How these numbers were read
Where each number below was read, and how it was checked.
-0.5493(1), 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/2505.20406-repo-final_energies_data.json and reproduces Table V's L = 6 energy and Table II's V-scores
Reported as. -0.5493287510342069 +/- 0.00009793047021398267 per site (S.S), Var(E/N) = 0.00009590376996331745, V-score 0.04576500475194352
github.com/mschuylermoss/HeisenbergRNN, commit 5f5cffa5cb (2025-10-28, HEAD; file last changed in d58a945cd4 2025-05-27, 'final plotting data and plotting notebooks'), Triangular/final_plotting/plotting_data/final_energies_data_plotting.pkl, sha256 bdc59e5bc3780fb4ea7444400dc9abd9984a6a259ab03f9299c1f80edb87d276: ['TriangularMS,periodicBC']['scale=4.0,rate=0.158,T=1.00'], L = 12. The paper's most accurate schedule (Table II, whose L = 12 V-score 4.6e-2 is this run's 0.0458); plotted in Fig. 4(a), not printed. TriangularMS is the 120-degree basis rotation U_tri, which changes the representation, not the Hamiltonian. Variance is Var(H)/N^2 in S.S units; total = var x N^2 x 16.
-0.54700(8), VMC (EPS)
Checked 2026-09-28 · source text of arXiv:1006.4480 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading by a reader agent, line checked in the committed text by the session
Reported as. -0.54700(8)
Table 2 (sources/1006.4480.txt L252-258), row N = 144; 'N plaquettes comprising 16 sites have been employed'. | Table 2: 'Variational GS energy per site (in units of Jxy) ... for the spin-1/2 AHM on a triangular lattice of size N with PBC' at the Heisenberg point Jxy = Jz = 1, S.S units; N = 324 is the 18 x 18 cluster and N = 144 the 12 x 12. Its N = 36 value -0.55420(5) is already carried on Heisenberg/triangular_36_P as a secondary row and is left to qmbl-verify.
-0.54624(2), 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. | -314.632 | 0.013 | 43.3 | 144 | 0 | VMC with Dirac+field+Jastrow | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/triangular_144_P/vmc_gutzwiller.sh) |
vmc_gutzwiller inputs: 2-site supercell x '12 0 0 6' = 12x12 triangular torus (basis at 120 degrees), nelup = neldo = 72 (Sz = 0), jperp = 1, uvv = 0.25 (H = sum S.S), Dirac spin liquid fluxes + 120-degree field at K (pitch 2/3, 1/sqrt(3) in datprep.in) + spin Jastrow (iesspin=17); binned error (ratio 2.02). Conversion is on record: first upload -2.18494 | 9e-05 | 2e-03 (5276be9, 2021-08-31; per site, already x4) -> -314.632 | 0.013 | 43.3 (5ee34f3, 2021-09-21): -2.18494 x 144 = -314.63136, 9e-05 x 144 = 0.01296, and 43.3/144^2 = 0.00209 (the per-site variance had been printed to one digit). 1.781 above the 2D-RNN record of Moss et al. Linked d186176 (2024-07-30).
-0.5199011, 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. | -299.46304457007847 | | 100.89209006595773 | 100 | 0 | DMRG (bond dimension = 512) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/triangular_144_P/dmrg.sh) |
dmrg.sh: heisenberg_2d.jl --peri --L 12 --J2 1 --zero_mag --max_B 512: 12x12 triangular torus (432 bonds), Pauli, J = 1, Sz = 0, 50 sweeps at D = 512. M2 (5.4% above the record): a torus of width 12 at D = 512 in a snake MPS; no constant shift. The DOF cell (VarBench 100) is already corrected to 144 on the row (ruling 2026-09-21). Entered c85080f (2022-09-03, Dian Wu), never changed; linked 2024-07-19.
-0.48966(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. | -282.04227 | 0.013 | 168.431 | 144 | 0 | RBM (alpha = 1) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/triangular_144_P/vmc_rbm.sh) |
scripts/Heisenberg/triangular_144_P/vmc_rbm.sh: 'python3 vmc.py --ham heis_tri --boundary peri --sign mars --ham_dim 2 --L 12 --zero_mag --net rbm --seed 123 --show_progress' -> 12x12 triangular lattice (ColoredJ1J2 back_diag=False: square grid + (i,j)-(i+1,j+1) diagonals), periodic; NetKet Heisenberg J = 1 in Pauli form with Marshall rotation on the square bonds only (sign_rule=[True, False], a unitary: the diagonals join same-sublattice sites); 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) = 1.026: the stored sigma is the error of that final estimate. 34.3711 above the 2D-RNN record -316.41336(5641) (Moss et al.). M2/M6 flags (if any): real float32 parameters make the amplitudes positive, so on this frustrated lattice the ansatz cannot carry the ground-state signs; no constant shift is involved and the energy is a valid bound. History: entered 10f5c12 (2022-07-11, Dian Wu) as -281.863 | 0.013 | 173.486 | RBM (alpha = 1); af1d48f (2022-09-01, 'Add Jastrow baseline'): energy -281.863 -> -282.04227; variance 173.486 -> 168.431; linked 5047bfd (2024-07-30). The value change sits in af1d48f, titled 'Add Jastrow baseline' (a re-run). Program files postdate the value (scripts added to varbench/methods 2024-07-30); they document the settings after the run.
-0.48552(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. | -279.66142 | 0.012 | 154.749 | 144 | 0 | Jastrow baseline | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/triangular_144_P/vmc_jastrow.sh) |
scripts/Heisenberg/triangular_144_P/vmc_jastrow.sh: 'python3 vmc.py --ham heis_tri --boundary peri --sign mars --ham_dim 2 --L 12 --zero_mag --net jas --seed 123 --show_progress' -> 12x12 triangular lattice (ColoredJ1J2 back_diag=False: square grid + (i,j)-(i+1,j+1) diagonals), periodic; NetKet Heisenberg J = 1 in Pauli form with Marshall rotation on the square bonds only (sign_rule=[True, False], a unitary: the diagonals join same-sublattice sites); 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) = 0.988: the stored sigma is the error of that final estimate. 36.7519 above the 2D-RNN record -316.41336(5641) (Moss et al.). M2/M6 flags (if any): real float32 parameters make the amplitudes positive, so on this frustrated lattice the ansatz cannot carry the ground-state signs; no constant shift is involved and the energy is a valid bound. History: entered af1d48f (2022-09-01, Dian Wu) as -277.77834 | 0.011 | 149.787 | Jastrow baseline; 59e621a (2022-09-13, 'Update Jastrow and RBM baselines'): energy -277.77834 -> -279.66142; sigma 0.011 -> 0.012; variance 149.787 -> 154.749; 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.