The Hamiltonian
H = −t ∑⟨ij⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t = 1, U = 8, N↑ = N↓ = 18 on 36 sites.
- lattice
- square
- sites
- 36
- boundary
- periodic
- Nf
- 18
- U
- 8
- energies
- 6
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 (5)
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.
-0.5278(6), 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 = -19.00(2) (total), 6 x 6, 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 6 x 6, column U = 8 PBC. Half filling: N_up = N_dn = 18. 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.
-0.5266, 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. -0.5266
Table II, row '6 × 6', column 'U = 8, K = 2' | Consistent with instance exact_per_site -0.5277777778 (own value lies above it, as expected variationally).
-0.5257, 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. -0.5257
Table III, n=1 block, 6x6 row, BW2 column | same as BW1 row above
-0.5255, 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. -0.5255
Table II, row '6 × 6', column 'U = 8, K = 1' | Consistent with instance exact_per_site -0.5277777778 (own value lies above it, as expected variationally).
-0.5186, 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. -0.5186
Table III, n=1 block, 6x6 row, BW1 column | 36 sites, Nf=18 (n=2*18/36=1, half filling), P boundary confirmed by caption 'Fermi-Hubbard model on rectangular lattices with PBC'; U=8 confirmed since the same row's AFQMC value matches instance exact_per_site -0.5277777778
-0.4797, 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. -0.4797
Table II, row '6 × 6', column 'U = 8, K = 0 (HF)' | Same row's quoted QMC value (-0.5278(3)) matches instance exact_per_site -0.5277777778 to displayed precision.