How these numbers were read
Where each number below was read, and how it was checked.
-0.67887215(2), SSE QMC
Checked 2026-09-15 · arXiv HTML parsed locally, one value per table cell; no LLM transcription
Reported as. e0 = -0.67887215(2) per spin (S.S), L = 6, 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 = 6. Pauli total = 4 N e0. The L = 6 value agrees with the exact-diagonalization row on square_36_P to 1.6e-8.
-0.6788721, ED
Checked 2026-09-29 · prior recomputation: E2 ed_sym.py (numba, table-based symmetric basis), output qmbl-runs/qmbl-verify/out/E2-exact-remote/results/sym36/Heisenberg__square_36_P__J2_0__k0-0__c4v-irrep0__z+.json (evidence cited by the plan: qmbl-verify/out/E2-exact-remote (verified))
Reported as. -97.75758959723629 (true residual ||Hx-Ex||/||x|| = 6.19e-11); S^z=0, k=0, C4v A1, spin inversion +1
Real recomputation (code and output file). |E - stored| = 5.97e-13 (relative 6.11e-15). Second determination: the stored value is the source's own ED (VarBench), a different code. Also read by VB3-varbench-code-heisenberg#50 (scope config: vendor/varbench/Heisenberg/square_36_P.md (snapshot 390a21e), row 'Exact diagonalization'; linked scripts/Heisenberg/square_36_P/ed_lattice_symmetries.sh).
-0.6788721, ED
Checked 2026-10-24 · arXiv:cond-mat/0412662: publisher PDF (Springer, scanned) Table 2.1, read from the page image rendered at 200 %; checked against the arXiv text (pypdf layout and plain modes); no LLM transcription of numbers; single reading + page-image check
Reported as. −24.4393974
Table 2.1 (p. 102), row N = 36, edge vectors (6, 0), (0, 6), column E0. The caption says the N = 36 results agree with Schulz et al. [55]; the stored ED energy -97.7575895972 is this value to its printed digits. | Spin-1/2 Heisenberg antiferromagnet H = J sum_<ij> S_i . S_j on the square lattice, J = 1, periodic boundaries on the finite lattice defined by the two edge vectors printed in the row (Betts construction, Sec. 2.4.1); E0 is the total singlet ground-state energy in S.S units, stored as the Pauli total 4 E0. Version read: publisher PDF (Springer, a scan), Table 2.1 on p. 102 read from the page image; identical to Table 1 of the arXiv text cond-mat/0412662. Peer review: a chapter of an edited lecture-notes volume, not a refereed journal.
-0.67887177(7), VMC (2D RNN)
Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription
Reported as. -0.67887177 (+/- 7e-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 "This work (zero-variance)". The paper states in the Table 3 caption that these extrapolated values can fall below the reference energies and that this is "an artifact of the zero-variance extrapolation" - not an achieved energy, so NOT a bound (RULES.md 4).
-0.678871(3), 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; adversarial re-read: confirmed
Reported as. -0.678871(3)
Table I, row (qx,qy,qsf)=(0,0,0), column |E0>, data row 'E/Ns': -0.678871(3) (source text lines 1012-1013). This is the global minimum across all 16 momentum/spin-flip sectors and all 4 excited states listed in Table I, confirming it is the true ground state (Gamma-point sector). | Own ED value on the same row (EED/Ns=-0.67887215) matches instance exact_per_site=-0.6788721499808035 to 8 decimals, fixing per-site S.S convention with no rescaling; 'square lattice geometries with periodic boundary conditions along both spatial directions' (Sec. V) plus the use of momentum labels (qx,qy) fixes this as the periodic (_P), not open (_O), 36-site instance.
-0.67887(2), VMC (2D RNN)
Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription
Reported as. -0.67887 (+/- 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)".
-0.67887, 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. -0.678870
Table I, row J2=0.00, column '6×6, 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.
-0.678868(2), 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. -0.678868 (+/- 0.000002) 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: "6x6 -0.678873(4) 2.51799(6) -0.678868(2) 2.51(2)" - Energy(QMC) beside Energy(RBM).
-0.67882(1), VMC (CNN)
Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription
Reported as. -0.67882 (+/- 0.00001) 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).
-0.6788, VMC (CNN)
Checked 2026-09-15 · source text of arXiv:2510.02051 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. -0.6788
S5.p7: 'At J2/J1=0, our work yields -0.6788, showing excellent agreement with the exact value [59] of -0.678872' | 6x6=36-site periodic square lattice, J2=0 reduces J1-J2 model to plain Heisenberg; matches catalogue Heisenberg/square_36_P by exact-value agreement.
-0.6786224, 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. | -97.72162414290578 | | 1.385137434483113 | 36 | 0 | DMRG (bond dimension = 2048) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_36_P/dmrg.sh) |
dmrg.sh: heisenberg_2d.jl --peri --L 6 --zero_mag --max_B 2048: 6x6 torus, Pauli, J = 1, Sz = 0, 50 sweeps at D = 2048. 0.0360 above the exact energy (a torus in a snake MPS); var/(E - E0) = 38.5. Entered 721e09e (2022-08-22), never changed; linked 2024-07-19.
-0.6785(2), VMC (EPS)
Checked 2026-09-15 · source text of arXiv:2510.02051 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. -0.6785(2)
S5.p7: 'VMC method [60] gives -0.6785(2)' | Same 36-site periodic Heisenberg instance.
-0.66974(3), 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. | -96.4427 | 0.0038 | 14.7488 | 36 | 0 | Jastrow baseline | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_36_P/vmc_jastrow.sh) |
scripts/Heisenberg/square_36_P/vmc_jastrow.sh: 'python3 vmc.py --ham heis --boundary peri --sign mars --ham_dim 2 --L 6 --zero_mag --net jas --seed 123 --show_progress' -> 6x6 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) = 1.013: the stored sigma is the error of that final estimate. 1.3149 above the exact -97.7575895972357. History: entered af1d48f (2022-09-01, Dian Wu) as -96.4427 | 0.0038 | 14.7488 | 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.
-0.66854(3), 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. | -96.2702 | 0.0037 | 13.9167 | 36 | 0 | RBM (alpha = 1) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_36_P/vmc_rbm.sh) |
scripts/Heisenberg/square_36_P/vmc_rbm.sh: 'python3 vmc.py --ham heis --boundary peri --sign mars --ham_dim 2 --L 6 --zero_mag --net rbm --seed 123 --show_progress' -> 6x6 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) = 1.016: the stored sigma is the error of that final estimate. 1.4874 above the exact -97.7575895972357. History: entered 721e09e (2022-08-22, Dian Wu) as -96.2702 | 0.0037 | 13.9167 | 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.