Hubbard square 16x16, U = 2, n = 1

Exact energy and best values (rules §6)

exact (stochastic)	-1.17420(2)	AFQMC Qin et al. (2016) · sign-problem-free QMC, exact within its error bar

Energies as E/site. Classes are different claims and are not ranked against each other.

The Hamiltonian

H = −t ∑_⟨ij⟩,σ (c^†_iσc_jσ + h.c.) + U ∑_i n_i↑n_i↓

t = 1, U = 2, N_↑ = N_↓ = 128 on 256 sites.

lattice: square
sites: 256
boundary: periodic
Nf: 128
U: 2
energies: 1

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.

E/site	distance to class best	σ	Var(E)	V-score	method	source	code	year	comment
-1.17420exact (stochastic)	0	2.3e-5	n/a	n/a	AFQMC peer reviewed	Qin et al. (2016)		2016

How these numbers were read

Where each number below was read, and how it was checked.

-1.17420(2), 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 = -300.596(6) (total), 16 x 16, U = 2, 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 16 x 16, column U = 2 PBC. Half filling: N_up = N_dn = 128. 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.

Coverage

Newest published number here: 2016.

This instance as data

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