Heisenberg square 6x6

Heisenberg/square_36_P

Record

-0.67887(2) E/N (S.S)

2D RNN wavefunction (best variational)

Moss et al. (2025) · lowest eligible strict variational bound (rules §6)

The Hamiltonian

H = ∑_⟨ij⟩ σ_i·σ_j

Nearest-neighbour Heisenberg antiferromagnet, J = 1, written with Pauli matrices.

lattice: square
sites: 36
boundary: periodic
energies: 9

Stored as a total energy in the convention of VarBench; 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. Ranking happens only inside the first group.

Strict variational upper bounds (6)

The record is the lowest eligible energy in this group, and only this group.

E/N (S.S)	σ	Var(E)	V-score	method	source	year
-0.67887(2)record	2.0e-5	n/a	n/a	2D RNN wavefunction (best variational) peer reviewed	Moss et al. (2025)	2025
-0.678868(2)	2.0e-6	n/a	n/a	RBM + Lanczos recursion	Chen et al. (2022)	2022
-0.67882(1)	1.0e-5	n/a	n/a	CNN quoted from another paper	Choo, Neupert & Carleo, Two-dime	n/a
-0.6786224	n/a	1.39e+0	5.2e-3	DMRG (bond dimension = 2048) VarBench reference run	run script, no paper cited	n/a
-0.66974(3)	2.6e-5	1.47e+1	5.7e-2	Jastrow baseline VarBench reference run	run script, no paper cited	n/a
-0.66854(3)	2.6e-5	1.39e+1	5.4e-2	RBM (alpha = 1) VarBench reference run	run script, no paper cited	n/a

Extrapolations (1)

Zero-variance, bond-dimension or Trotter-error extrapolations. Not a bound: no ansatz ever reached the number, so it cannot hold the record.

E/N (S.S)	σ	Var(E)	V-score	method	source	year
-0.67887177(7)	7.0e-8	n/a	n/a	2D RNN wavefunction, zero-variance extrapolation peer reviewed	Moss et al. (2025)	2025

Numerically exact (2)

Exact diagonalization, an exact solution, or sign-problem-free QMC where that is established: the answer, not a claim about it.

E/N (S.S)	σ	Var(E)	V-score	method	source	year
-0.67887215(2)	2.0e-8	n/a	n/a	SSE QMC (stochastic series expansion), T -> 0 converged at beta/L = 32 and 64 preprint	Sandvik (2026)	2026
-0.6788721	n/a	n/a	n/a	Exact diagonalization VarBench reference run	run script, no paper cited	n/a

How these numbers were read

Rows added after the VarBench import record where the number came from.

-0.67887215(2) — SSE QMC (stochastic series expansion), T -> 0 converged at...

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.

-0.67887177(7) — 2D RNN wavefunction, zero-variance extrapolation

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).

-0.67887(2) — 2D RNN wavefunction (best variational)

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)".

-0.678868(2) — RBM + Lanczos recursion

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) — 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).

Coverage

Newest published number here: 2026. 5 rows added from the 2026-09 literature sweep, on top of the VarBench snapshot of 2024-10-22.

This instance as data

JSON · source file on GitHub · open an issue about this instance