How these numbers were read
Where each number below was read, and how it was checked.
-0.5603734, ED
Checked 2026-09-29 · prior recomputation: E2 ed_sym.py, output qmbl-runs/qmbl-verify/out/E2-exact-remote/results/sym36/Heisenberg__triangular_36_P__J2_1__k0-0__c6v-irrep0__z+.json (evidence cited by the plan: qmbl-verify/out/E2-exact-remote (verified))
Reported as. -80.69376896124292 (true residual ||Hx-Ex||/||x|| = 1.75e-10); S^z=0, Gamma, C6v A1, spin inversion +1
Real recomputation (code and output file). |E - stored| = 3.27e-13 (relative 4.05e-15). Second determination: the stored value is the source's own ED (VarBench), a different code. Also read by VB3-varbench-code-heisenberg#78 (scope config: vendor/varbench/Heisenberg/triangular_36_P.md (snapshot 390a21e), row 'Exact diagonalization'; linked scripts/Heisenberg/triangular_36_P/ed_lattice_symmetries.sh).
-0.560313(3), VMC (GCNN)
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. -0.560313 (+/- 0.000003) per site in S.S units
Read from Table 5 of arXiv:2505.20406 (TLAHM, L=6 triangular lattice, PBC, E/N in S.S units), parsed from the arXiv HTML. Row "Group Convolutional Neural Network [32]" = arXiv:2211.07749. Sits 6.0e-5 ABOVE the exact -0.5603734 already on this instance, as a variational bound must. CONFIRMED IN PRIMARY SOURCE - arXiv:2211.07749, results table: "J2/J1 N GCNN ... Exact/Interpolated ... 0 36 -0.560313(3) - -0.5603734" - the GCNN energy at J2/J1 = 0, N = 36, printed beside the exact value.
-0.5601(4), VMC (LCN)
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. -0.5601 (+/- 0.0004) per site in S.S units
Read from Table 5 of arXiv:2505.20406 (TLAHM, L=6 triangular lattice, PBC, E/N in S.S units), parsed from the arXiv HTML. Row "Lattice Convolutional Network [49]" = arXiv:2206.07370. CONFIRMED IN PRIMARY SOURCE - arXiv:2206.07370v1, Table 1: "Triangular 36 0 - -0.55889 -0.5601(4) -0.5603734" - the LCN energy at J2 = 0 on the 36-site triangular lattice, with the regular kernel (Table 3 prints -0.5601(4) for the regular and -0.5501(6) for the special kernel).
-0.55922, VMC (GCNN)
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. -0.55922 per site in S.S units
Read from Table 5 of arXiv:2505.20406 (TLAHM, L=6 triangular lattice, PBC, E/N in S.S units), parsed from the arXiv HTML. Row "Group Convolutional Neural Network [104]" = arXiv:2104.05085. No error bar is given in the table. CONFIRMED IN PRIMARY SOURCE - arXiv:2104.05085, results table: "J2 = 0 ... ED [28] -0.5603734 ... G-CNN -0.55922".
-0.559214(6), VMC (aCNN)
Checked 2026-09-17 · 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. -0.559214(6)
Table 4 of arXiv:2308.09664 (arXiv HTML, sources/2308.09664.html): row "aCNN", column HMTL (Heisenberg model on the 36-site triangular lattice), energy per site; caption: "The aCNN is optimized with the exact sign structure from ED". ED -0.560374 in the same table agrees with the carried exact row -0.5603734 within the last printed digit. | ED value in the same column agrees with Heisenberg/triangular_36_P's exact row within 1e-6 (S.S per site).
-0.55874(4), VMC (GCNN)
Checked 2026-09-24 · checks/cost/run_nqs.py (NetKet 3.22.4, jax 0.8.3) on 1 x NVIDIA A100 80GB PCIe, protocol checks/cost/README.md
Reported as. energy -80.45855704311515 sigma 0.006354516711348047 energy_variance 5.45144124948146 tau_corr 0 r_hat 1.0037788694987775 | wall_hms 00:28:22 (wall_seconds 1701.7986515802331) on 1 x NVIDIA A100 80GB PCIe, Slurm job 15097829 | parameters 7232, steps 2000 x n_samples 4096, eval samples 131072
Final evaluation, the training chains carried on: 131072 samples, 1024 chains, 64 discarded per chain; tau_corr 0.00, R_hat 1.004. Trained 2000 steps of SR (QGTJacobianDense, conjugate gradients, 300 iterations max, diag_shift 1e-06, diag_scale 0.01), lr 0.01 after a 200-step linear warmup, 4096 samples per step, MetropolisExchange, d_max 2, seed 20260921. Divergence rule applied: energy -1.04e+8 at step 153, outside the Hamiltonian's bound, parameters restored from step 150 (E = -73.15), learning rate 0.005 from there. Per-step trace and final parameters beside the results file.
-0.5581022, 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. | -80.36672248793934 | | 7.068731007373572 | 36 | 0 | DMRG (bond dimension = 2048) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/triangular_36_P/dmrg.sh) |
dmrg.sh: heisenberg_2d.jl --peri --L 6 --J2 1 --zero_mag --max_B 2048: 6x6 triangular torus, Pauli, J = 1, Sz = 0, 50 sweeps at D = 2048. 0.327 above the exact energy (0.4%): a torus in a snake MPS; QMBL's own chi = 2000 DMRG cost run on this instance gives -80.35309, consistent. Entered 721e09e (2022-08-22), never changed; linked 2024-07-19.
-0.5580076, DMRG
Checked 2026-09-24 · checks/cost/run_dmrg.py (TeNPy 1.1.1) on 8 cores, protocol checks/cost/README.md
Reported as. chi_max 2000 chi_reached 2000 energy -80.35309162427853 (energy_SS -20.088272906069633) sweeps 16 max_trunc_err 0.00023684509156278683 | wall_hms 03:03:20 (wall_seconds 10999.921602776274) on 8 cores, Slurm job 15056281 | cpu_core_hours 24.44427022839172
two-site DMRG, TeNPy, finite MPS on the torus with long-range couplings, Sz conserved, Neel product start, mixer on, a new engine per rung on the previous rung's state; MPO bond dimension 41; stopping rule max_sweeps 15, max_E_err 0.000001 (TeNPy also requires |Delta S| < 1e-5 by default and stops only once the sweep count exceeds max_sweeps); this rung ran 16 sweeps, the cap. Energy computed in S.S units and stored x4 (Pauli, the instance's convention).
-0.5562(2), VMC (2D RNN)
Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription
Reported as. -0.5562 (+/- 0.0002) per site in S.S units
Read from Table 5 of arXiv:2505.20406 (TLAHM, L=6 triangular lattice, PBC, E/N in S.S units), parsed from the arXiv HTML. The citing paper's own result.
-0.55549(5), VMC (ViT)
Checked 2026-09-24 · checks/cost/run_nqs.py (NetKet 3.22.4, jax 0.8.3) on 1 x NVIDIA A100 80GB PCIe, protocol checks/cost/README.md
Reported as. energy -79.9911596136248 sigma 0.006754502141950846 energy_variance 5.852950728342892 tau_corr 0.010848061807489207 r_hat 1.0042366987297604 | wall_hms 02:40:42 (wall_seconds 9641.68374238303) on 1 x NVIDIA A100 80GB PCIe, Slurm job 15056279 | parameters 154800, steps 2000 x n_samples 4096, eval samples 131072
Final evaluation, the training chains carried on: 131072 samples, 1024 chains, 64 discarded per chain; tau_corr 0.01, R_hat 1.004. Trained 2000 steps of SR (QGTJacobianDense, conjugate gradients, 300 iterations max, diag_shift 1e-06, diag_scale 0.01), lr 0.01 after a 200-step linear warmup, 4096 samples per step, MetropolisExchange, d_max 2, seed 20260921. Per-step trace and final parameters beside the results file.
-0.55519(4), VMC (PP)
Checked 2026-09-15 · source text of arXiv:2104.05085 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. -0.55519
Table I, J2=0 column, row 'VMC [30]': -0.55519. | Same table as ED row.
-0.5550453, DMRG
Checked 2026-09-24 · checks/cost/run_dmrg.py (TeNPy 1.1.1) on 8 cores, protocol checks/cost/README.md
Reported as. chi_max 1000 chi_reached 1000 energy -79.92652217664256 (energy_SS -19.98163054416064) sweeps 16 max_trunc_err 0.00037311592377740595 | wall_hms 01:10:45 (wall_seconds 4244.665265383199) on 8 cores, Slurm job 15056281 | cpu_core_hours 9.43258947862933
two-site DMRG, TeNPy, finite MPS on the torus with long-range couplings, Sz conserved, Neel product start, mixer on, a new engine per rung on the previous rung's state; MPO bond dimension 41; stopping rule max_sweeps 15, max_E_err 0.000001 (TeNPy also requires |Delta S| < 1e-5 by default and stops only once the sweep count exceeds max_sweeps); this rung ran 16 sweeps, the cap. Energy computed in S.S units and stored x4 (Pauli, the instance's convention).
-0.55420(5), VMC (EPS)
Checked 2026-09-15 · source text of arXiv:2104.05085 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. -0.55420
Table I, J2=0 column, row 'VMC [31]': -0.55420 (harvested cell rounded this to -0.5542; full value read directly from source text). | Same table as ED row.
-0.553, VMC (Neural Gutzwiller)
Checked 2026-09-15 · source text of arXiv:2104.05085 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. -0.553
Table I, J2=0 column, row 'NN + Gutzwiller [12]': -0.553, N/A error. Not present as a separate row in the harvested cells (its value appears merged into another row's garbled col_header text); read directly from the source table. | Same table as ED row.
-0.55148(5), VMC (Gutzwiller)
Checked 2026-09-15 · source text of arXiv:2505.20406 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. -0.55148(5)
TABLE V | same as above
-0.5500876, DMRG
Checked 2026-09-24 · checks/cost/run_dmrg.py (TeNPy 1.1.1) on 8 cores, protocol checks/cost/README.md
Reported as. chi_max 500 chi_reached 500 energy -79.2126086267484 (energy_SS -19.8031521566871) sweeps 16 max_trunc_err 0.0006064990325569687 | wall_hms 00:27:35 (wall_seconds 1654.652795038186) on 8 cores, Slurm job 15056281 | cpu_core_hours 3.6770062111959687
two-site DMRG, TeNPy, finite MPS on the torus with long-range couplings, Sz conserved, Neel product start, mixer on, a new engine per rung on the previous rung's state; MPO bond dimension 41; stopping rule max_sweeps 15, max_E_err 0.000001 (TeNPy also requires |Delta S| < 1e-5 by default and stops only once the sweep count exceeds max_sweeps); this rung ran 16 sweeps, the cap. Energy computed in S.S units and stored x4 (Pauli, the instance's convention).
-0.548025(3), VMC (Gutzwiller)
Checked 2026-09-15 · source text of arXiv:2104.05085 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. -0.548025
Table I, J2=0 column, row 'VMC [27]': -0.548025. | Same table as ED row.
-0.543(1), VMC (Gutzwiller)
Checked 2026-09-15 · source text of arXiv:2505.20406 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. -0.543(1)
TABLE V. Estimates of the ground-state energy per spin for the TLAHM on an L=6 lattice with PBC. | per-site convention fixed by exact-diagonalization and record rows in the same table
-0.5320(1), VMC (RBM)
Checked 2026-09-24 · checks/cost/run_nqs.py (NetKet 3.22.4, jax 0.8.3) on 1 x NVIDIA A100 80GB PCIe, protocol checks/cost/README.md
Reported as. energy -76.60531652120582 sigma 0.016821273919707346 energy_variance 35.74488444047138 tau_corr 0.018780686130572932 r_hat 1.0040647022141393 | wall_hms 00:06:08 (wall_seconds 368.4269102141261) on 1 x NVIDIA A100 80GB PCIe, Slurm job 15056266 | parameters 1368, steps 2000 x n_samples 4096, eval samples 131072
Final evaluation, the training chains carried on: 131072 samples, 1024 chains, 64 discarded per chain; tau_corr 0.02, R_hat 1.004. Trained 2000 steps of SR (QGTJacobianDense, Cholesky on the dense S, diag_shift 1e-06, diag_scale 0.01), lr 0.01 after a 200-step linear warmup, 4096 samples per step, MetropolisExchange, d_max 2, seed 20260921. Per-step trace and final parameters beside the results file.
-0.5100(2), VMC (RBM)
Checked 2026-09-24 · checks/cost/run_nqs.py (NetKet 3.22.4, jax 0.8.3) on 1 x NVIDIA A100 80GB PCIe, protocol checks/cost/README.md
Reported as. energy -73.44638853443263 sigma 0.02482466794726441 energy_variance 80.29017754419445 tau_corr 0.0030190221099478665 r_hat 1.0041322644078527 | wall_hms 00:07:07 (wall_seconds 427.04232935328037) on 1 x NVIDIA A100 80GB PCIe, Slurm job 15056270 | parameters 149, steps 2000 x n_samples 4096, eval samples 131072
Final evaluation, the training chains carried on: 131072 samples, 1024 chains, 64 discarded per chain; tau_corr 0.00, R_hat 1.004. Trained 2000 steps of SR (QGTJacobianDense, Cholesky on the dense S, diag_shift 1e-06, diag_scale 0.01), lr 0.01 after a 200-step linear warmup, 4096 samples per step, MetropolisExchange, d_max 2, seed 20260921. Per-step trace and final parameters beside the results file.
-0.49161(5), 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. | -70.7915 | 0.0065 | 43.4753 | 36 | 0 | RBM (alpha = 1) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/triangular_36_P/vmc_rbm.sh) |
scripts/Heisenberg/triangular_36_P/vmc_rbm.sh: 'python3 vmc.py --ham heis_tri --boundary peri --sign mars --ham_dim 2 --L 6 --zero_mag --net rbm --seed 123 --show_progress' -> 6x6 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) = 1.009: the stored sigma is the error of that final estimate. 9.9023 above the exact -80.6937689612426. 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 721e09e (2022-08-22, Dian Wu) as -70.7915 | 0.0065 | 43.4753 | 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.
-0.48678(4), 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. | -70.0967 | 0.0060 | 37.0349 | 36 | 0 | Jastrow baseline | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/triangular_36_P/vmc_jastrow.sh) |
scripts/Heisenberg/triangular_36_P/vmc_jastrow.sh: 'python3 vmc.py --ham heis_tri --boundary peri --sign mars --ham_dim 2 --L 6 --zero_mag --net jas --seed 123 --show_progress' -> 6x6 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) = 1.010: the stored sigma is the error of that final estimate. 10.5971 above the exact -80.6937689612426. 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 -70.0967 | 0.0060 | 37.0349 | 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.