The Hamiltonian
H = −t ∑⟨ij⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t = 1, U = 8, N↑ = N↓ = 50 on 100 sites.
- lattice
- square
- sites
- 100
- boundary
- periodic
- Nf
- 50
- U
- 8
- energies
- 7
Stored as a total energy; quoted here as
E/site, 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.
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.
How these numbers were read
Where each number below was read, and how it was checked.
-0.5254(3), AFQMC
Checked 2026-09-15 · arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription
Reported as. E = -52.54(3) (total), 10 x 10, U = 8, PBC
Table IV, "Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit." Row 10 x 10, column U = 8 PBC. Half filling: N_up = N_dn = 50. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.
-0.52492, VMC (transformer BF)
Checked 2026-09-15 · source text of arXiv:2507.02644 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. -0.52492
Table S1: Benchmark energy in pure Hubbard model at half-filling with PBC, System 10x10, NQS row: -0.52492 | Same table's AFQMC row for the identical system (-0.5254(3)) equals the instance's exact_per_site exactly; per-site E/N convention, no rescale needed.
-0.524, VMC (HB)
Checked 2026-09-15 · source text of arXiv:2606.00924 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. -0.5240
Table II, row '10 × 10', column 'U = 8, K = 2' | Consistent with instance exact_per_site -0.5254 (own value lies above it, as expected variationally).
-0.523, VMC (tensor BF)
Checked 2026-09-15 · source text of arXiv:2308.11823 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. -0.5230
Table III, n=1, 10x10 row, BW2 column | Same instance.
-0.5226, VMC (HB)
Checked 2026-09-15 · source text of arXiv:2606.00924 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. -0.5226
Table II, row '10 × 10', column 'U = 8, K = 1' | Consistent with instance exact_per_site -0.5254 (own value lies above it, as expected variationally).
-0.5181, VMC (tensor BF)
Checked 2026-09-15 · source text of arXiv:2308.11823 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. -0.5181
Table III, n=1, 10x10 row, BW1 column | Half filling, Nf=50, matches catalogue Nf=50,U=8.
-0.469, HF
Checked 2026-09-15 · source text of arXiv:2606.00924 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. -0.4690
Table II, row '10 × 10', column 'U = 8, K = 0 (HF)' | Same row's quoted QMC value (-0.5254(3)) matches instance exact_per_site -0.5254 exactly.