The Hamiltonian
H = −t ∑⟨ij⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t = 1, U = 2, N↑ = N↓ = 18 on 36 sites.
- lattice
- square
- sites
- 36
- boundary
- periodic/antiperiodic
- Nf
- 18
- U
- 2
- energies
- 2
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 (1)
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.
-1.20831(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 = -43.499(2) (total), 6 x 6, U = 2, PBC-APBC
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 = 2 PBC-APBC. 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.
-1.2079(1), VMC (HFDS)
Checked 2026-09-15 · PNAS supplementary PDF (via Europe PMC, PMC9371695) extracted locally, cross-read against the arXiv version; no LLM transcription
Reported as. -1.2079(1) per site, no variance
PNAS SI Table 5 (identical to arXiv SI Table V), "Variational energy per site in the L x L Hubbard model at half filling with periodic boundary conditions along one of the sides of the square and anti-periodic boundary conditions along the other side", row L = 6, column U = 2. SI Sec. 6: N_hidden = 8, alpha = 78 for 6x6; no projection or constraint stated, nor what the error bar is. Column alignment checked: the VarBench HFDS rows for 6x6 at U = 4, 6, 8 and 8x8 at U = 2, 4, 8 match the printed digits. As a total, -43.4844 sits 3.4e-4 (relative) above the sign-free AFQMC value -43.499(2) for this lattice (Qin, Shi & Zhang, PRB 94, 085103 (2016), Table IV, PBC-APBC) and above the non-interacting bound -59.712813.