J1-J2 square 6x6, J2 = 1

Exact energy and best values (rules §6)

exact	-0.7143604	ED Schulz et al. (1996) · the instance is solved
best variational	-0.71429(1)	VMC (RBM) Chen et al. (2022) · the lowest unflagged variational energy, a strict variational bound

Energies as E/N (S.S). Classes are different claims and are not ranked against each other.

The Hamiltonian

H = ∑_⟨ij⟩ σ_i·σ_j + J₂ ∑_{⟨⟨ij⟩⟩} σ_i·σ_j

J₁ = 1, J₂ = 1, nearest and next-nearest neighbours, written with Pauli matrices.

lattice: square
sites: 36
boundary: periodic
J2: 1
energies: 7

Stored as a total energy; quoted here as E/N (S.S), which is what the papers report (how the conversion works).

Every published energy

Grouped by what the number is. The exact energy comes first where there is one; in every other group the lowest unflagged energy is its best value, and each row's distance is to that best (rules §6).

Numerically exact (1)

Exact diagonalization, an exact solution, or sign-problem-free QMC where that is established. An exact energy is shown first, before the best of each other class. QMC is exact only within its error bar, which it must state, and reads exact (stochastic); an exact diagonalization outranks it. For a diagonalization resolved by symmetry sector only the ground state's sector is listed; the lowest energies of the other sectors are the instance's spectrum, in its JSON, and are not shown as the exact energy.

E/N (S.S)	distance to class best	σ	Var(E)	V-score	method	source	code	year	comment
-0.7143604exact	0	n/a	n/a	n/a	ED VarBench reference run	Schulz et al. (1996)	run script	1996

Strict variational upper bounds (6)

Energies that a variational state reached (VMC, DMRG and MPS at finite bond dimension, PEPS, NQS, VQE). Each is an upper bound on the ground-state energy.

E/N (S.S)	distance to class best	σ	Var(E)	V-score	method	source	code	year
-0.71429best variational	0	1.0e-5	n/a	n/a	VMC (RBM) peer reviewed	Chen et al. (2022)		2022
-0.71351	7.8e-4	1.0e-5	n/a	n/a	VMC (CNN) peer reviewed	Choo et al. (2019)	uses NetKet	2019
-0.7130465	1.2e-3	n/a	7.16e+0	2.4e-2	DMRG VarBench reference run	Wu et al. (2024)	run script	2024
-0.712413	1.9e-3	n/a	n/a	n/a	VMC (ViT) preprint	Golubev et al. (2026)	uses NetKet	2026
-0.70921	5.1e-3	3.1e-5	2.05e+1	7.1e-2	VMC (RBM) VarBench reference run	Wu et al. (2024)	run script	2024
-0.70016	1.4e-2	3.5e-5	2.53e+1	9.0e-2	VMC (Jastrow) VarBench reference run	Wu et al. (2024)	run script	2024

How these numbers were read

Where each number below was read, and how it was checked.

-0.7143604, ED

Checked 2026-09-29 · prior recomputation: E2 ed_sym.py; RE scan + second basis, output qmbl-runs/qmbl-verify/out/E2-exact-remote/results/sym36/J1J2__square_36_P__J2_1__k0-0__c4v-irrep2__z+.json (evidence cited by the plan: qmbl-verify-2026-09-17-j1j2-36/out/RE-exact-recompute (verified))

Reported as. -102.86790231498468 (true residual ||Hx-Ex||/||x|| = 6.19e-10); S^z=0, k=0, C4v B1, z=+1

Real recomputation (code and output file). |E - stored| = 3.27e-13 (relative 3.18e-15). Second determination: the stored value is the source's own ED (VarBench), a different code. RE scanned every z=+1 sector at J2 = 1; RE's c2v_axes (C2=+1, s=+1) run, a second symmetric basis, gives -102.8679023149708 (1.4e-11 from E2). Also read by VJ2-journal-other#37 (scope value: Table II (journal p. 681 = PDF p. 7), "The ground state energy per site for different clusters and different values of J2 (J1 is normalized to unity)", row 1.00, column 36(B1)).

-0.71429(1), VMC (RBM)

Checked 2026-09-14 · arXiv PDF text extracted locally with pypdf, no LLM transcription

Reported as. -0.71429 (+/- 0.00001) per site in S.S units

Read from Table 5 of arXiv:2206.14307 (J1-J2 on 6x6 and 10x10 square lattices, PBC, E/N in S.S units), extracted from the PDF with pypdf.

-0.71351(1), VMC (CNN)

Checked 2026-09-14 · arXiv PDF text extracted locally with pypdf, no LLM transcription

Reported as. -0.71351 (+/- 0.00001) per site in S.S units

Read from Table 5 of arXiv:2206.14307 (J1-J2 on 6x6 and 10x10 square lattices, PBC, E/N in S.S units), extracted from the PDF with pypdf.

-0.7130465, DMRG

Checked 2026-09-29 · scripts/J1J2/square_36_P_1/dmrg.sh + programs/dmrg_itensors (heisenberg_2d.jl, args.jl, common.jl; ITensors 0.6.16); varbench/methods@ed31bb0 (shallow clone), VarBench data history (varbench-history, partial clone at 390a21e)

Reported as. julia --project heisenberg_2d.jl --peri --L 6 --J2 1 --J22 1 --zero_mag --max_B 2048 --seed 123

Script builds the 6x6 PBC J1-J2 at J2 = J22 = 1 (Pauli, no Marshall rotation, no shift), S^z = 0, max bond dimension 2048 = method string, 50 sweeps, cutoff 1e-12, seed 123; final dmrg() energy: variational at stated bond dimension (no extrapolation), sigma not applicable. 1.892e-1 above the ED row; variance within the Bhatia-Davis bound (ratio 0.154). History: entered 300eb2e (2022-10-17, Dian, 'Add ED and DMRG for J1-J2 L=6 scanning J2'); unchanged since; code link added f88f9ba (2024-07-19).

-0.712413, 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. -0.712413

Table I, row J2=1.00, 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.

-0.70921(3), VMC (RBM)

Checked 2026-09-29 · scripts/J1J2/square_36_P_1/vmc_rbm.sh + programs/vmc_netket (vmc.py, args_parser.py, ham.py), NetKet 3.13.0 source; varbench/methods@ed31bb0 (shallow clone), VarBench data history (varbench-history, partial clone at 390a21e)

Reported as. python3 vmc.py --ham j1j2 --boundary peri --sign mars --ham_dim 2 --L 6 --J2 1 --zero_mag --net rbm --seed 123 --show_progress

Script builds the 6x6 PBC J1-J2 at J2 = 1 (NetKet Heisenberg, Pauli, J = [1, 1], Marshall rotation on J1 bonds only: isospectral), S^z = 0, MetropolisExchange; nk.models.RBM(alpha = 1, log_cosh) with real float32 parameters; 10^4 SR steps, then a 2^20-sample estimate: stored sigma / sqrt(Var/2^20) = 1.02, i.e. the NetKet error_of_mean of that estimate; variance = Var(H) total. Variance within the Bhatia-Davis bound (ratio 0.112 of (E-E0)(Emax-E)). No constant shift in the Hamiltonian (NetKet bond op sz_sz +- exchange), so the M2 gap to the record is the ansatz's: real parameters give positive amplitudes in the Marshall basis. Upstream value changes are listed in the history below; the stored value is the latest. History: entered 300eb2e (2022-10-17, Dian, 'Add ED and DMRG for J1-J2 L=6 scanning J2'); value history: 300eb2e 2022-10-17 'Add ED and DMRG for J1-J2 L=6 scanning J2': E= S=- V=- -> 08e1899 2022-10-17 'Add RBM and Jastrow for J1-J2 L=6 scanning J2': E=-102.12601 S=0.0045 V=20.4796; code link added 5047bfd (2024-07-30).

-0.70016(3), VMC (Jastrow)

Checked 2026-09-29 · scripts/J1J2/square_36_P_1/vmc_jastrow.sh + programs/vmc_netket (vmc.py, args_parser.py, ham.py), NetKet 3.13.0 source; varbench/methods@ed31bb0 (shallow clone), VarBench data history (varbench-history, partial clone at 390a21e)

Reported as. python3 vmc.py --ham j1j2 --boundary peri --sign mars --ham_dim 2 --L 6 --J2 1 --zero_mag --net jas --seed 123 --show_progress

Script builds the 6x6 PBC J1-J2 at J2 = 1 (NetKet Heisenberg, Pauli, J = [1, 1], Marshall rotation on J1 bonds only: isospectral), S^z = 0, MetropolisExchange; nk.models.Jastrow with real float32 parameters; 10^4 SR steps, then a 2^20-sample estimate: stored sigma / sqrt(Var/2^20) = 1.02, i.e. the NetKet error_of_mean of that estimate; variance = Var(H) total. Variance within the Bhatia-Davis bound (ratio 0.051 of (E-E0)(Emax-E)). No constant shift in the Hamiltonian (NetKet bond op sz_sz +- exchange), so the M2 gap to the record is the ansatz's: real parameters give positive amplitudes in the Marshall basis. Upstream value changes are listed in the history below; the stored value is the latest. History: entered 300eb2e (2022-10-17, Dian, 'Add ED and DMRG for J1-J2 L=6 scanning J2'); value history: 300eb2e 2022-10-17 'Add ED and DMRG for J1-J2 L=6 scanning J2': E= S=- V=- -> 08e1899 2022-10-17 'Add RBM and Jastrow for J1-J2 L=6 scanning J2': E=-100.82252 S=0.0050 V=25.3075; code link added fdb4bfb (2024-07-30).

Coverage

Newest published number here: 2026. Literature last checked 2026-09-15 (table harvest of cached arXiv sources (match_tables.mjs --all), every matched paper read for this instance): 5 papers read, 0 of them with an energy on this page.

This instance as data

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