Hubbard square 8x8, periodic/antiperiodic, U = 2, n = 1

Exact energy and best values (rules §6)

exact (stochastic)	-1.19231(5)	AFQMC Qin et al. (2016) · sign-problem-free QMC, exact within its error bar
best variational	-1.19003(9)	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 = 2, N_↑ = N_↓ = 32 on 64 sites.

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

E/site	distance to class best	σ	Var(E)	V-score	method	source	code	year	comment
-1.19231exact (stochastic)	0	4.7e-5	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
-1.19003best variational	0	9.4e-5	1.40e-1	7.7e-4	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.

-1.19231(5), 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 = -76.308(3) (total), 8 x 8, 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 8 x 8, column U = 2 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.

-1.19003(9), VMC (HFDS)

Checked 2026-09-29 · PNAS 119(32), e2122059119 (2022), SI Appendix PDF (PMC9371695; sources/pnas.2122059119.sapp.PMC9371695.pdf, sha256 c6e62d9c...), pypdf layout mode, re-extracted 2026-09-29 (byte-identical to the earlier extraction); same cell read in arXiv:2111.10420v2 (18 Jun 2022) Appendix H (sources/2111.10420v2.pdf, pypdf layout)

Reported as. -1.1900(2) (energy per site, units of t)

Record row (the only variational row, hence the best variational bound; the instance record is the sign-free AFQMC -76.308(3), Qin, Shi & Zhang, PRB 94, 085103, Table IV, PBC-APBC, 0.146 below). Energy verified; sigma differs (ambiguous entry). 8 x 8, 'periodic boundary conditions along one of the sides of the square and anti-periodic boundary conditions along the other side' (SI Sec. 8 D and Sec. 6) = PA; half filling 32 + 32, U = 2, t = 1, no shift (einf 32). N~ = 16 and alpha = 1 for 8 x 8 (SI Sec. 6): method string right; the 'Soft mean-field constraint for Neel order' is the author's upload wording, not stated in the paper and not contradicted by it. SI Fig. 4 (vector marks, re-read 2026-09-29) plots this point at relative error 1.917e-3 against -76.308, i.e. -76.1617 total. Upload c810bdc (jrm874, 2022-07-16): -76.162 | 0.006 | 0.14(2), never changed; energy_variance upload-only. M6/M8 settled from the source: a sampled VMC energy ('General expectation values and gradients of the objective function are computed using Markov chain Monte Carlo sampling', main text Sec. 2 C, p. 5) whose stored sigma is the uploader's Monte Carlo error; units.mjs counts the row deterministic only because 'Soft mean-field constraint' matches /mean[- ]field/ (units.mjs line 20), which changes nothing while sigma is stored (isSampled is consulted only when sigma is null). Venue PNAS, peer reviewed ('Edited by Angel Rubio ... received December 6, 2021; accepted June 28, 2022'; 'This article is a PNAS Direct Submission'); the imported row carries no peer_reviewed field (loader-level, see report). No duplicate on the instance (its other row is AFQMC); the identical number on square_36_PA_18_2 is the copied upload flagged there.

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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