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All instances / Hubbard

Hubbard square 4x4, U = 7.74263683, n = 0.625

Record

-1.1002332 E/site

ED

Moreno et al. (2022) · exact energy: the instance is solved, and this is the state of the art on it (rules §6)

The Hamiltonian

H = −t ∑⟨ij⟩,σ (cc + h.c.) + U ∑i ni↑ni↓

t = 1, U = 7.74263683, N = N = 5 on 16 sites.

Upstream this instance is Hubbard/square_16_P_5_7.74264. The name's 7.74264 rounds the coupling of this Hamiltonian, 7.74263683 (10^(8/9)). The exact diagonalization, -17.60373155517909, is the ground state there; at U = 7.74264 the ground state is 1.18e-6 higher, so the stored exact energy sat below the ground state of the instance as named. Evidence: checks/hubbard-u-labels.

lattice
square
sites
16
boundary
periodic
Nf
5
U
7.74263683
energies
2

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 (1)

Where the instance is not solved, the record is the lowest eligible energy in this group; where it is, the lowest eligible energy here is the best variational bound and the closest challenger to the exact one.

E/siteσVar(E)V-score methodsourceyear
-1.1001250(6) 6.3e-7 3.70e-2 4.2e-4 HFDS (10 hidden fermions, 1-layer FFN with α = 64, C4 and K = 0 projections) 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, and therefore the record wherever one exists. QMC is exact only within its statistical error bar, which it must state, and its rows read exact (stochastic); an exact diagonalization outranks it. Sector-resolved diagonalizations state the lowest energy in one symmetry sector and do not hold it.

E/siteσVar(E)V-score methodsourceyear
-1.1002332record n/a n/a n/a ED Moreno et al. (2022) 2022

How these numbers were read

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

-1.1002332 — ED

Checked 2026-09-16 · recomputed: Lanczos ED over the full fixed-(N_up, N_dn) space by two independent codes (checks/hubbard-u-labels/)

Reported as. -17.60373155517896 (ed_check.py) | -17.603731555119207 (ed_lib.mjs)

Reproduced at the grid U the instance now carries, not at the U of VarBench's name (relabels.mjs).

Coverage

Newest published number here: 2022. Nothing has been added to this instance since the VarBench snapshot of 2024-10-22. 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

JSON · source file on GitHub · open an issue about this instance