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.045123)", "energy_per_site": -0.48835 }, "no_record_reason": null }, { "model": "J1J2", "lattice": "triangular", "n_sites": 108, "boundary": "P", "params": { "J2": 0.125 }, "instance_id": "J1J2/triangular_108_P_0.125", "rows": [ { "energy": -221.47776, "sigma": 0.03888, "energy_variance": null, "dof": 108, "einf": 0, "v_score": null, "method": "GCNN + Lanczos step", "bound_type": "variational", "bound_type_reason": "assigned during source verification (RULES.md 4)", "reference": "Roth, Szabo & MacDonald, High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks, Phys. Rev. B 108, 054410 (2023), arXiv:2211.07749", "peer_reviewed": true, "source": "sweep-pdf-2026-09-14", "provenance": "primary", "verified": { "checked_on": "2026-09-14", "method": "arXiv PDF extracted locally with pypdf in layout mode, no LLM transcription", "reported_as": "-0.51268 (+/- 0.00009) per site in S.S units", "note": "Read from Table III of arXiv:2211.07749 (J1-J2 triangular lattice, periodic, energies in units of J1 per spin, i.e. the S.S per-site convention), extracted from the PDF with pypdf in layout mode so the column positions are unambiguous. Column \"GCNN+LS\" at J2/J1 = 1/8, N = 108. A Lanczos step on a variational state stays variational (RULES.md 4).", "secondary_of": null } }, { "energy": -221.076, "sigma": 0.03024, "energy_variance": null, "dof": 108, "einf": 0, "v_score": null, "method": "GCNN (deep group-equivariant CNN)", "bound_type": "variational", "bound_type_reason": "assigned during source verification (RULES.md 4)", "reference": "Roth, Szabo & MacDonald, High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks, Phys. Rev. B 108, 054410 (2023), arXiv:2211.07749", "peer_reviewed": true, "source": "sweep-pdf-2026-09-14", "provenance": "primary", "verified": { "checked_on": "2026-09-14", "method": "arXiv PDF extracted locally with pypdf in layout mode, no LLM transcription", "reported_as": "-0.51175 (+/- 0.00007) per site in S.S units", "note": "Read from Table III of arXiv:2211.07749 (J1-J2 triangular lattice, periodic, energies in units of J1 per spin, i.e. the S.S per-site convention), extracted from the PDF with pypdf in layout mode so the column positions are unambiguous. Row J2/J1 = 1/8, N = 108.", "secondary_of": null } }, { "energy": -218.9808, "sigma": 0.3456, "energy_variance": null, "dof": 108, "einf": 0, "v_score": null, "method": "Graph neural network", "bound_type": "variational", "bound_type_reason": "assigned during source verification (RULES.md 4)", "reference": "Kochkov, Pfaff, Sanchez-Gonzalez, Battaglia & Clark, Learning ground states of quantum Hamiltonians with graph networks, arXiv:2110.06390 (2021)", "peer_reviewed": null, "source": "sweep-pdf-2026-09-14", "provenance": "secondary", "verified": { "checked_on": "2026-09-14", "method": "arXiv PDF extracted locally with pypdf in layout mode, no LLM transcription", "reported_as": "-0.5069 (+/- 0.0008) per site in S.S units", "note": "Read from Table III of arXiv:2211.07749 (J1-J2 triangular lattice, periodic, energies in units of J1 per spin, i.e. the S.S per-site convention), extracted from the PDF with pypdf in layout mode so the column positions are unambiguous. Column \"Graph NN [64]\".", "secondary_of": "Roth, Szabo & MacDonald, High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks, Phys. Rev. B 108, 054410 (2023), arXiv:2211.07749" } }, { "energy": -221.60304, "sigma": null, "energy_variance": null, "dof": 108, "einf": 0, "v_score": null, "method": "Thermodynamic-limit estimate interpolated to this size (1/L^3)", "bound_type": "extrapolated", "bound_type_reason": "assigned during source verification (RULES.md 4)", "reference": "Iqbal, Hu, Thomale, Poilblanc & Becca, Spin liquid nature in the Heisenberg J1-J2 triangular antiferromagnet, Phys. Rev. B 93, 144411 (2016)", "peer_reviewed": null, "source": "sweep-pdf-2026-09-14", "provenance": "secondary", "verified": { "checked_on": "2026-09-14", "method": "arXiv PDF extracted locally with pypdf in layout mode, no LLM transcription", "reported_as": "-0.51297 per site in S.S units", "note": "Read from Table III of arXiv:2211.07749 (J1-J2 triangular lattice, periodic, energies in units of J1 per spin, i.e. the S.S per-site convention), extracted from the PDF with pypdf in layout mode so the column positions are unambiguous. Column \"Exact/Interpolated [10,63]\". Per the caption this is NOT an exact energy above 36 sites: it is built from the thermodynamic-limit estimate of Ref. [10] and the exact 36-site energy, assuming finite-size effects scale as 1/L^3. Carried as `extrapolated` (RULES.md 4).", "secondary_of": "Roth, Szabo & MacDonald, High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks, Phys. Rev. B 108, 054410 (2023), arXiv:2211.07749" } } ], "url": "https://qmbl.org/i/J1J2/triangular_108_P_0.125/", "per_site_divisor": 432, "per_site_label": "E/N (S.S)", "record": { "energy": -221.47776, "sigma": 0.03888, "method": "GCNN + Lanczos step", "reference": "Roth, Szabo & MacDonald, High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks, Phys. Rev. B 108, 054410 (2023), arXiv:2211.07749", "energy_per_site": -0.51268 }, "no_record_reason": null }, { "model": "J1J2", "lattice": "triangular", "n_sites": 144, "boundary": "P", "params": { "J2": 0.125 }, "instance_id": "J1J2/triangular_144_P_0.125", "rows": [ { "energy": -295.01568, "sigma": 0.05184, "energy_variance": null, "dof": 144, "einf": 0, "v_score": null, "method": "GCNN + Lanczos step", "bound_type": "variational", "bound_type_reason": "assigned during source verification (RULES.md 4)", "reference": "Roth, Szabo & MacDonald, High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks, Phys. Rev. B 108, 054410 (2023), arXiv:2211.07749", "peer_reviewed": true, "source": "sweep-pdf-2026-09-14", "provenance": "primary", "verified": { "checked_on": "2026-09-14", "method": "arXiv PDF extracted locally with pypdf in layout mode, no LLM transcription", "reported_as": "-0.51218 (+/- 0.00009) per site in S.S units", "note": "Read from Table III of arXiv:2211.07749 (J1-J2 triangular lattice, periodic, energies in units of J1 per spin, i.e. the S.S per-site convention), extracted from the PDF with pypdf in layout mode so the column positions are unambiguous. Column \"GCNN+LS\" at J2/J1 = 1/8, N = 144.", "secondary_of": null } }, { "energy": -294.34176, "sigma": 0.03456, "energy_variance": null, "dof": 144, "einf": 0, "v_score": null, "method": "GCNN (deep group-equivariant CNN)", "bound_type": "variational", "bound_type_reason": "assigned during source verification (RULES.md 4)", "reference": "Roth, Szabo & MacDonald, High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks, Phys. Rev. B 108, 054410 (2023), arXiv:2211.07749", "peer_reviewed": true, "source": "sweep-pdf-2026-09-14", "provenance": "primary", "verified": { "checked_on": "2026-09-14", "method": "arXiv PDF extracted locally with pypdf in layout mode, no LLM transcription", "reported_as": "-0.51101 (+/- 0.00006) per site in S.S units", "note": "Read from Table III of arXiv:2211.07749 (J1-J2 triangular lattice, periodic, energies in units of J1 per spin, i.e. the S.S per-site convention), extracted from the PDF with pypdf in layout mode so the column positions are unambiguous. Row J2/J1 = 1/8, N = 144.", "secondary_of": null } }, { "energy": -294.081408, "sigma": 0.00288, "energy_variance": null, "dof": 144, "einf": 0, "v_score": null, "method": "Gutzwiller-projected fermionic state + Lanczos step", "bound_type": "variational", "bound_type_reason": "assigned during source verification (RULES.md 4)", "reference": "Iqbal, Hu, Thomale, Poilblanc & Becca, Spin liquid nature in the Heisenberg J1-J2 triangular antiferromagnet, Phys. Rev. B 93, 144411 (2016)", "peer_reviewed": null, "source": "sweep-pdf-2026-09-14", "provenance": "secondary", "verified": { "checked_on": "2026-09-14", "method": "arXiv PDF extracted locally with pypdf in layout mode, no LLM transcription", "reported_as": "-0.510558 (+/- 0.000005) per site in S.S units", "note": "Read from Table III of arXiv:2211.07749 (J1-J2 triangular lattice, periodic, energies in units of J1 per spin, i.e. the S.S per-site convention), extracted from the PDF with pypdf in layout mode so the column positions are unambiguous. Column \"Gutzwiller+LS [10]\" = Phys. Rev. B 93, 144411 (2016).", "secondary_of": "Roth, Szabo & MacDonald, High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks, Phys. Rev. B 108, 054410 (2023), arXiv:2211.07749" } }, { "energy": -295.344, "sigma": null, "energy_variance": null, "dof": 144, "einf": 0, "v_score": null, "method": "Thermodynamic-limit estimate interpolated to this size (1/L^3)", "bound_type": "extrapolated", "bound_type_reason": "assigned during source verification (RULES.md 4)", "reference": "Iqbal, Hu, Thomale, Poilblanc & Becca, Spin liquid nature in the Heisenberg J1-J2 triangular antiferromagnet, Phys. Rev. B 93, 144411 (2016)", "peer_reviewed": null, "source": "sweep-pdf-2026-09-14", "provenance": "secondary", "verified": { "checked_on": "2026-09-14", "method": "arXiv PDF extracted locally with pypdf in layout mode, no LLM transcription", "reported_as": "-0.51275 per site in S.S units", "note": "Read from Table III of arXiv:2211.07749 (J1-J2 triangular lattice, periodic, energies in units of J1 per spin, i.e. the S.S per-site convention), extracted from the PDF with pypdf in layout mode so the column positions are unambiguous. Column \"Exact/Interpolated [10,63]\". Per the caption this is NOT an exact energy above 36 sites: it is built from the thermodynamic-limit estimate of Ref. [10] and the exact 36-site energy, assuming finite-size effects scale as 1/L^3. Carried as `extrapolated` (RULES.md 4).", "secondary_of": "Roth, Szabo & MacDonald, High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks, Phys. Rev. B 108, 054410 (2023), arXiv:2211.07749" } } ], "url": "https://qmbl.org/i/J1J2/triangular_144_P_0.125/", "per_site_divisor": 576, "per_site_label": "E/N (S.S)", "record": { "energy": -295.01568, "sigma": 0.05184, "method": "GCNN + Lanczos step", "reference": "Roth, Szabo & MacDonald, High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks, Phys. Rev. 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