Flagged rows
A flagged row stays listed in rank order but cannot be the best of its class.
Rules §10 says how a flag is lifted or upheld.
below exact energy, PEPS, -0.603535(1)
Finding. Correctly transcribed from arXiv:1611.09467v3 (posted 2017-05-04; the accepted version of Phys. Rev. B 95, 195154, received 2016-12-11, published 2017-05-24) Table II, row 'D=10', column 'L=6': -0.603535(1) per site, read in three extractions (pypdf layout, pypdf plain, pdftotext -layout) and on a 300 dpi render of page 5; the cell first appears in v3 (v1 and v2 have no D=10 row). It sits 1.32e-5 per site (1.90e-3 total) below the instance's exact energy -0.6035218345 (VarBench ED, Dian Wu 2022-08-23, lattice_symmetries, S^z = 0, all point-group sectors 0, unchanged since), 1.28e-5 below Sandvik's SSE -0.60352217(17) and 1.32e-5 below the paper's own L=6 reference, Huang et al.'s 4096-state DMRG printed as -0.6035218 with the citation superscript 34 (primary: -21.7267859/36 = -0.60352183, arXiv:1611.09574 Fig. 4 caption). That is 13.2 printed sigma (12.7 combined with the SSE bar), a violation of RULES.md 9.4 by any reading, and cannot be sampling noise at the printed bar. The '(1)' is the Monte Carlo sampling error only ('The MC sampling error is order of 10^-6', Fig. 5 caption; 'The error bars are too small to show', Fig. 3 caption); the cut-off systematic is treated separately in Appendix A, whose only number, 'absolute error 8x10^-6' between Dc=16 and Dc=20, is for D=8 on 10x10 and is 1.6x smaller than this dip; D=10/Dc=20 and the 6x6 cluster are not tested there, so the paper gives no accuracy claim that covers this cell. It does not comment on the L=6 cells lying below its own 'Exact' row; Sec. IV calls the agreement 'remarkable', Table III claims relative error ~1e-4 for MC+GO. No convention shift applies: H = J1 sum S_i.S_j, S_i.S_i = 3/4 (Eq. 7), per site; Table I's 4x4 'Exact' -0.57432544 is reproduced by a local S.S exact diagonalization to 1.6e-9 (checks/peps-4x4-open-ed/ed_4x4_open.py); a per-bond mislabel (60 bonds) would change the value by a factor 60/36 and an S.S-1/4 shift by 0.4167. Against the paper's own trend the cell is off by at least 2e-5 per site: GO at D=10 has relative error +4.2e-6 on 4x4, +2.5e-6 on 4x6 (J2=0) and +8.7e-5 on 10x10 (Table I and the text of Sec. IV, all above their exact values), whereas this cell is -2.2e-5; the D=8 to D=10 change at L=6 (-1.2e-5) has the size of the genuine improvements at L=8..16 (-2.0e-5 to -9.4e-5), but at L=6 the D=8 value already sits at the exact energy and there is no room below it. The authors' public code (github.com/Shaojun-Dong/libTNSP, the TNSPackage library) contains no PEPS-GO run scripts, tensors or energies for this paper; the PRB full text is paywalled and unread. The three candidate causes (truncation entering the local-energy ratios W(S')/W(S) differently from the sampling weights so the estimate is not a Rayleigh quotient; a minimum over the noisy GO trace, cf. the 5e-4 band of Fig. 2(b); a misprint, e.g. of -0.603515) cannot be told apart from the sources. The authors' final-measurement output for the 6x6 D=10 run, or an exact re-contraction of their tensors, would settle it.
below exact energy, PEPS, -0.603523(1)
Finding. Correctly transcribed from arXiv:1611.09467v3 (posted 2017-05-04; the accepted version of Phys. Rev. B 95, 195154, received 2016-12-11, published 2017-05-24) Table II, row 'D=8', column 'L=6': -0.603523(1) per site, read in three extractions (pypdf layout, pypdf plain, pdftotext -layout) and on a 300 dpi render of page 5. It sits 1.17e-6 per site (1.68e-4 total) below the instance's exact energy -0.6035218345 (VarBench ED, uploaded by Dian Wu 2022-08-23 with the instance, unchanged since; ed_ls.py, lattice_symmetries, S^z = 0, all point-group sectors 0), 8.3e-7 below Sandvik's SSE -0.60352217(17) (this instance, J. Stat. Mech. 2026 Table 2) and 1.17e-6 below the paper's own L=6 reference, Huang et al.'s 4096-state DMRG, printed as -0.6035218 with the citation superscript 34 (Table II 'Exact'; primary: E = -21.7267859 total, arXiv:1611.09574 Fig. 4 caption, = -0.60352183 per site). That is 1.2 printed sigma (0.8 sigma combined with the SSE bar): below the 3 sigma of RULES.md 9.4, so not a validator violation, but as stored the row lies below every exact-level reference and would otherwise be listed as the instance's best variational bound (6.1). The printed '(1)' is the Monte Carlo sampling error only: the paper's only statements about errors are 'The MC sampling error is order of 10^-6' (Fig. 5 caption) and 'The error bars are too small to show' (Fig. 3 caption); the bond-dimension-cut-off systematic is treated separately in Appendix A, which tests D=8 on 10x10 (Dc=16 against Dc=20: 'absolute error 8x10^-6') and 4x6 at J2=0.5, never 6x6, and is not folded into any bar. No convention shift can produce a 1e-6 dip: H = J1 sum S_i.S_j with S_i.S_i = 3/4 (Eq. 7), energies per site; the paper's 4x4 'Exact' -0.57432544 (Table I) is reproduced by a local S.S exact diagonalization to 1.6e-9 (checks/peps-4x4-open-ed/ed_4x4_open.py: -0.5743254416), a per-bond mislabel (60 bonds) would move the value by a factor 60/36 and an S.S-1/4 shift by 0.4167 per site. Against the paper's own accuracy trend the cell is anomalous as well: GO at D=8 has relative error +1.1e-5 on 4x4, +1.3e-5 on 4x6 (J2=0) and +2.4e-4 on 10x10 (Table I, every cell above its exact value), so a 6x6 D=8 energy within 2e-6 of the exact value, let alone below it, is at least 1e-5 per site outside that trend. Version history: v1 (2016-11-29) and v2 (2016-12-13) print this cell as -0.6034(3) and -0.6034(8), above the exact value; v3 replaced Table II with re-optimised runs and 1e-6 bars. The authors released their tensor library (github.com/Shaojun-Dong/libTNSP, TNSPackage, Comput. Phys. Commun. 228, 163) but no PEPS-GO run scripts, tensors or energies for this paper; the PRB full text is paywalled and unread. Whether the dip is the truncated boundary-MPS contraction entering the local-energy ratios W(S')/W(S) differently from the weights (the estimator is a Rayleigh quotient of a well-defined state only if both use the same truncated contraction, which the paper does not state), a minimum taken over the noisy GO trace (Fig. 2(b) shows a 5e-4 band on 10x10), or a misprint, cannot be decided from the sources; the authors' final-measurement output for the 6x6 runs, or an exact re-contraction of their D=8 tensors (feasible at 6x6), would settle it.
How these numbers were read
Where each number below was read, and how it was checked.
-0.603535(1), PEPS
Checked 2026-09-15 · source text of arXiv:1611.09467 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; adversarial re-read: refuted
Reported as. -0.603535(1)
Table II, row 'D=10', column 'L=6' (36 sites); the cell is not in Fig. 4(a), whose caption and the fit in Sec. IV cover L = 8, 10, 12, 14, 16 only; first printed in v3 (v1/v2 have no D=10 row). | Per-site S.S units, same check as the D=8 row above.
-0.603523(1), PEPS
Checked 2026-09-15 · source text of arXiv:1611.09467 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; adversarial re-read: refuted
Reported as. -0.603523(1)
Table II (caption: 'Comparison of the ground state energies of the Heisenberg model calculated by PEPS ... for L = 6, 8, 10, 12, 14, 16 ... D=8, and Dc=16'), row 'D=8', column 'L=6' (36 sites). | Per-site S.S units; Table II's own Exact row for L=6 is printed -0.6035218 followed by the citation superscript 34 (Ref. [34], Huang et al., 4096-state DMRG; the text extraction glues it into -0.603521834), which matches the instance's exact_per_site (-0.6035218345) to 7 sig figs, confirming direct convention match, no rescale; the 4x4 'Exact' -0.57432544 of Table I is reproduced by an S.S exact diagonalization to 1.6e-9.
-0.6035222(2), SSE QMC
Checked 2026-09-15 · 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
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.
-0.6035218, ED
Checked 2026-09-29 · this pass on an AMD EPYC 9654 server: E2 ed_sym.py --pbc 0 (open-boundary path validated by E2 on the open 4x4), C4v A1, z=+1
Reported as. -86.90714416997474 (true residual 2.06e-10)
Pauli H = sum_NN sigma.sigma on the open 6x6 square (60 bonds). Marshall's sign rule with N/2 = 18 even puts the unique singlet ground state in C4v A1 with z = +1. |E - stored| = 1.56e-12 (relative 1.80e-14). Second determination: the stored value is VarBench's own ED (lattice-symmetries, a different code and basis), which agrees to this precision. Also read by VB3-varbench-code-heisenberg#46 (scope config: vendor/varbench/Heisenberg/square_36_O.md (snapshot 390a21e), row 'Exact diagonalization'; linked scripts/Heisenberg/square_36_O/ed_lattice_symmetries.sh).
-0.6035218, DMRG
Checked 2026-09-15 · source text of arXiv:1611.09574 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; two readings (reader and skeptic, qmbl-verify 2026-09-19-flagged)
Reported as. E = -21.7267859 (total, S.S units, 6x6 open lattice, DMRG keeping 4096 states)
arXiv:1611.09574 Fig. 4 caption: "The dashed lines are the corresponding results obtained using DMRG by keeping 4096 states (E = -21.7267859 and S = 2.802147)". The row was first read on 2026-09-15 as the 'Exact' L=6 cell of arXiv:1611.09467 Table II (a quote of this value, printed -0.6035218 with the citation superscript 34, which the text extraction glued into -0.603521834); the primary was read on 2026-09-19 (qmbl-verify 2026-09-19-flagged, FP2 reader and skeptic) and the row now carries the primary value and reference. | -21.7267859 / 36 = -0.6035218306 per site, S.S units; the instance exact_per_site is -0.6035218345, so the 4096-state DMRG sits 3.9e-9 per site above the exact energy
-0.6035218, DMRG
Checked 2026-09-29 · VarBench snapshot 390a21e (vendor/varbench) and full upstream history (varbench-history, git log --follow on the file, every version); varbench/methods ed31bb0 run script and program read as text; no output log or result file exists in the methods repo for any Heisenberg row
Reported as. | -86.90713263791655 | | 0.00045932030843687244 | 36 | 0 | DMRG (bond dimension = 2048) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_36_O/dmrg.sh) |
dmrg.sh: heisenberg_2d.jl --L 6 --zero_mag --max_B 2048 (open), Pauli, J = 1, Sz = 0, 50 sweeps at D = 2048. 1.15e-5 above the exact energy; var/(E - E0) = 39.8. M13 (same number as Huang et al.'s 'DMRG keeping 4096 states' -86.9071436064): not the same number, -0.6035217544 vs -0.6035218306 per site (1.1e-5 apart in total); they coincide only at 6 printed decimals; different runs (D = 2048 here, D = 4096 there); not a duplicate. Entered 9401341 (2022-08-23), never changed; linked 2024-07-19.
-0.603519(2), VMC (minGRU)
Checked 2026-10-27 · arXiv:2605.13807: arXiv:2605.13807v1 (only version; no journal reference on 2026-10-27): pypdf layout text sources/2605.13807.txt read locally, checked against Table 6 of the arXiv HTML sources/2605.13807.html; no LLM transcription; single reading
Reported as. -0.603519(2)
TABLE VI, row N = 6^2 (line 682 of sources/2605.13807.txt) | energy per site in S.S units, J = 1, open boundaries; the same table row prints the QMC value equal to the instance's stored SSE row
-0.603517(6), VMC (2D RNN)
Checked 2026-09-15 · source text of arXiv:2502.17144 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -0.603517(6)
Table V, 'open 6x6' block, row 'This work (best RNN)' | slightly weaker than instance record (DMRG bond dim 2048, -0.603521754429976, a different paper) and than exact_per_site -0.6035218345137243, consistent with a valid variational bound
-0.603516(1), VMC (RNN)
Checked 2026-09-15 · source text of arXiv:2502.17144 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -0.603516(1)
Table V, 'open 6x6' block, row 'RNN [38]' | S.S convention, no rescale
-0.60351496(1), VMC (2D RNN)
Checked 2026-09-15 · source text of arXiv:2502.17144 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -0.60351496(1)
Table V, 'open 6x6' block, row 'This work (zero-variance)' | above exact_per_site -0.6035218345137243, consistent with paper's claim of no real variational-principle violation
-0.603497(2), VMC (Adaptive RNN)
Checked 2026-09-15 · source text of arXiv:2507.18700 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -21.72589(8)
TABLE II | total/36 = per-site, just above exact_per_site; this is the paper's best of the three own values | variance reported as 0.0001775, not stored: convention not verified
-0.603486(3), VMC (Adaptive RNN)
Checked 2026-09-15 · source text of arXiv:2507.18700 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -21.72551(9)
TABLE II | total/36 = per-site | variance reported as 0.0002107, not stored: convention not verified
-0.603483(3), VMC (RNN)
Checked 2026-09-15 · source text of arXiv:2507.18700 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -21.7254(1)
TABLE II. Comparison of Static RNN and Adaptive RNN models on the 2D square-lattice Heisenberg model, N=6x6, OBC (Eq. 7). | total/36 = per-site, just above exact_per_site | variance reported as 0.0002811, not stored: convention not verified
-0.603417, VMC (LSTM)
Checked 2026-09-15 · source text of arXiv:2502.17144 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -0.603417
Table V, 'open 6x6' block, row 'LSTM [35]' | S.S convention, no rescale
-0.59484(2), VMC (Jastrow)
Checked 2026-09-29 · VarBench snapshot 390a21e (vendor/varbench) and full upstream history (varbench-history, git log --follow on the file, every version); varbench/methods ed31bb0 run script and program read as text; no output log or result file exists in the methods repo for any Heisenberg row
Reported as. | -85.6567 | 0.0036 | 13.4570 | 36 | 0 | Jastrow baseline | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_36_O/vmc_jastrow.sh) |
scripts/Heisenberg/square_36_O/vmc_jastrow.sh: 'python3 vmc.py --ham heis --boundary open --sign mars --ham_dim 2 --L 6 --zero_mag --net jas --seed 123 --show_progress' -> 6x6 square lattice, open (nk.graph.Grid); NetKet Heisenberg J = 1 in Pauli form with Marshall sign rule (a sublattice rotation on a bipartite lattice, spectrum unchanged); Sz = 0 (total_sz=0, MetropolisExchange); real float32 Jastrow (nk.models.Jastrow); 10^4 SR steps; final estimate over 2^20 samples (vmc.py lines 268-269). sigma/sqrt(Var/2^20) = 1.005: the stored sigma is the error of that final estimate. 1.2504 above the exact -86.9071441699763. History: entered af1d48f (2022-09-01, Dian Wu) as -85.6567 | 0.0036 | 13.4570 | Jastrow baseline; linked fdb4bfb (2024-07-30). Program files postdate the value (scripts added to varbench/methods 2024-07-30); they document the settings after the run.
-0.59365(2), VMC (RBM)
Checked 2026-09-29 · VarBench snapshot 390a21e (vendor/varbench) and full upstream history (varbench-history, git log --follow on the file, every version); varbench/methods ed31bb0 run script and program read as text; no output log or result file exists in the methods repo for any Heisenberg row
Reported as. | -85.4857 | 0.0035 | 12.2737 | 36 | 0 | RBM (alpha = 1) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_36_O/vmc_rbm.sh) |
scripts/Heisenberg/square_36_O/vmc_rbm.sh: 'python3 vmc.py --ham heis --boundary open --sign mars --ham_dim 2 --L 6 --zero_mag --net rbm --seed 123 --show_progress' -> 6x6 square lattice, open (nk.graph.Grid); NetKet Heisenberg J = 1 in Pauli form with Marshall sign rule (a sublattice rotation on a bipartite lattice, spectrum unchanged); Sz = 0 (total_sz=0, MetropolisExchange); real float32 RBM, alpha = 1 (nk.models.RBM, log_cosh); 10^4 SR steps; final estimate over 2^20 samples (vmc.py lines 268-269). sigma/sqrt(Var/2^20) = 1.023: the stored sigma is the error of that final estimate. 1.4214 above the exact -86.9071441699763. History: entered 9401341 (2022-08-23, Dian Wu) as -85.4857 | 0.0035 | 12.2737 | RBM (alpha = 1); linked 5047bfd (2024-07-30). Program files postdate the value (scripts added to varbench/methods 2024-07-30); they document the settings after the run.