All instances / Hubbard
Hubbard square 4x4, U = 3.59381366, n = 0.625
The Hamiltonian
H = −t ∑⟨ij⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t = 1, U = 3.59381366, N↑ = N↓ = 5 on 16 sites.
Upstream this instance is Hubbard/square_16_P_5_3.5981. The name's 3.5981 has two digits swapped: the coupling of this Hamiltonian is 3.59381366 (10^(5/9)). The exact diagonalization, -19.891637199948672, is the ground state there; at U = 3.5981 the ground state is 3.41e-3 higher, and the HFDS row would sit 369 sigma below it. Evidence: checks/hubbard-u-labels.
- lattice
- square
- sites
- 16
- boundary
- periodic
- Nf
- 5
- U
- 3.59381366
- 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.
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.
How these numbers were read
Rows added after the VarBench import record where the number came from.
-1.2432273 — 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. -19.891637199948637 (ed_check.py) | -19.89163719987478 (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.