Hubbard square 8x8, U = 4, n = 1

Hubbard/square_64_P_32_4

Record

-0.859156(8) E/site

mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos, (U=4), alpha = 4

run script, no paper cited · 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 = 4, N_↑ = N_↓ = 32 on 64 sites.

lattice: square
sites: 64
boundary: periodic
Nf: 32
U: 4
energies: 5

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
-0.859156(8)record	7.7e-6	4.70e-1	2.1e-3	mVMC with SU(2) and momentum projections (gamma point) + RBM + Lanczos, (U=4), alpha = 4 VarBench reference run	run script, no paper cited	n/a
-0.85651(1)	1.1e-5	8.63e-1	3.9e-3	VMC with Neel AF (+Jastrow and backflow) VarBench reference run	run script, no paper cited	n/a

Projected or fixed-node estimates (1)

Variational only within a constraint: fixed-node, constrained-path, GFMC on a trial state. Node- or constraint-dependent, so not cleanly comparable to each other or to the group above, and never the record.

E/site	σ	Var(E)	V-score	method	source	year
-0.857653(9)	9.4e-6	n/a	n/a	FN on the state above VarBench reference run	run script, no paper cited	n/a

Numerically exact (2)

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
-0.86036(6)	6.3e-5	n/a	n/a	AFQMC (Metropolis, Trotter error extrapolated), numerically exact	Qin et al. (2016)	2016
-0.8602(2)	1.6e-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

How these numbers were read

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

-0.8602(2) — 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 = -55.05(1) (total), 8 x 8, U = 4, 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 8 x 8, column U = 4 PBC. 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.

Coverage

Newest published number here: 2016. 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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