How these numbers were read
Where each number below was read, and how it was checked.
-0.6715950(5), VMC (2D RNN)
Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription
Reported as. -0.671595 (+/- 5e-7) per site in S.S units
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).
-0.67155266(2), SSE QMC
Checked 2026-09-15 · 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
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.
-0.67155260(3), VMC (ResNet)
Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription
Reported as. -0.6715526 (+/- 3e-8) per site in S.S units
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).
-0.671544(4), VMC (CNN BF + RBM)
Checked 2026-09-15 · source text of arXiv:2507.10287 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. -0.671544(4)
Table II, row (qx,qy,qsf)=(0,0,0), column |E0>, data row 'E/Ns': -0.671544(4) (source text line 1056). Body text: 'For the ground state in the q = Gamma sector, we also report the Quantum Monte Carlo (QMC) estimate for the energy [41], EQMC/Ns = -0.671549(4), which is in excellent agreement with our result E/Ns = -0.671544(4)' (lines 682-686). This cell was missed entirely by the automated harvester (not present anywhere in the .cells.tsv despite exhaustive grep) and was recovered by direct reading of the source table. | No finite-size exact value exists for 100 sites (exact_per_site is null in the instance file), so the convention is fixed instead by direct agreement with the paper's own quoted Sandvik(1997) SSE-QMC value (-0.671549(4)) for the identical lattice/sector, and by proximity to the instance's established record_per_site (-0.6715526, CNN+MinSR); all three numbers are in the same S.S per-site units of Eq. 30, no rescaling applied.
-0.671519(4), VMC (RBM)
Checked 2026-09-14 · 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
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).
-0.67151(2), VMC (2D RNN)
Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription
Reported as. -0.67151 (+/- 0.00002) per site in S.S units
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)".
-0.671476, VMC (ViT)
Checked 2026-09-15 · source text of arXiv:2606.04558 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. -0.671476
Table I, row J2=0.00, column '10×10, ViT (this work)' | Table I value is already printed per-site (not total): e.g. 6×6 ED at J2=0.50 prints -0.503809, exactly matching catalog J1J2/square_36_P_0.5 exact_per_site -0.5038096539; the J2=0.00 quoted QMC column value -0.67349005 [29] exactly equals catalog Heisenberg/square_64_P exact_per_site -0.67349005 (and -0.67155266 [29] equals Heisenberg/square_100_P exact_per_site -0.67155266) — confirms direct S·S per-site match, no Pauli conversion needed.
-0.671431(2), VMC (aCNN)
Checked 2026-09-15 · source text of arXiv:2308.09664 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. -0.671431(2)
Table I, L=10 column, row aCNN(C4v) | same as QMC row above; aCNN value sits correctly above exact_per_site -0.67155266
-0.67135(1), VMC (CNN)
Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription
Reported as. -0.67135 per site in S.S units
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.
-0.671134(1), 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. | -268.4538 | 0.0005 | 0.5409 | 100 | 0 | RNN | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_100_P/vmc_rnn.sh) |
vmc_rnn.sh: periodic 10x10 square lattice, Pauli, J = 1, Marshall sign rule (bipartite, L even), full Hilbert space, FastLSTMNet 2 x 16, Adam 10^5 steps. The stored (sigma, Var) imply 2.2e6 independent samples (ratio 0.696): from the 2022 run (5af37ba, 2022-03-04, Dian Wu), not the linked 2024 script. 0.1673 above Sandvik's SSE: a valid bound.
-0.66935(1), VMC (Gutzwiller)
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. | -267.741 | 0.004 | 12.2 | 100 | 0 | VMC with fermions (flux+neel+Jastrow) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_100_P/vmc_gutzwiller.sh) |
vmc_gutzwiller inputs: 4-site 2x2 supercell x '5 0 0 5' = 10x10 periodic spin torus, nelup = neldo = 50 (Sz = 0), jperp = 1, uvv = 0.25 (H = sum S.S), pi-flux hopping (wf.in) + Neel field (iesaf=1) + spin Jastrow (iesspin=18); binned error (sigma/sqrt(Var/2^20) = 1.17). Stored = 4N x <H>/N. 0.880 above Sandvik's SSE. Entered 21ff9bc (2022-02-15, F. Ferrari), never changed; linked d186176 (2024-07-30).
-0.6648562, 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. | -265.9424657801722 | | 42.496814964441 | 100 | 0 | DMRG (bond dimension = 1024) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_100_P/dmrg.sh) |
dmrg.sh: heisenberg_2d.jl --peri --L 10 --zero_mag --max_B 1024: 10x10 torus, Pauli, J = 1, no diagonal, Sz = 0, 50 sweeps at D = 1024, cutoff 1e-12. 2.679 above Sandvik's SSE (1%): a torus in a snake MPS at D = 1024. History: replaced the D = 80 value -261.3172227195609 (var 94.47555607618415; 3c108e3, 2022-07-11) in c85080f (2022-09-03, Dian Wu), the commit that also typed the square_196_P/triangular_144_P DOF = 100 (already corrected); nothing restored. Linked f88f9ba/cd362c4 (2024-07-19).
-0.66443(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. | -265.7726 | 0.0060 | 37.11238 | 100 | 0 | RBM (alpha = 1) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_100_P/vmc_rbm.sh) |
scripts/Heisenberg/square_100_P/vmc_rbm.sh: 'python3 vmc.py --ham heis --boundary peri --sign mars --ham_dim 2 --L 10 --zero_mag --net rbm --seed 123 --show_progress' -> 10x10 square lattice, periodic (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) = 1.009: the stored sigma is the error of that final estimate. 2.8485 above Sandvik's SSE -268.621064(8). History: entered 10f5c12 (2022-07-11, Dian Wu) as -265.773 | 0.0060 | 37.1123 | RBM (alpha = 1); af1d48f (2022-09-01, 'Add Jastrow baseline'): energy -265.773 -> -265.7726; variance 37.1123 -> 37.11238; 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.
-0.66421(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. | -265.6846 | 0.0056 | 31.98062 | 100 | 0 | Jastrow baseline | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_100_P/vmc_jastrow.sh) |
scripts/Heisenberg/square_100_P/vmc_jastrow.sh: 'python3 vmc.py --ham heis --boundary peri --sign mars --ham_dim 2 --L 10 --zero_mag --net jas --seed 123 --show_progress' -> 10x10 square lattice, periodic (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) = 1.014: the stored sigma is the error of that final estimate. 2.9365 above Sandvik's SSE -268.621064(8). History: entered af1d48f (2022-09-01, Dian Wu) as -265.6846 | 0.0056 | 31.98062 | 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.