Hubbard square 8x8, periodic/antiperiodic, U = 8, n = 1

Exact energy and best values (rules §6)

exact (stochastic)	-0.5259(3)	AFQMC Qin et al. (2016) · sign-problem-free QMC, exact within its error bar
best variational	-0.5244(2)	VMC (HFDS) Moreno et al. (2022) · the lowest unflagged variational energy, a strict variational bound

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 = 8, N_↑ = N_↓ = 32 on 64 sites.

lattice: square
sites: 64
boundary: periodic/antiperiodic
Nf: 32
U: 8
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.

E/site	distance to class best	σ	Var(E)	V-score	method	source	code	year	comment
-0.5259exact (stochastic)	0	3.1e-4	n/a	n/a	AFQMC peer reviewed	Qin et al. (2016)		2016

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.

E/site	distance to class best	σ	Var(E)	V-score	method	source	code	year	comment
-0.5244best variational	0	1.6e-4	8.00e-1	2.0e-3	VMC (HFDS)	Moreno et al. (2022)	uses NetKet	2022

How these numbers were read

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

-0.5259(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 = -33.66(2) (total), 8 x 8, U = 8, 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 8 x 8, column U = 8 PBC-APBC. Half filling: N_up = N_dn = 32. 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.

-0.5244(2), VMC (HFDS)

Checked 2026-10-15 · published SI Appendix PDF, PNAS 119(32), e2122059119 (2022) (Europe PMC PMC9371695 supplementary bundle; pnas.org PDF and Europe PMC render return Cloudflare 403), sources/journal/pnas119-e2122059119-si.pdf, sha256 c6e62d9c (byte-identical to the SI read 2026-09-29), pypdf layout + plain; main text: published full-text XML (Europe PMC), sources/journal/pnas119-e2122059119.fulltext.xml, sha256 761defc3; arXiv compared: 2111.10420v2 (Appendix H, via the 2026-09-29 extraction)

Reported as. -0.5244(6) (energy per site, units of t)

8 x 8 PA, 32 + 32, U = 8, N~ = 16, alpha = 1. Journal = arXiv v2. Printed bar 0.0384 vs stored 0.01: Stored sigma is the first author's VarBench upload (Robledo Moreno, 2022-07/08), not the printed bar; kept under the 2026-10-02 handoff ruling R4/E2 ('the author's finer uploaded sigma stays where the printed bar is display rounding (HFDS x12)'), so no new correction. Venue PNAS, peer reviewed ('Edited by Angel Rubio ... received December 6, 2021; accepted June 28, 2022').

Coverage

Newest published number here: 2022. Literature last checked 2026-09-15 (citation screen of Robledo Moreno et al. 2022 (PNAS, HFDS): citing works from OpenAlex and Semantic Scholar (scripts/cited_by.mjs)): 29 papers read, 0 of them with an energy on this page.

This instance as data

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