{ "model": "tJ", "lattice": "square", "n_sites": 26, "boundary": "P", "params": { "t": 1, "J": 0.5, "N_up": 12, "N_dn": 12, "nn_term": true }, "cluster": "square-shaped periodic cluster of 26 sites, edge vectors (5, 1) and (-1, 5), the only square-shaped cluster with N = 26 (RULES.md 2.1). Inferred, not printed: arXiv:cond-mat/0606659 describes its clusters as 'a square lattice with L sites and periodic boundary conditions rotated by 45 degrees such that L = 2l^2, l being an odd integer', which does not include L = 26, and states no other shape; arXiv:cond-mat/0009149 lists 13 shells for its 26-site cluster, as the (5, 1) cluster has (SWEEP.md, 2026-10-22). Confirmed by QMBL's diagonalization of the (5, 1) cluster (checks/tj-exact-recompute-2026-10/, 2026-10-23), which reproduces every printed 26-site exact energy to its printed digits: with 2 holes in all 14 momentum and C4 sectors, with 4 holes in the zero-momentum, C4-even sector, the lowest of the 1 to 6 of 14 sectors computed", "instance_id": "tJ/square_26_P_h2_J0.5", "rows": [ { "energy": -17.4317, "sigma": 0.00026, "energy_variance": null, "dof": 24, "einf": null, "v_score": null, "method": "VMC", "ansatz": "RVB", "method_detail": "RVB (projected BCS + Jastrow, no antiferromagnetic field)", "method_as_published": "E_VMC^RVB", "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": "Lugas, Spanu, Becca & Sorella, Finite compressibility in the low-doping region of the two-dimensional t-J model, Phys. Rev. B 74, 165122 (2006), doi:10.1103/PhysRevB.74.165122, arXiv:cond-mat/0606659", "peer_reviewed": true, "source": "sweep-allresults-2026-10-22", "provenance": "primary", "verified": { "checked_on": "2026-10-22", "method": "source text of arXiv:cond-mat/0606659 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; first reading and an adversarial second reading (2026-10-22)", "reported_as": "-0.67045(1)", "note": "Table I, J/t = 0.5, column E_VMC^RVB (line 385); version read: arXiv text (sources/cond-mat-0606659.txt); the journal version was not reached from this session (the publisher returned 403), to be read on the verification pass (RULES.md 8) | Eq. (1): H = -t sum (c+c + h.c.) + J sum (S_i.S_j - n_i n_j / 4), t = 1; Tables I and II print E/L, L = 26, for 2 and 4 holes. The Lanczos column agrees with the exact column of Sorella, Phys. Rev. B 64, 024512, Table I at J/t = 0.3, 0.5, 1.0 (4 holes), which fixes one convention for both papers (that paper is Sorella, Phys. Rev. B 64, 024512).", "secondary_of": null } }, { "energy": -17.70366, "sigma": 0.00026, "energy_variance": null, "dof": 24, "einf": null, "v_score": null, "method": "FN-GFMC", "ansatz": "RVB", "method_detail": "lattice fixed node; projected BCS + Jastrow", "method_as_published": "E_FN^RVB", "family": "AFQMC / GFMC", "bound_type": "projected", "bound_type_reason": "projection (GFMC/fixed-node) on a trial state; not a strict variational bound (assigned during source reading)", "reference": "Lugas, Spanu, Becca & Sorella, Finite compressibility in the low-doping region of the two-dimensional t-J model, Phys. Rev. B 74, 165122 (2006), doi:10.1103/PhysRevB.74.165122, arXiv:cond-mat/0606659", "peer_reviewed": true, "source": "sweep-allresults-2026-10-22", "provenance": "primary", "verified": { "checked_on": "2026-10-22", "method": "source text of arXiv:cond-mat/0606659 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; first reading and an adversarial second reading (2026-10-22)", "reported_as": "-0.68091(1)", "note": "Table I, J/t = 0.5, column E_FN^RVB (line 385); version read: arXiv text (sources/cond-mat-0606659.txt); the journal version was not reached from this session (the publisher returned 403), to be read on the verification pass (RULES.md 8) | Eq. (1): H = -t sum (c+c + h.c.) + J sum (S_i.S_j - n_i n_j / 4), t = 1; Tables I and II print E/L, L = 26, for 2 and 4 holes. The Lanczos column agrees with the exact column of Sorella, Phys. Rev. B 64, 024512, Table I at J/t = 0.3, 0.5, 1.0 (4 holes), which fixes one convention for both papers (that paper is Sorella, Phys. Rev. B 64, 024512).", "secondary_of": null } }, { "energy": -17.56768, "sigma": 0.00026, "energy_variance": null, "dof": 24, "einf": null, "v_score": null, "method": "VMC", "ansatz": "Pfaffian", "method_detail": "projected BCS + antiferromagnetic field + Jastrow", "method_as_published": "E_VMC^Pfaff", "family": "backflow / Pfaffian", "bound_type": "variational", "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)", "reference": "Lugas, Spanu, Becca & Sorella, Finite compressibility in the low-doping region of the two-dimensional t-J model, Phys. Rev. B 74, 165122 (2006), doi:10.1103/PhysRevB.74.165122, arXiv:cond-mat/0606659", "peer_reviewed": true, "source": "sweep-allresults-2026-10-22", "provenance": "primary", "verified": { "checked_on": "2026-10-22", "method": "source text of arXiv:cond-mat/0606659 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; first reading and an adversarial second reading (2026-10-22)", "reported_as": "-0.67568(1)", "note": "Table I, J/t = 0.5, column E_VMC^Pfaff (line 385); version read: arXiv text (sources/cond-mat-0606659.txt); the journal version was not reached from this session (the publisher returned 403), to be read on the verification pass (RULES.md 8) | Eq. (1): H = -t sum (c+c + h.c.) + J sum (S_i.S_j - n_i n_j / 4), t = 1; Tables I and II print E/L, L = 26, for 2 and 4 holes. The Lanczos column agrees with the exact column of Sorella, Phys. Rev. B 64, 024512, Table I at J/t = 0.3, 0.5, 1.0 (4 holes), which fixes one convention for both papers (that paper is Sorella, Phys. Rev. B 64, 024512).", "secondary_of": null } }, { "energy": -17.76502, "sigma": 0.00026, "energy_variance": null, "dof": 24, "einf": null, "v_score": null, "method": "FN-GFMC", "ansatz": "Pfaffian", "method_detail": "lattice fixed node; projected BCS + antiferromagnetic field + Jastrow", "method_as_published": "E_FN^Pfaff", "family": "AFQMC / GFMC", "bound_type": "projected", "bound_type_reason": "projection (GFMC/fixed-node) on a trial state; not a strict variational bound (assigned during source reading)", "reference": "Lugas, Spanu, Becca & Sorella, Finite compressibility in the low-doping region of the two-dimensional t-J model, Phys. Rev. B 74, 165122 (2006), doi:10.1103/PhysRevB.74.165122, arXiv:cond-mat/0606659", "peer_reviewed": true, "source": "sweep-allresults-2026-10-22", "provenance": "primary", "verified": { "checked_on": "2026-10-22", "method": "source text of arXiv:cond-mat/0606659 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; first reading and an adversarial second reading (2026-10-22)", "reported_as": "-0.68327(1)", "note": "Table I, J/t = 0.5, column E_FN^Pfaff (line 385); version read: arXiv text (sources/cond-mat-0606659.txt); the journal version was not reached from this session (the publisher returned 403), to be read on the verification pass (RULES.md 8) | Eq. (1): H = -t sum (c+c + h.c.) + J sum (S_i.S_j - n_i n_j / 4), t = 1; Tables I and II print E/L, L = 26, for 2 and 4 holes. The Lanczos column agrees with the exact column of Sorella, Phys. Rev. B 64, 024512, Table I at J/t = 0.3, 0.5, 1.0 (4 holes), which fixes one convention for both papers (that paper is Sorella, Phys. Rev. B 64, 024512).", "secondary_of": null } }, { "energy": -17.79284, "sigma": 0.00078, "energy_variance": null, "dof": 24, "einf": null, "v_score": null, "method": "FN-GFMC", "ansatz": "Pfaffian", "method_detail": "Eq. (15) finite-difference estimate, from fixed-node runs at γ = 0 and γ̃ = 1, of the expectation value of H over the γ = 0 fixed-node ground state (an upper bound to it)", "method_as_published": "Ẽ_Ψ0^γ=0 (Eq. 15, Pfaff)", "family": "AFQMC / GFMC", "bound_type": "projected", "bound_type_reason": "finite-difference estimate (Eq. 15) from fixed-node runs at γ = 0 and γ̃ = 1 of the expectation value of H over the γ = 0 fixed-node ground state, an upper bound within fixed node: 'Ẽ^{γ=0}_{Ψ0} certainly improves the standard FN upper bound of the energy and still Ẽ^{γ=0}_{Ψ0} ≥ E^{γ=0}_{Ψ0}' and 'we have computed Ẽ^{γ=0}_{Ψ0} using γ̃ = 1' (text after Fig. 9); not an extrapolation (ruled 2026-10-24)", "reference": "Lugas, Spanu, Becca & Sorella, Finite compressibility in the low-doping region of the two-dimensional t-J model, Phys. Rev. B 74, 165122 (2006), doi:10.1103/PhysRevB.74.165122, arXiv:cond-mat/0606659", "peer_reviewed": true, "source": "sweep-allresults-2026-10-22", "provenance": "primary", "verified": { "checked_on": "2026-10-22", "method": "source text of arXiv:cond-mat/0606659 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; first reading and an adversarial second reading (2026-10-22)", "reported_as": "-0.68434(3)", "note": "Table I, J/t = 0.5, column Ẽ_Ψ0^γ=0 (Eq. 15, Pfaff) (line 385); version read: arXiv text (sources/cond-mat-0606659.txt); the journal version was not reached from this session (the publisher returned 403), to be read on the verification pass (RULES.md 8) | Eq. (1): H = -t sum (c+c + h.c.) + J sum (S_i.S_j - n_i n_j / 4), t = 1; Tables I and II print E/L, L = 26, for 2 and 4 holes. The Lanczos column agrees with the exact column of Sorella, Phys. Rev. B 64, 024512, Table I at J/t = 0.3, 0.5, 1.0 (4 holes), which fixes one convention for both papers (that paper is Sorella, Phys. Rev. B 64, 024512).", "secondary_of": null } }, { "energy": -17.925669794311, "sigma": null, "energy_variance": null, "dof": 24, "einf": null, "v_score": null, "method": "ED", "ansatz": null, "method_detail": "Lanczos", "method_as_published": "E_ex (Lanczos)", "family": "other", "bound_type": "exact", "bound_type_reason": "numerically exact reference as stated by the source (assigned during source reading)", "reference": "Lugas, Spanu, Becca & Sorella, Finite compressibility in the low-doping region of the two-dimensional t-J model, Phys. Rev. B 74, 165122 (2006), doi:10.1103/PhysRevB.74.165122, arXiv:cond-mat/0606659", "peer_reviewed": true, "source": "sweep-allresults-2026-10-22", "provenance": "primary", "verified": { "checked_on": "2026-10-22", "method": "source text of arXiv:cond-mat/0606659 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; first reading and an adversarial second reading (2026-10-22)", "reported_as": "-0.68945", "note": "Table I, J/t = 0.5, column E_ex (Lanczos) (line 385); the paper names no source for its Lanczos column, and its acknowledgements name only funding; the 4-hole values at J/t = 0.3, 0.5 and 1 equal digit for digit the Exact column of Sorella, Phys. Rev. B 64, 024512, Table I, whose acknowledgements credit D. Poilblanc's unpublished 26-site data (arXiv:cond-mat/0009149, lines 829-831); the row stays primary to this paper (ruled 2026-10-24); version read: arXiv text (sources/cond-mat-0606659.txt); the journal version was not reached from this session (the publisher returned 403), to be read on the verification pass (RULES.md 8) | Eq. (1): H = -t sum (c+c + h.c.) + J sum (S_i.S_j - n_i n_j / 4), t = 1; Tables I and II print E/L, L = 26, for 2 and 4 holes. The Lanczos column agrees with the exact column of Sorella, Phys. Rev. B 64, 024512, Table I at J/t = 0.3, 0.5, 1.0 (4 holes), which fixes one convention for both papers (that paper is Sorella, Phys. Rev. B 64, 024512).", "secondary_of": null, "second_read": { "checked_on": "2026-10-23", "scope": "value", "method": "independent exact diagonalization written for this check (checks/tj-exact-recompute-2026-10/tjk.c: C/OpenMP, basis adapted to lattice translations and point-group operations with 1D characters, Lanczos to |dE| < 1e-13 per step; clusters and sectors from gen.py), run 2026-10-23; validated against a brute-force full-space ED (bf.py) on 12 small clusters (agreement to 1e-10)", "location": "Table I, J/t = 0.5, column E_ex (Lanczos) (line 385)", "reported_as": "-0.68945", "conversion": "E = -17.9256697943 (t = 1); printed form E/L: -0.68945 vs printed -0.68945", "note": "26-site square-shaped periodic cluster, edge vectors (5, 1) and (-1, 5) (52 bonds); 14 sectors in all: Gamma and M with the four C4 characters, six C4 stars of 4 momenta with translations only; lowest of 14 sector(s) covering the whole electron-number space (ground state), best sector sq26P_h2_J0.5_k0-0_l2_star1, dim 8450824; 11395 s wall on 16 threads over all sectors; not among the six reproduced by checks/tj-ed-2026-10/; the computed energy on the (5, 1) cluster reproduces the printed value, which confirms the inferred cluster shape; the lowest sector is Gamma with C4 eigenvalue -1 (d-wave-like, l = 2)", "source_entry": "tjE-exact-recompute#10 (qmbl-verify 2026-10-23)" } }, "corrections": [ { "field": "energy", "to": -17.925669794311, "reported_as": "-0.68945 (E/L)", "location": "Table I, J/t = 0.5, column E_ex (Lanczos) (line 385)", "version_read": "recomputation: checks/tj-exact-recompute-2026-10/ (tjk.c, 2026-10-23); the printed value as the generator read it (scripts/tj_2026_10_22.mjs)", "conversion": "-0.68945 x 26 = -17.9257 (stored); recomputed -17.925669794311 = -0.6894488382 in the printed form, which rounds to the printed digits", "checked_on": "2026-10-23", "reason": "Rule for exact rows QMBL recomputed (2026-09-30; applied to the t-J rows, ruled 2026-10-24): the printed value is QMBL's value rounded to the printed digits, so the row keeps its citation and carries QMBL's digits; the printed value stays here. Recomputed by tjk.c (symmetry-adapted Lanczos, |dE| < 1e-13 per step; 14 of 14 symmetry sectors, lowest sector sq26P_h2_J0.5_k0-0_l2_star1, dimension 8450824), whose code agrees with the brute-force bf.py to 1e-10 on 12 small clusters; codes and outputs in checks/tj-exact-recompute-2026-10/.", "source_entry": "tjE-exact-recompute#10 (qmbl-verify 2026-10-23); rule for recomputed exact rows, ruled 2026-10-24", "from": -17.9257 }, { "field": "sampled", "to": false, "reported_as": "-0.68945", "location": "Table I, J/t = 0.5, column E_ex (Lanczos) (line 385)", "version_read": "the row's published method string; the exact diagonalization is recomputed in checks/tj-exact-recompute-2026-10/", "conversion": "none", "checked_on": "2026-10-24", "reason": "Deterministic: an exact (Lanczos) diagonalization carries no statistical error, but the label \"E_ex (Lanczos)\" matches none of isSampled's deterministic markers, so the row read as a sampled energy without an error bar (the README marked it); ruled 2026-10-24.", "source_entry": "tjE-exact-recompute#10 (qmbl-verify 2026-10-23); ruled 2026-10-24 (exact rows whose label reads as sampled)", "from": true } ], "sampled": false } ], "url": "https://qmbl.org/i/tJ/square_26_P_h2_J0.5/", "per_site_divisor": 26, "per_site_label": "E/site", "best": { "exact": { "energy": -17.925669794311, "sigma": null, "method": "ED", "ansatz": null, "method_detail": "Lanczos", "reference": "Lugas, Spanu, Becca & Sorella, Finite compressibility in the low-doping region of the two-dimensional t-J model, Phys. Rev. B 74, 165122 (2006), doi:10.1103/PhysRevB.74.165122, arXiv:cond-mat/0606659", "bound_type": "exact", "energy_per_site": -0.6894488382427307 }, "variational": { "energy": -17.56768, "sigma": 0.00026, "method": "VMC", "ansatz": "Pfaffian", "method_detail": "projected BCS + antiferromagnetic field + Jastrow", "reference": "Lugas, Spanu, Becca & Sorella, Finite compressibility in the low-doping region of the two-dimensional t-J model, Phys. Rev. B 74, 165122 (2006), doi:10.1103/PhysRevB.74.165122, arXiv:cond-mat/0606659", "bound_type": "variational", "energy_per_site": -0.67568 }, "projected": { "energy": -17.79284, "sigma": 0.00078, "method": "FN-GFMC", "ansatz": "Pfaffian", "method_detail": "Eq. (15) finite-difference estimate, from fixed-node runs at γ = 0 and γ̃ = 1, of the expectation value of H over the γ = 0 fixed-node ground state (an upper bound to it)", "reference": "Lugas, Spanu, Becca & Sorella, Finite compressibility in the low-doping region of the two-dimensional t-J model, Phys. Rev. B 74, 165122 (2006), doi:10.1103/PhysRevB.74.165122, arXiv:cond-mat/0606659", "bound_type": "projected", "energy_per_site": -0.6843400000000001 }, "extrapolated": null, "approximate": null, "below_exact": [] }, "no_best_reason": null, "record": { "energy": -17.925669794311, "sigma": null, "method": "ED", "ansatz": null, "method_detail": "Lanczos", "reference": "Lugas, Spanu, Becca & Sorella, Finite compressibility in the low-doping region of the two-dimensional t-J model, Phys. Rev. B 74, 165122 (2006), doi:10.1103/PhysRevB.74.165122, arXiv:cond-mat/0606659", "bound_type": "exact", "energy_per_site": -0.6894488382427307 }, "no_record_reason": null }