The Hamiltonian
H = ∑⟨ij⟩ σi·σj + J2 ∑⟨⟨ij⟩⟩ σi·σj
J1 = 1, J2 = 0.2, nearest and next-nearest neighbours, written with Pauli matrices.
- lattice
- square
- sites
- 36
- boundary
- periodic
- J2
- 0.2
- energies
- 4
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 (3)
Where the instance is not solved, the record is the lowest eligible energy in this group; where it is, the lowest eligible energy here is the best variational bound and the closest challenger to the exact one.
Numerically exact (1)
Exact diagonalization, an exact solution, or sign-problem-free QMC where that is established: the answer, not a claim about it, and therefore the record wherever one exists. QMC is exact only within its statistical error bar, which it must state, and its rows read exact (stochastic); an exact diagonalization outranks it. Sector-resolved diagonalizations state the lowest energy in one symmetry sector and do not hold it.
How these numbers were read
Rows added after the VarBench import record where the number came from.
-0.5990460 — ED
Checked 2026-09-28 · source text of arXiv:cond-mat/9402061 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. -0.599046
Table II, row J2 = 0.20, column 36(A1). QMBL's Lanczos (qmbl-runs/qmbl-ed-2026-09-28: E2's ed_sym.py, C4v one-dimensional sectors with even spin at k = 0 (A1, B1)) gives -0.599046322 per site, so the printed value is the correctly rounded ground-state energy. Golubev et al. (arXiv:2606.04558 Table I, 6x6 ED) print -0.599046, also correctly rounded | Ground-state energy per site, J1 = 1, S.S units; the same column reproduces QMBL's exact 36-site rows at J2 = 0, 0.3, 0.4, 0.5, 0.6 to 1e-6.
-0.5990440 — ViT
Checked 2026-09-28 · 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. -0.599044
Table I, row J2=0.20, column '6×6, ViT (this work)' | Table I prints energy per site in S.S units: its J2 = 0 QMC column equals Sandvik's exact values on Heisenberg/square_64_P and square_100_P.
-0.599044(3) — RBM
Checked 2026-09-28 · source text of arXiv:2206.14307 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. 599044(3)
Table 5, row 'Energy(RBM)', 6x6 J2 = 0.2 | Energy per site E/N in S.S units (table captions); the 6x6 exact column equals the exact rows QMBL carries.
-0.59895(1) — CNN
Checked 2026-09-28 · source text of arXiv:2206.14307 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. -0.59895(1)
Table 5, row 'Energy(CNN)', 6x6 J2 = 0.2 | Energy per site E/N in S.S units (table captions); the 6x6 exact column equals the exact rows QMBL carries.