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Heisenberg square 10x10, open

Heisenberg/square_100_O

Record

-0.628656(9) E/N (S.S)

2D tensorized RNN (symmetry + annealing)

Hibat-Allah et al. (2022) · lowest eligible strict variational bound (rules §6)

The Hamiltonian

H = ∑⟨ij⟩ σi·σj

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

lattice
square
sites
100
boundary
open
energies
16

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 (13)

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

E/N (S.S)σVar(E)V-score methodsourceyear
-0.628656(9)record 9.0e-6 n/a n/a 2D tensorized RNN (symmetry + annealing)
quoted from another paper
 Hibat-Allah et al. (2022) 2022
-0.628649(2) 1.7e-6 1.95e-2 3.1e-5 RNN + translational symmetry
VarBench reference run
 run script, no paper cited n/a
-0.628638(1) 1.1e-6 1.94e-1 3.1e-4 2D Gated RNN Hibat-Allah et al. (2022) 2022
-0.628637(4) 4.0e-6 n/a n/a 2D minGRU (3 layers, c4v symmetry, parallel scan)
preprint
 Merali et al. (2026) 2026
-0.628627(1) 1.0e-6 n/a n/a PixelCNN (deep autoregressive)
quoted from another paper
 Sharir, Levine, Wies, Carleo & S n/a
-0.628607(3) 2.5e-6 1.10e-1 1.7e-4 RNN
VarBench reference run
 run script, no paper cited n/a
-0.6286010 ○ n/a n/a n/a PEPS (bond dimension = 10) Liu et al. (2017) 2017
-0.628601(2) 2.0e-6 n/a n/a Finite PEPS, gradient optimization
quoted from another paper
 Liu, Dong, Han, Guo & He, Gradie n/a
-0.628528(4) 3.8e-6 2.29e+0 3.6e-3 Tensor-RNN (bond dimension = 40) Wu et al. (2023) 2023
-0.627697(6) 5.5e-6 4.93e+0 7.8e-3 2D MPS-RNN (bond dimension = 40) Wu et al. (2023) 2023
-0.62587(1) 1.0e-5 1.68e+1 2.7e-2 1D MPS-RNN (bond dimension = 40) Wu et al. (2023) 2023
-0.62276(1) 1.3e-5 2.61e+1 4.2e-2 RBM (alpha = 1)
VarBench reference run
 run script, no paper cited n/a
-0.62027(1) 1.4e-5 3.17e+1 5.2e-2 Jastrow baseline
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)σVar(E)V-score methodsourceyear
-0.6286335 n/a 3.20e-1 5.1e-4 DMRG (MaxTruncError ~1.87E-6, MaxBondDim=10000, Extrapolated Energy = - 251.4628 +/- 0.0019)
VarBench reference run
 run script, no paper cited n/a

Numerically exact (1)

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)σVar(E)V-score methodsourceyear
-0.6286561(2) 2.0e-7 n/a n/a QMC (stochastic series expansion)
quoted from another paper
 Sandvik (2026) 2026

Not yet classified (1)

The method string does not say whether the energy is sign-problem-free or constrained, so no bound_type could be assigned without guessing.

E/N (S.S)σVar(E)V-score methodsourceyear
-0.6286562 ○ n/a n/a n/a QMC Lubasch et al. (2014) 2014

How these numbers were read

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

-0.6286561(2) — QMC (stochastic series expansion)

Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription

Reported as. -0.6286561 (+/- 2e-7) per site in S.S units

Read from Table 2 of arXiv:2605.13807 (2D Heisenberg, square lattice, OBC, E/N in S.S units), parsed from the arXiv HTML. Column "QMC [49]" = arXiv:2601.20189. The square-lattice Heisenberg AFM is bipartite and sign-problem-free, so SSE QMC is numerically exact here (RULES.md 4).

-0.628656(9) — 2D tensorized RNN (symmetry + annealing)

Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription

Reported as. -0.628656 (+/- 0.000009) per site in S.S units

Read from Table 2 of arXiv:2605.13807 (2D Heisenberg, square lattice, OBC, E/N in S.S units), parsed from the arXiv HTML. Column "2D TRNN [40]". The same value appears in arXiv:2502.17144 Table 5 as "This work (best RNN)", OBC 10x10. It sits 1e-7 below the QMC reference, well inside its own 9e-6 error bar.

-0.628637(4) — 2D minGRU (3 layers, c4v symmetry, parallel scan)

Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription

Reported as. -0.628637 (+/- 0.000004) per site in S.S units

Read from Table 2 of arXiv:2605.13807 (2D Heisenberg, square lattice, OBC, E/N in S.S units), parsed from the arXiv HTML. Column "2D minGRU (ours)".

-0.628627(1) — PixelCNN (deep autoregressive)

Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription

Reported as. -0.628627 (+/- 0.000001) per site in S.S units

Read from Table 2 of arXiv:2605.13807 (2D Heisenberg, square lattice, OBC, E/N in S.S units), parsed from the arXiv HTML. Column "PixelCNN [52]" = Phys. Rev. Lett. 124, 020503 (2020).

-0.628601(2) — Finite PEPS, gradient optimization

Checked 2026-09-14 · arXiv HTML parsed locally from a benchmark comparison table, no LLM transcription

Reported as. -0.628601 (+/- 0.000002) per site in S.S units

Read from Table 2 of arXiv:2605.13807 (2D Heisenberg, square lattice, OBC, E/N in S.S units), parsed from the arXiv HTML. Column "PEPS [33]" = Phys. Rev. B 95, 195154 (2017). Finite PEPS on the 10x10 lattice, not an infinite-lattice value.

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