{ "model": "Heisenberg", "lattice": "triangular", "n_sites": 324, "boundary": "P", "params": {}, "instance_id": "Heisenberg/triangular_324_P", "rows": [ { "energy": -715.007088, "sigma": 0.001296, "energy_variance": null, "dof": 324, "einf": 0, "v_score": null, "method": "ViT", "method_detail": "spatial attention, b = 3, translations + C6v projection", "method_as_published": "ViT with Spatial Attention, translations + C6v projection", "family": "transformer / ViT", "bound_type": "variational", "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)", "reference": "Viteritti, Rende, Sachdev & Carleo, Approaching the Thermodynamic Limit with Neural-Network Quantum States, arXiv:2602.02665", "peer_reviewed": false, "source": "sweep-allresults-2026-09-28", "provenance": "primary", "verified": { "checked_on": "2026-09-28", "method": "source text of arXiv:2602.02665 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading, parsed from the committed HTML table and matched against the PDF text", "reported_as": "-0.551703(1)", "note": "Table 1, 18 × 18 block, row 'ViT', printed '-0.551703(1)', 'This work'; caption: 'The results of this work are obtained by enforcing translational and C6v point-group symmetries (refer to Section V.2)'. | Table 1 prints ground-state energies per site of the spin-1/2 triangular Heisenberg model on periodic L x L clusters (Sec. III: 'All calculations are performed on periodic L x L clusters with L chosen as a multiple of 3', H = J sum S_i.S_j), S.S units. Magnitude check: the values lie between -0.5517 and -0.5520, next to the 12 x 12 value -0.55315 (GCNN, per site, S.S) on Heisenberg/triangular_144_P, not four times it (Pauli) or a total.", "secondary_of": null }, "compute": { "parameters": 450000, "gpu_hours": 25000, "device": "NVIDIA GH200", "n_devices": null, "samples": 16384, "wall_clock": null, "cpu_core_hours": null, "bond_dimension": null, "iterations": 5600, "evaluation": { "stages": [ { "iterations": 5000, "samples": null, "parameters": null, "evaluations_per_amplitude": 1 }, { "iterations": 500, "samples": null, "parameters": null, "evaluations_per_amplitude": 9 }, { "iterations": 100, "samples": null, "parameters": null, "evaluations_per_amplitude": 108 } ], "sr": { "kind": "minsr" } }, "reported_as": "The Vision Transformer architecture used in this work comprises h = 12 attention heads, n_l = 8 layers, and an embedding dimension of d = 72, resulting in approximately P ≈ 4.5 × 10^5 trainable parameters for different system sizes, from L = 18 to L = 42. [...] we set b = 3 for the triangular lattice [...] (C6v group for the triangular lattice [...]) [...] Variational Monte Carlo (VMC) optimizations employing M = 2^14 samples for the stochastic estimates were carried out using SR with the linear algebra trick being in the regime P ≫ M enhanced with SPRING. The optimization proceeded in three stages: 5000 steps without symmetry summation, followed by 500 steps including translational symmetries, and a final 100 steps with both rotational and reflection symmetries restored. || Acknowledgments: The numerical experiments performed in this work required 25000 hours on GH200 GPUs.", "source": "Sec. V B Architecture and Optimization Details; Acknowledgments, arXiv:2602.02665 (mining pass 2026-09-28)", "scope": "paper", "confidence": "medium", "note": "gpu_hours = 25000 is the total for ALL numerical experiments in the paper (triangular L = 18-42 and the 20x20 J1-J2 run), not for this row. parameters 'approximately 4.5 x 10^5', stated for every triangular size. iterations = 5000 + 500 + 100 stated stages; samples = 2^14 per step. Stages: 1 evaluation per amplitude unsymmetrised; b^2 = 9 translations within a 3x3 patch; those 9 x the 12 C6v elements = 108. sr minsr (the P >> M linear-algebra trick with SPRING)." } }, { "energy": -715.392, "sigma": 0.01296, "energy_variance": null, "dof": 324, "einf": 0, "v_score": null, "method": "ViT", "method_detail": "spatial attention, b = 3, variance → 0", "method_as_published": "ViT with Spatial Attention, zero-variance extrapolation (triangular)", "family": "transformer / ViT", "bound_type": "extrapolated", "bound_type_reason": "zero-variance or bond-dimension extrapolation; not an upper bound (assigned during source reading)", "reference": "Viteritti, Rende, Sachdev & Carleo, Approaching the Thermodynamic Limit with Neural-Network Quantum States, arXiv:2602.02665", "peer_reviewed": false, "source": "sweep-allresults-2026-09-28", "provenance": "primary", "verified": { "checked_on": "2026-09-28", "method": "source text of arXiv:2602.02665 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading, parsed from the committed HTML table and matched against the PDF text", "reported_as": "-0.55200(1)", "note": "Table 1, 18 × 18 block, row 'Zero Variance', printed '-0.55200(1)', 'This work'; caption: 'For each system size the zero-variance extrapolated energy is also reported'. Sec. V.3 / Fig. 6: linear fit of energy against variance per site over the unprojected, translation-projected and fully projected ViT states at this L; an extrapolation of the same runs, not a separate calculation. | Table 1 prints ground-state energies per site of the spin-1/2 triangular Heisenberg model on periodic L x L clusters (Sec. III: 'All calculations are performed on periodic L x L clusters with L chosen as a multiple of 3', H = J sum S_i.S_j), S.S units. Magnitude check: the values lie between -0.5517 and -0.5520, next to the 12 x 12 value -0.55315 (GCNN, per site, S.S) on Heisenberg/triangular_144_P, not four times it (Pauli) or a total.", "secondary_of": null }, "compute": { "parameters": 450000, "gpu_hours": 25000, "device": "NVIDIA GH200", "n_devices": null, "samples": 16384, "wall_clock": null, "cpu_core_hours": null, "bond_dimension": null, "iterations": 5600, "evaluation": { "stages": [ { "iterations": 5000, "samples": null, "parameters": null, "evaluations_per_amplitude": 1 }, { "iterations": 500, "samples": null, "parameters": null, "evaluations_per_amplitude": 9 }, { "iterations": 100, "samples": null, "parameters": null, "evaluations_per_amplitude": 108 } ], "sr": { "kind": "minsr" } }, "reported_as": "The Vision Transformer architecture used in this work comprises h = 12 attention heads, n_l = 8 layers, and an embedding dimension of d = 72, resulting in approximately P ≈ 4.5 × 10^5 trainable parameters for different system sizes, from L = 18 to L = 42. [...] we set b = 3 for the triangular lattice [...] (C6v group for the triangular lattice [...]) [...] Variational Monte Carlo (VMC) optimizations employing M = 2^14 samples for the stochastic estimates were carried out using SR with the linear algebra trick being in the regime P ≫ M enhanced with SPRING. The optimization proceeded in three stages: 5000 steps without symmetry summation, followed by 500 steps including translational symmetries, and a final 100 steps with both rotational and reflection symmetries restored. || Acknowledgments: The numerical experiments performed in this work required 25000 hours on GH200 GPUs.", "source": "Sec. V B Architecture and Optimization Details; Acknowledgments, arXiv:2602.02665 (mining pass 2026-09-28)", "scope": "paper", "confidence": "medium", "note": "gpu_hours = 25000 is the total for ALL numerical experiments in the paper (triangular L = 18-42 and the 20x20 J1-J2 run), not for this row. parameters 'approximately 4.5 x 10^5', stated for every triangular size. iterations = 5000 + 500 + 100 stated stages; samples = 2^14 per step. Stages: 1 evaluation per amplitude unsymmetrised; b^2 = 9 translations within a 3x3 patch; those 9 x the 12 C6v elements = 108. sr minsr (the P >> M linear-algebra trick with SPRING). This row is the zero-variance extrapolation of the same three stages (Fig. 6), not a separate run, so it draws the same cost again." } }, { "energy": -706.86432, "sigma": 0.01296, "energy_variance": null, "dof": 324, "einf": 0, "v_score": null, "method": "VMC", "method_detail": "120° order, π-flux hopping", "method_as_published": "Gutzwiller-projected 120° magnetic state (π-flux hopping)", "family": "classic VMC", "bound_type": "variational", "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)", "reference": "Ghorbani, Tocchio & Becca, Variational wave functions for the S = 1/2 Heisenberg model on the anisotropic triangular lattice: Spin liquids and spiral orders, Phys. Rev. B 93, 085111 (2016), arXiv:1512.03356", "peer_reviewed": true, "source": "sweep-allresults-2026-09-28", "provenance": "primary", "verified": { "checked_on": "2026-09-28", "method": "source text of arXiv:1512.03356 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading by a reader agent, line checked in the committed text by the session", "reported_as": "-0.54542(1)", "note": "Table I (sources/1512.03356.txt L366-377), row J'/J = 1.0, column 'E/J (Magnetic state)'; repeated in Table II at J/J' = 1.0. At the isotropic point the best magnetic state has 120° order (text, L336-340) with π-fluxes threading up triangles. Quoted as 'Jastrow-Gutzwiller -0.54542(1) [71]' in Table 1 of arXiv:2602.02665. | Table I: E/J on the 18 x 18 cluster, periodic boundary conditions (Sec. II), H = J sum S_i.S_j + J' sum S_i.S_j; row J'/J = 1.0 is the isotropic triangular Heisenberg model, S.S units.", "secondary_of": null } }, { "energy": -694.2024, "sigma": 0.02592, "energy_variance": null, "dof": 324, "einf": 0, "v_score": null, "method": "VMC", "method_detail": "U(1) Dirac spin liquid", "method_as_published": "Gutzwiller-projected U(1) Dirac spin liquid", "family": "classic VMC", "bound_type": "variational", "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)", "reference": "Ghorbani, Tocchio & Becca, Variational wave functions for the S = 1/2 Heisenberg model on the anisotropic triangular lattice: Spin liquids and spiral orders, Phys. Rev. B 93, 085111 (2016), arXiv:1512.03356", "peer_reviewed": true, "source": "sweep-allresults-2026-09-28", "provenance": "primary", "verified": { "checked_on": "2026-09-28", "method": "source text of arXiv:1512.03356 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading by a reader agent, line checked in the committed text by the session", "reported_as": "-0.53565(2)", "note": "Table I (sources/1512.03356.txt L366-377), row J'/J = 1.0, column 'E/J (Spin liquid)', the paper's best spin liquid; for J'/J >~ 0.85 that is the U(1) Dirac state (text L283-286: the Z2Ad Ansatz is lower only for J'/J <~ 0.85). | Table I: E/J on the 18 x 18 cluster, periodic boundary conditions (Sec. II), H = J sum S_i.S_j + J' sum S_i.S_j; row J'/J = 1.0 is the isotropic triangular Heisenberg model, S.S units.", "secondary_of": null } }, { "energy": -707.4864, "sigma": 0.1296, "energy_variance": null, "dof": 324, "einf": 0, "v_score": null, "method": "EPS", "method_detail": "16-site plaquettes", "method_as_published": "Entangled-plaquette state (EPS), 16-site plaquettes", "family": "classic VMC", "bound_type": "variational", "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)", "reference": "Mezzacapo & Cirac, Ground-state properties of the spin-1/2 antiferromagnetic Heisenberg model on the triangular lattice: a variational study based on entangled-plaquette states, New J. Phys. 12, 103039 (2010), arXiv:1006.4480", "peer_reviewed": true, "source": "sweep-allresults-2026-09-28", "provenance": "primary", "verified": { "checked_on": "2026-09-28", "method": "source text of arXiv:1006.4480 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading by a reader agent, line checked in the committed text by the session", "reported_as": "-0.5459(1)", "note": "Table 2 (sources/1006.4480.txt L252-259), row N = 324; 'N plaquettes comprising 16 sites have been employed'. Quoted as 'EPS -0.5459(1) [72]' in Table 1 of arXiv:2602.02665. | Table 2: 'Variational GS energy per site (in units of Jxy) ... for the spin-1/2 AHM on a triangular lattice of size N with PBC' at the Heisenberg point Jxy = Jz = 1, S.S units; N = 324 is the 18 x 18 cluster and N = 144 the 12 x 12. Its N = 36 value -0.55420(5) is already carried on Heisenberg/triangular_36_P as a secondary row and is left to qmbl-verify.", "secondary_of": null } }, { "energy": -694.2672, "sigma": 0.01296, "energy_variance": null, "dof": 324, "einf": 0, "v_score": null, "method": "VMC", "method_detail": "projected BCS + spin Jastrow", "method_as_published": "p-BCS (projected BCS + spin Jastrow)", "family": "classic VMC", "bound_type": "variational", "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)", "reference": "Yunoki & Sorella, Two spin liquid phases in the spatially anisotropic triangular Heisenberg model, Phys. Rev. B 74, 014408 (2006), arXiv:cond-mat/0602180 (quoted in arXiv:2602.02665 and arXiv:1512.03356)", "peer_reviewed": true, "source": "sweep-allresults-2026-09-28", "provenance": "secondary", "verified": { "checked_on": "2026-09-28", "method": "source text of arXiv:2602.02665 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading of the quotes; primary not read", "reported_as": "−0.53570(1)", "note": "Table 1 of arXiv:2602.02665, 18 × 18 block, row 'p-BCS', printed '−0.53570(1)' [70]; the same string is Table II of arXiv:1512.03356 (L569), column 'E/J' (Ref. 33)', J/J' = 1.0, 'on the 18 × 18 cluster'. Primary not read: arxiv.org/pdf/cond-mat/0602180 returned HTTP 500 on 2026-09-28, so no committed text; a reader saw '-0.5357 ± 0.0001' in the primary's Table 2 on the arXiv HTML render, not committed and not used here. | Table 1 of arXiv:2602.02665 prints energies per site of the triangular Heisenberg model on periodic L x L clusters, S.S units.", "secondary_of": "arXiv:2602.02665" } }, { "energy": -709.11936, "sigma": 0.03888, "energy_variance": null, "dof": 324, "einf": 0, "v_score": null, "method": "VMC", "method_detail": "RVB", "method_as_published": "RVB wave function", "family": "classic VMC", "bound_type": "variational", "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)", "reference": "Heidarian, Sorella & Becca, Spin-1/2 Heisenberg model on the anisotropic triangular lattice: From magnetism to a one-dimensional spin liquid, Phys. Rev. B 80, 012404 (2009), arXiv:0902.2338 (quoted in arXiv:2602.02665 and arXiv:1512.03356)", "peer_reviewed": true, "source": "sweep-allresults-2026-09-28", "provenance": "secondary", "verified": { "checked_on": "2026-09-28", "method": "source text of arXiv:2602.02665 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading of the quotes; primary read, value not located", "reported_as": "−0.54716(3)", "note": "Table 1 of arXiv:2602.02665, 18 × 18 block, row 'RVB', printed '−0.54716(3)' [73]; the same string is Table II of arXiv:1512.03356 (L569), column 'E/J' (Ref. 35)', J/J' = 1.0, 'on the 18 × 18 cluster' (Becca co-authored both). Not located in the committed text of arXiv:0902.2338 (searched '547', 'Table'; its finite-size energies appear as Fig. 3 points; the text prints -0.5470(1) as the L -> infinity value). | Table 1 of arXiv:2602.02665 prints energies per site of the triangular Heisenberg model on periodic L x L clusters, S.S units.", "secondary_of": "arXiv:2602.02665" } }, { "energy": -711.001403809, "sigma": 0.0885369, "energy_variance": 78.387794, "dof": 324, "einf": 0, "v_score": 0.05024033613898959, "method": "2D RNN", "method_detail": "iterative retraining, s = 4.0, r = 0.158", "method_as_published": "2D RNN wavefunction (iterative retraining, s=4.0, r=0.158)", "family": "RNN", "bound_type": "variational", "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)", "reference": "Moss, Wiersema, Hibat-Allah, Carrasquilla & Melko, arXiv:2505.20406v3 (2025-10-13)", "peer_reviewed": false, "source": "repo-data-2026-09-28", "provenance": "primary", "verified": { "checked_on": "2026-09-28", "method": "authors' data release read locally (pickle via numpy), values copied by key, no transcription; the file is committed as sources/2505.20406-repo-final_energies_data.json and reproduces Table V's L = 6 energy and Table II's V-scores", "reported_as": "-0.5486121942967545 +/- 0.0000683154952622206 per site (S.S), Var(E/N) = 0.00004667006892922484, V-score 0.05024033645606475", "note": "github.com/mschuylermoss/HeisenbergRNN, commit 5f5cffa5cb (2025-10-28, HEAD; file last changed in d58a945cd4 2025-05-27, 'final plotting data and plotting notebooks'), Triangular/final_plotting/plotting_data/final_energies_data_plotting.pkl, sha256 bdc59e5bc3780fb4ea7444400dc9abd9984a6a259ab03f9299c1f80edb87d276: ['TriangularMS,periodicBC']['scale=4.0,rate=0.158,T=1.00'], L = 18. The paper's most accurate schedule (Table II, whose L = 18 V-score 5.0e-2 is this run's 0.0502); plotted in Fig. 4(a), not printed; quoted as -0.54861(1) in Table 1 of arXiv:2602.02665, whose error bar is not the primary's. TriangularMS is the 120-degree basis rotation U_tri, which changes the representation, not the Hamiltonian. Variance is Var(H)/N^2 in S.S units; total = var x N^2 x 16.", "secondary_of": null }, "compute": { "parameters": null, "gpu_hours": 1700, "device": "NVIDIA H200", "n_devices": 2, "samples": 10000, "wall_clock": null, "cpu_core_hours": null, "bond_dimension": null, "iterations": null, "reported_as": "Sec. IV Discussion: The longest simulation reported in this work took 1,700 GPU hours and produced energies for six different system sizes up to 30x30, albeit with more modern hardware (see Appendix E). Appendix E, Figs. 10-11 captions: ... using two H200 GPUs. Fig. 4 caption: Each of our variational energies is estimated with 10x10^3 samples.", "source": "Sec. IV Discussion, Appendix E Figs. 10-11 captions, Fig. 4 caption, arXiv:2505.20406 (compute pass 2026-09-16, block reused for the repository row)", "scope": "ansatz", "confidence": "medium", "note": "1,700 GPU-hours is the paper's total for its longest iterative-retraining chain (L = 6 ... 30), read as the s = 4.0, r = 0.158 schedule this row comes from; the paper does not split it per size. No parameter count stated." } } ], "url": "https://qmbl.org/i/Heisenberg/triangular_324_P/", "per_site_divisor": 1296, "per_site_label": "E/N (S.S)", "record": { "energy": -715.007088, "sigma": 0.001296, "method": "ViT", "method_detail": "spatial attention, b = 3, translations + C6v projection", "reference": "Viteritti, Rende, Sachdev & Carleo, Approaching the Thermodynamic Limit with Neural-Network Quantum States, arXiv:2602.02665", "bound_type": "variational", "energy_per_site": -0.5517029999999999 }, "no_record_reason": null }