The Hamiltonian
H = −t ∑⟨ij⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t = 1, U = 7.74263683, N↑ = N↓ = 4 on 16 sites.
Upstream this instance is Hubbard/square_16_P_4_7.74264. The name's 7.74264 rounds the coupling of this Hamiltonian, 7.74263683 (10^(8/9)). The exact diagonalization, -16.509154825265142, is the ground state there; at U = 7.74264 the ground state is 6.1e-7 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
- 4
- U
- 7.74263683
- 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)
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.
-1.0318222 — 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. -16.50915482526524 (ed_check.py) | -16.509154825269405 (ed_lib.mjs)
Reproduced at the grid U the instance now carries, not at the U of VarBench's name (relabels.mjs).
-1.0318221 — DMRG (bond dimension ~3200)
Checked 2026-09-16 · run script varbench/methods programs/dmrg_itensors_hubbard/square_16_P_4_7.74264.jl read in full
Reported as. U = 10
The linked script is a copy of the U = 10 file, so it did not produce this number. The energy sits 1.2e-7 above the ground state at U = 7.74264 and 7.3e-7 above the one at 7.74263683: a valid bound at either, and too close to both to say which it was run at, so it stays on this instance.