Hubbard square 16x16, periodic/antiperiodic, U = 8, n = 1

Hubbard/square_256_PA_128_8

Record

-0.52427(4) E/site

VAFQMC

Sorella (2023) · lowest eligible strict variational bound (rules §6)

The Hamiltonian

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

t = 1, U = 8, N_↑ = N_↓ = 128 on 256 sites.

lattice: square
sites: 256
boundary: periodic/antiperiodic
Nf: 128
U: 8
energies: 4

Stored as a total energy in the convention of VarBench; quoted here as E/site, which is what the papers report (how the conversion works).

Every published energy

Grouped by what the number is. Ranking happens only inside the first group.

Strict variational upper bounds (2)

The record is the lowest eligible energy in this group, and only this group.

E/site	σ	Var(E)	V-score	method	source	year
-0.52427(4)record	3.6e-5	1.00e+0	6.1e-4	VAFQMC	Sorella (2023)	2023
-0.51898(2)	1.7e-5	8.10e+0	5.0e-3	mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos, (U=8) VarBench reference run	run script, no paper cited	n/a

Numerically exact (1)

Exact diagonalization, an exact solution, or sign-problem-free QMC where that is established: the answer, not a claim about it.

E/site	σ	Var(E)	V-score	method	source	year
-0.5244(1)	1.2e-4	n/a	n/a	AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error) peer reviewed	Qin, Shi & Zhang, Benchmark stud	n/a

Not yet classified (1)

The method string does not say whether the energy is sign-problem-free or constrained, so no bound_type could be assigned without guessing.

E/site	σ	Var(E)	V-score	method	source	year
-0.5244(1)	1.2e-4	n/a	n/a	AFQMC	Qin et al. (2016)	2016

How these numbers were read

Rows added after the VarBench import record where the number came from.

-0.5244(1) — AFQMC, sign-problem-free at half filling (tau = 0.01...

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 = -134.25(3) (total), 16 x 16, 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 16 x 16, column U = 8 PBC-APBC. 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: 2023. 1 row added from the 2026-09 literature sweep, on top of the VarBench snapshot of 2024-10-22.

This instance as data

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