How these numbers were read
Where each number below was read, and how it was checked.
-0.6703092(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. -0.6703092(5)
Table III (Appendix H), row L=14, column 'RNN zero-variance E/N' | more negative than the instance's currently listed variational record (-0.6683456632653061, 'VMC with fermions'); the paper's own note that zero-variance extrapolation is not a strict bound applies here too
-0.67023225(2), SSE QMC
Checked 2026-09-15 · arXiv HTML parsed locally, one value per table cell; no LLM transcription
Reported as. e0 = -0.67023225(2) per spin (S.S), L = 14, 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 = 14. Pauli total = 4 N e0. The L = 6 value agrees with the exact-diagonalization row on square_36_P to 1.6e-8.
-0.67014(2), VMC (2D RNN)
Checked 2026-09-16 · authors' data release read locally (pickle via numpy), values copied by key, no transcription; the file is committed as sources/2502.17144-repo-final_energy_data.json and reproduces the four values Table V prints
Reported as. -0.6701383006816007 +/- 0.00002145833623584729 per site (S.S), Var(E/N) = 0.00000460460194010677
github.com/mschuylermoss/HeisenbergRNN, commit 29bf62ac28 (2025-02-24, 'reproduce figures'), Square/final_plotting/final_data/final_energy_data_plotting.pkl: ['periodic']['rate=0.25']['scale=4'], L = 14; the lowest of the ten training schedules (s = 4, r = 0.25). Plotted as a point in Fig. 3(a), not printed in the paper; Table III prints the zero-variance extrapolation over the s >= 1 runs at this L, carried on this instance as an `extrapolated` row. Variance is Var(H)/N^2 in S.S units (see script header); total = var x N^2 x 16.
-0.668346(8), 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. | -523.983 | 0.006 | 22.07 | 196 | 0 | VMC with fermions (flux+neel+Jastrow) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_196_P/vmc_gutzwiller.sh) |
vmc_gutzwiller inputs: 2x2 supercell x '7 0 0 7' = 14x14 periodic spin torus, nelup = neldo = 98 (Sz = 0), jperp = 1, uvv = 0.25 (H = sum S.S), pi-flux + Neel field + spin Jastrow (iesspin=32); binned error (ratio 1.31). Stored = 4N x <H>/N. 1.479 above Sandvik's SSE. Entered 32046ef (2022-02-15, F. Ferrari), never changed; linked d186176 (2024-07-30).
-0.66381(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. | -520.4234 | 0.0079 | 63.70537 | 196 | 0 | RBM (alpha = 1) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_196_P/vmc_rbm.sh) |
scripts/Heisenberg/square_196_P/vmc_rbm.sh: 'python3 vmc.py --ham heis --boundary peri --sign mars --ham_dim 2 --L 14 --zero_mag --net rbm --seed 123 --show_progress' -> 14x14 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) = 1.014: the stored sigma is the error of that final estimate. 5.0387 above Sandvik's SSE -525.462084(16). History: entered 10f5c12 (2022-07-11, Dian Wu) as -517.807 | 0.0096 | 94.5612 | RBM (alpha = 1); af1d48f (2022-09-01, 'Add Jastrow baseline'): energy -517.807 -> -520.4234; sigma 0.0096 -> 0.0079; variance 94.5612 -> 63.70537; 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.
-0.66256(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. | -519.4475 | 0.0081 | 66.74661 | 196 | 0 | Jastrow baseline | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_196_P/vmc_jastrow.sh) |
scripts/Heisenberg/square_196_P/vmc_jastrow.sh: 'python3 vmc.py --ham heis --boundary peri --sign mars --ham_dim 2 --L 14 --zero_mag --net jas --seed 123 --show_progress' -> 14x14 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) = 1.015: the stored sigma is the error of that final estimate. 6.0146 above Sandvik's SSE -525.462084(16). History: entered af1d48f (2022-09-01, Dian Wu) as -519.4475 | 0.0081 | 66.74661 | 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.
-0.6586622, 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. | -516.3911695223306 | | 127.79517553787447 | 100 | 0 | DMRG (bond dimension = 512) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_196_P/dmrg.sh) |
dmrg.sh: heisenberg_2d.jl --peri --L 14 --zero_mag --max_B 512: 14x14 torus, Pauli, J = 1, Sz = 0, 50 sweeps at D = 512. 9.071 above Sandvik's SSE (1.7%). The DOF cell (VarBench 100, typed by hand in c85080f) is already corrected to 196 on the row (ruling 2026-09-21), v_score recomputed; the VarBench snapshot still prints 100. Entered c85080f (2022-09-03, Dian Wu), never changed; linked 2024-07-19.