How these numbers were read
Where each number below was read, and how it was checked.
-0.628657(5), 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. 'Extrapolated Energy = - 251.4628 +/- 0.0019' inside the method cell of null
QMBL split row: the number is printed inside VarBench's method string, 'Extrapolated Energy = - 251.4628 +/- 0.0019' (c342a72, 2022-03-27, J. M. Silvester; earlier '- 251.46(3) +/- 0.002' (dd105fb) and '- 251.463 +/- 0.002' (2892ab4); a first 'extrapolated energy: -251.32413735' of 2021-12-05 was removed the same day, 7567499). M6 (deterministic but sigma 0.0019): the +/- is the uploader's uncertainty of a truncation-error extrapolation, not a statistical error; square_100_O.jl contains no extrapolation and the fit is not documented. bound_type extrapolated is right. Agrees with the exact (stochastic) QMC -251.46248(80) (0.2 combined sigma).
-0.628656(2), QMC
Checked 2026-09-18 · arXiv:1405.3259v2 PDF, pypdf layout mode (Lubasch, Cirac & Banuls, PRB 90, 064425); v1 checked for the digit
Reported as. -0.628656(2) (per site, S.S units)
open 10x10, S = 1/2 Heisenberg, loop algorithm at T = 1e-4; Liu et al. PRB 95, 195154 and Sharir et al. quote the same -0.628656(2)
-0.6286561(2), SSE QMC
Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription
Reported as. -0.6286561 (+/- 2e-7) per site in S.S units
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).
-0.628656(9), VMC (2D RNN)
Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription
Reported as. -0.628656 (+/- 0.000009) per site in S.S units
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]". Bibliography item [40] of arXiv:2605.13807 is Moss et al. 2025 (arXiv:2502.17144), whose Table 5 prints this value as "This work (best RNN)", OBC 10x10; until 2026-09-16 this row credited Hibat-Allah, Melko & Carrasquilla 2022, a misread of the citation number (the arXiv HTML numbers its bibliography differently from the printed text). It sits 1e-7 below the QMC reference, well inside its own 9e-6 error bar.
-0.62864114(5), 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. -0.62864114(5)
Table V, 'open 10x10' block, row 'This work (zero-variance)' | S.S convention, no rescale
-0.628638(1), VMC (2D RNN)
Checked 2026-09-29 · arXiv:2207.14314v2 (12 Jan 2024) and v1 (28 Jul 2022) PDFs, NeurIPS ML4PS 2021 workshop PDF (ml4physicalsciences.github.io/2021/files/NeurIPS_ML4PS_2021_92.pdf); pypdf plain + layout; Fig. 2 (/Im2 p. 4) and Fig. 3 (/Im3 p. 5) vector streams parsed with work/cairo_parse.py, decoded by work/rnn/decode_rnn.mjs, tick labels read from the pdftoppm render
Reported as. band centre -0.628638445 per site (E/N, S.S units), band half-height 1.099e-6 (decoded from the vector data; no printed digits in the paper)
Instance, convention (H = 1/4 sum sigma.sigma = S.S), OBC, U(1) + C4v sector and variational class agree; the number is the paper's own (first-author VarBench upload 1ab3fe8, 2021-12-13, identical to the plotted band). Moss, Wiersema, Hibat-Allah, Carrasquilla & Melko (arXiv:2502.17144v3 = PRB 112, 134450, Table V, 'open 10 x 10') quote it as 'RNN [38] -0.628638(4)': same digits, a bar 3.6x the plotted 1.1e-6; the stored bar follows the primary figure and the upload (sqrt(Var/N_s) with Var = 0.19397/16 in S.S units implies N_s = 1.0e6). energy_variance 0.19397 is printed in no version (upload only). Venue: NeurIPS 2021 ML4PS workshop, non-archival (peer_reviewed question already raised by VP16 on square_36_O). Not a duplicate: no other row on square_100_O carries -0.62863845 (Moss's own best RNN is -251.4624, the minGRU -251.4548).
-0.628637(4), VMC (minGRU)
Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription
Reported as. -0.628637 (+/- 0.000004) per site in S.S units
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)".
-0.6286335, 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. | -251.4534 | | 0.3203 | 100 | 0 | DMRG (MaxTruncError ~1.87E-6, MaxBondDim=10000, Extrapolated Energy = - 251.4628 +/- 0.0019) | [code](https://github.com/varbench/methods/blob/main/programs/dmrg_itensors_heisenberg/square_100_O.jl) |
Program square_100_O.jl: 10x10, xperiodic = yperiodic = false (lines 99-103), nearest-neighbour edges once, lattice_heisenberg_MPO in Pauli form (lines 80-82), Neel start with conserve_qns (Sz = 0), maxdim 4 ... 10000, two sweeps each (line 113). M8 (method text reads extrapolated): answered, -251.4534 is the energy the MPS reached at MaxBondDim = 10000 (the program has no extrapolation); variational is right, and the extrapolated number is carried on its own row. VarBench SM: 'we only report full-variance calculations and forgo extrapolations to allow comparisons with other methods' (arXiv:2302.04919 SM S3, sources/2302.04919.txt lines 765-767). 0.0091 above the exact (stochastic) QMC -251.46248(80). Variance: entered 0.02002 (21a4642, 2022-03-26) and multiplied by 16 the next day (0.32032, 626f8ca; 0.3203 in d871a56): the run printed S.S units (the file's unused chain MPO is S.S). 0.3203 (Pauli) gives var/(E - E0) = 35, in line with VarBench's 6x6 square DMRG rows (38-40); 0.02002 would give 2.2. History of the energy: -251.4437712 (a0766e7, 2021-12-05, 'max trunc error ~4E-6', MaxBondDim 9442) replaced by -251.4534 (dd105fb, 2022-02-22); nothing restored. Program linked 2024-08-08 (d944d80).
-0.628627(1), VMC (NAQS)
Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription
Reported as. -0.628627 (+/- 0.000001) per site in S.S units
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).
-0.628607(3), VMC (RNN)
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. | -251.4430 | 0.0010 | 0.1102 | 100 | 0 | RNN | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_100_O/vmc_rnn.sh) |
vmc_rnn.sh: open 10x10 square lattice, Pauli, J = 1, Marshall sign rule (bipartite), full Hilbert space, exact autoregressive sampling, FastLSTMNet 2 x 16, Adam 10^5 steps. The stored (sigma, Var) imply 1.1e5 independent samples (sigma/sqrt(Var/2^20) = 3.09): from the 2021 run (c12e2c8, 2021-10-14, Dian Wu, data-repository cell 'To be published'), not the linked 2024 script (d54f7c0). 0.0195 above the QMC reference: a valid bound.
-0.628605(6), VMC (minGRU)
Checked 2026-09-15 · source text of arXiv:2605.13807 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. -0.628605(6)
TABLE VI, N=10^2 row | same per-site convention confirmed via the row's own QMC quote -0.6286561(2)
-0.628601(2), PEPS
Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription
Reported as. -0.628601 (+/- 0.000002) per site in S.S units
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.
-0.628507(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; single reading
Reported as. -0.628507(1)
Table II, row 'D=8', column 'L=10' (100 sites); identical value in Table I, column '10x10 J2/J1=0.0', GO sub-column, D=8 row. Also the explicit worked example of Fig. 2 (see compute fields). | Per-site S.S units; instance exact_per_site (-0.6286560999999999) matches Table II's Exact row for L=10 (-0.628656(2)) to 6 sig figs.
-0.628486(6), VMC (pRNN)
Checked 2026-09-15 · source text of arXiv:2608.18065 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. -0.628486(6)
Table 2, row '2D Heisenberg', column E_minSR/N | Same table's quoted reference row (-0.628656 [80]) matches catalogue exact_per_site -0.6286561 for square_100_O to 6 sig figs, fixing N=100, open boundary, S·S per-site convention.
-0.62840(2), VMC (pRNN)
Checked 2026-09-15 · source text of arXiv:2608.18065 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. -0.62840(2)
Table 2, row '2D Heisenberg', column E_Adam/N | Same table's quoted reference row (-0.628656 [80]) matches catalogue exact_per_site -0.6286561 for square_100_O to 6 sig figs, fixing N=100, open boundary, S·S per-site convention.
-0.627697(6), VMC (2D MPS-RNN)
Checked 2026-09-30 · Phys. Rev. Research 5, L032001 (2023), published PDF (sources/journal/prr5-L032001.pdf, 6 pp., sha256 0e96ec27..., Distiller 2023-07-03, CC-BY; main text and references only, the Supplemental Material is a separate file at link.aps.org/supplemental and is not local); text by pypdf plain + layout (work/vj2_prr5-L032001.*); Fig. 4 is vector content inline in the page-4 stream, parsed with work/vecparse.py (every painted path with its CTM, colour and bbox) and decoded by work/decode_prr_fig4.mjs on the journal's own major ticks (panel a: 10^1/10^2 at x = 128.936/181.922, 10^-4..10^-1 at y = 651.542/678.017/704.492/730.967; panel b: 10^1/10^2 at x = 334.524/387.509, 10^-2/10^-1 at y = 681.108/724.604; decade labels identified from the outlined superscript glyphs); compared marker by marker with the VA5 decodes of arXiv:2206.12363v1/v2
Reported as. Fig. 4(a) marker, no printed digits: relative error 1.5257e-3 with respect to QMC [42]
The published figure carries the same point as arXiv v2 and the HAL deposit (VA5 decode v2 a|2D MPS-RNN|40 -> the same total to 1.5e-4; positions agree to 3e-5 decades), so the earlier arXiv-based verification stands in the journal. Journal text: "We perform numerical experiments on a 10 x 10 square lattice with open boundary conditions (OBCs)" / "the AFHM on a 10 x 10 triangular lattice with OBC"; VMC-optimised variational energies (bound_type variational). sigma and energy_variance are printed nowhere in the journal PDF (SM Sec. S7, not local, is where the 10^6-sample evaluation is described in the arXiv versions); they remain the author's VarBench upload values. peer reviewed: PRR. Also read by VA5-hubbard-heisenberg-tfising-j1j2#24 (scope value: Fig. 4(a), green '2D MPS-RNN' marker at chi = 40; identical in v1 and v2).
-0.62611(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; single reading
Reported as. -0.62611(1)
Table I, column '10x10 J2/J1=0.0', SU sub-column, D=10 row (final/most-converged SU value in the sweep D=2..10). | Per-site S.S units, same check as GO rows above; SU is a weaker (less accurate) variational method than GO per the paper's own text ('error is approximately 1e-2 at D=8, which changes only a little for up to D=10').
-0.62587(1), VMC (1D MPS-RNN)
Checked 2026-09-30 · Phys. Rev. Research 5, L032001 (2023), published PDF (sources/journal/prr5-L032001.pdf, 6 pp., sha256 0e96ec27..., Distiller 2023-07-03, CC-BY; main text and references only, the Supplemental Material is a separate file at link.aps.org/supplemental and is not local); text by pypdf plain + layout (work/vj2_prr5-L032001.*); Fig. 4 is vector content inline in the page-4 stream, parsed with work/vecparse.py (every painted path with its CTM, colour and bbox) and decoded by work/decode_prr_fig4.mjs on the journal's own major ticks (panel a: 10^1/10^2 at x = 128.936/181.922, 10^-4..10^-1 at y = 651.542/678.017/704.492/730.967; panel b: 10^1/10^2 at x = 334.524/387.509, 10^-2/10^-1 at y = 681.108/724.604; decade labels identified from the outlined superscript glyphs); compared marker by marker with the VA5 decodes of arXiv:2206.12363v1/v2
Reported as. Fig. 4(a) marker, no printed digits: relative error 4.4398e-3 with respect to QMC [42]
The published figure carries the same point as arXiv v2 and the HAL deposit (VA5 decode v2 a|1D MPS-RNN|40 -> the same total to 1.5e-4; positions agree to 3e-5 decades), so the earlier arXiv-based verification stands in the journal. Journal text: "We perform numerical experiments on a 10 x 10 square lattice with open boundary conditions (OBCs)" / "the AFHM on a 10 x 10 triangular lattice with OBC"; VMC-optimised variational energies (bound_type variational). sigma and energy_variance are printed nowhere in the journal PDF (SM Sec. S7, not local, is where the 10^6-sample evaluation is described in the arXiv versions); they remain the author's VarBench upload values. peer reviewed: PRR. Also read by VA5-hubbard-heisenberg-tfising-j1j2#23 (scope value: Fig. 4(a) (10 x 10 square, OBC), orange '1D MPS-RNN' marker at chi = 40 (x = 142.320 pt); identical in v1 and v2).
-0.62276(1), 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. | -249.1033 | 0.0051 | 26.13060 | 100 | 0 | RBM (alpha = 1) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_100_O/vmc_rbm.sh) |
scripts/Heisenberg/square_100_O/vmc_rbm.sh: 'python3 vmc.py --ham heis --boundary open --sign mars --ham_dim 2 --L 10 --zero_mag --net rbm --seed 123 --show_progress' -> 10x10 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) = 1.022: the stored sigma is the error of that final estimate. 2.3592 above the exact (stochastic) QMC -251.46248(80). History: entered 10f5c12 (2022-07-11, Dian Wu) as -248.367 | 0.0059 | 35.6700 | RBM (alpha = 1); af1d48f (2022-09-01, 'Add Jastrow baseline'): energy -248.367 -> -249.1033; sigma 0.0059 -> 0.0051; variance 35.6700 -> 26.13060; linked 5047bfd (2024-07-30). The value change sits in af1d48f, titled 'Add Jastrow baseline' (a re-run: lower energy and variance). Program files postdate the value (scripts added to varbench/methods 2024-07-30); they document the settings after the run.
-0.62027(1), 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. | -248.1073 | 0.0056 | 31.73600 | 100 | 0 | Jastrow baseline | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_100_O/vmc_jastrow.sh) |
scripts/Heisenberg/square_100_O/vmc_jastrow.sh: 'python3 vmc.py --ham heis --boundary open --sign mars --ham_dim 2 --L 10 --zero_mag --net jas --seed 123 --show_progress' -> 10x10 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) = 1.018: the stored sigma is the error of that final estimate. 3.3552 above the exact (stochastic) QMC -251.46248(80). History: entered af1d48f (2022-09-01, Dian Wu) as -247.9477 | 0.0057 | 31.69845 | Jastrow baseline; 59e621a (2022-09-13, 'Update Jastrow and RBM baselines'): energy -247.9477 -> -248.1073; sigma 0.0057 -> 0.0056; variance 31.69845 -> 31.73600; 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.