Hubbard square 6x6, periodic/antiperiodic, U = 2, n = 1

Hubbard/square_36_PA_18_2

Record

-1.2079(1) E/site

VMC Hidden Fermion Determinant State Ansatz (N_hidden = 8, fully parametrized hidden sub-matrix, hidden-unit density alpha = 78)

Moreno et al. (2022) · 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 = 2, N_↑ = N_↓ = 18 on 36 sites.

lattice: square
sites: 36
boundary: periodic/antiperiodic
Nf: 18
U: 2
energies: 3

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
-2.1156(2)	1.7e-4	1.40e-1	5.7e-4	VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel order. flagged: wrong-instance	Moreno et al. (2022)	2022
-1.2079(1)record	1.0e-4	n/a	n/a	VMC Hidden Fermion Determinant State Ansatz (N_hidden = 8, fully parametrized hidden sub-matrix, hidden-unit density alpha = 78) peer reviewed	Moreno et al. (2022)	2022

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
-1.20831(6)	5.6e-5	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

Flagged rows

A flag withholds the record and nothing else: the row stays listed, in rank order. Rules §10 is how a flag is lifted or upheld.

wrong-instance — VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16...., -2.1156(2)

Finding. Energy, sigma and variance (-76.162, 0.006, 0.14(2)) are identical to the 8x8 row on square_64_PA_32_2. The HFDS first author uploaded them to the 6x6 file 67 minutes after the 8x8 file (VarBench commits c810bdcd22, then 21f00e54f6), and they never changed. -76.162 / 64 = -1.19003 is the paper's 8x8, U = 2 value, -1.1900(2) (PNAS SI Table 5, arXiv SI Table V). As a 36-site total it is -2.1156 per site: 16.45 below the non-interacting ground state of this lattice, -59.712813, which a U >= 0 Hamiltonian cannot go below, and 32.7 below the sign-free AFQMC total -43.499(2) (Qin, Shi & Zhang, PRB 94, 085103 (2016), Table IV). The paper's 6x6 value, -1.2079(1), is carried as a separate row.

How these numbers were read

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

-1.20831(6) — 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 = -43.499(2) (total), 6 x 6, 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 6 x 6, column U = 2 PBC-APBC. Half filling: N_up = N_dn = 18. 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.

-1.2079(1) — VMC Hidden Fermion Determinant State Ansatz (N_hidden = 8...

Checked 2026-09-15 · PNAS supplementary PDF (via Europe PMC, PMC9371695) extracted locally, cross-read against the arXiv version; no LLM transcription

Reported as. -1.2079(1) per site, no variance

PNAS SI Table 5 (identical to arXiv SI Table V), "Variational energy per site in the L x L Hubbard model at half filling with periodic boundary conditions along one of the sides of the square and anti-periodic boundary conditions along the other side", row L = 6, column U = 2. SI Sec. 6: N_hidden = 8, alpha = 78 for 6x6; no projection or constraint stated, nor what the error bar is. Column alignment checked: the VarBench HFDS rows for 6x6 at U = 4, 6, 8 and 8x8 at U = 2, 4, 8 match the printed digits. As a total, -43.4844 sits 3.4e-4 (relative) above the sign-free AFQMC value -43.499(2) for this lattice (Qin, Shi & Zhang, PRB 94, 085103 (2016), Table IV, PBC-APBC) and above the non-interacting bound -59.712813.

Coverage

Newest published number here: 2022. 2 rows 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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