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J1-J2 square 8x8, J2 = 0.5

J1J2/square_64_P_0.5

Record

-0.498963(2) E/N (S.S)

RBM+PP with momentum (K=0), spin-parity (even S), and point-group (A1) projections, 16 hidden units

Nomura & Imada (2021) · lowest eligible strict variational bound (rules §6)

The Hamiltonian

H = ∑⟨ij⟩ σi·σj + J2⟨⟨ij⟩⟩ σi·σj

J1 = 1, J2 = 0.5, nearest and next-nearest neighbours, written with Pauli matrices.

lattice
square
sites
64
boundary
periodic
J2
0.5
energies
8

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 (8)

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

E/N (S.S)σVar(E)V-score methodsourceyear
-0.5001(1) 1.0e-4 1.43e+0 5.6e-3 Holographic Quantum Transformer (HQT)
flagged: energy-variance-inconsistent peer reviewed
Guo et al. (2026) 2026
-0.498963(2)record 1.9e-6 2.50e-1 9.8e-4 RBM+PP with momentum (K=0), spin-parity (even S), and point-group (A1) projections, 16 hidden units Nomura & Imada (2021) 2021
-0.498666(2) 2.3e-6 1.85e+0 7.3e-3 RBM with momentum (K=0), spin-parity (even S), and point-group (A1) projections, 96 hidden units Nomura (2021) 2021
-0.49857(5) 5.0e-5 1.21e+0 4.7e-3 ClebschTree Vieijra & Nys (2021) 2021
-0.49654(2) 1.6e-5 6.92e+0 2.7e-2 VMC with projected BCS (Z2 spin liquid)
VarBench reference run
run script, no paper cited n/a
-0.4940154 n/a 8.65e+0 3.5e-2 DMRG (bond dimension = 1024)
VarBench reference run
run script, no paper cited n/a
-0.48270(2) 1.6e-5 1.72e+1 7.2e-2 RBM (alpha = 1)
VarBench reference run
run script, no paper cited n/a
-0.47459(3) 2.6e-5 4.58e+1 2.0e-1 Jastrow baseline
VarBench reference run
run script, no paper cited n/a

Flagged rows

A flag withholds the record and nothing else: the row stays listed, in rank order. Rules §10 is how a flag is lifted or upheld.

energy-variance-inconsistent — Holographic Quantum Transformer (HQT), -0.5001(1)

Finding. Claims E/N = -0.5001, below RBM+PP's -0.4989635, while reporting a variance that gives a V-score of 5.6e-3 against RBM+PP's 9.81e-4 - 5.7x worse. Taken at face value a state further from an eigenstate should not be lower in energy: by the V-score calibration (rel. err ~ V/63) this energy would sit ~1.2e-3 ABOVE where it is reported. That is grounds for objection, not proof of error. The inference assumes the lower-variance competitor sits near the ground state; a state pinned near a competing, excited configuration can have a small variance and a large energy error (RULES.md 9, the HFDS stripe case). Nothing suggests RBM+PP is such a state here, but the variance alone cannot rule it out.

Diagnosis. The reported energy is inconsistent with the paper's OWN reported variance. At 8x8 the claimed E/N = -0.5001 beats RBM+PP's -0.4989635 while the reported variance (sigma^2 = 1.4e-3 per site, S.S units) gives a V-score of 5.6e-3 against RBM+PP's 9.81e-4 - 5.7x worse. Energy and variance move together - a state further from an eigenstate cannot be lower in energy - so by the V-score calibration this energy should be ~1.2e-3 higher than claimed.

Ruled out. NOT 'below the exact ground state'. There is no exact reference at 8x8 or 10x10: ED for this model reaches about 6x6, and the paper correctly uses 6x6 ED (-0.5038) as its only exact anchor. Chen & Heyl's -0.497715(9) is a zero-variance extrapolation, not a bound, so a lower variational energy would only mean the extrapolation carries systematic error. The sigma^2 is per site, so the V-score is 5.6e-3 and not a range. Separately, the paper is internally inconsistent about it: Table 1 gives sigma^2 = 1.4e-3 for the 8x8 run while the J2 scan lists 0.0034 at J2 = 0.50, which would make the V-score 1.4e-2 and the contradiction larger. No variance is reported at 10x10 at all, so that row's flag rests only on it being an unclaimed record.

Evidence. arXiv:2607.00398 Tables 1-3; V-scores recomputed from this instance's own rows.

How these numbers were read

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

-0.5001(1) — Holographic Quantum Transformer (HQT)

Checked 2026-09-13 · arXiv HTML parsed locally, no LLM transcription

Reported as. -0.5001 (+/- 0.0001) per site in S.S units

Abstract: "HQT reaches a ground-state energy per site of -0.5001(1)" on 8x8 at J2=0.5. That is 1.1e-3 below the best known variational energy for this instance (RBM+PP, -0.4989635), on a well-studied lattice. See the defect flag.

Coverage

Newest published number here: 2026. 1 row added from the 2026-09 literature sweep, on top of the VarBench snapshot of 2024-10-22.

This instance as data

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