The Hamiltonian
H = −t ∑⟨ij⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t = 1, U = 8, N↑ = N↓ = 4 on 16 sites.
- lattice
- square
- sites
- 16
- boundary
- periodic
- Nf
- 4
- U
- 8
- energies
- 2
Stored as a total energy; quoted here as
E/site, which is what the papers report
(how the conversion works).
Every published energy
Grouped by what the number is. The exact energy comes first where there is one; in every other group the lowest unflagged energy is its best value, and each row's distance is to that best (rules §6).
Numerically exact (1)
Exact diagonalization, an exact solution, or sign-problem-free QMC where that is established. An exact energy is shown first, before the best of each other class. QMC is exact only within its error bar, which it must state, and reads exact (stochastic); an exact diagonalization outranks it. For a diagonalization resolved by symmetry sector only the ground state's sector is listed; the lowest energies of the other sectors are the instance's spectrum, in its JSON, and are not shown as the exact energy.
Strict variational upper bounds (1)
Energies that a variational state reached (VMC, DMRG and MPS at finite bond dimension, PEPS, NQS, VQE). Each is an upper bound on the ground-state energy.
How these numbers were read
Where each number below was read, and how it was checked.
-1.0287892, ED
Checked 2026-09-29 · QMBL ED cost run checks/cost/ed/results/Hubbard--square_16_P_4_8.json (checks/cost/ed/run_ed.py: NetKet 3.22.4 operator, sparse CSR built in blocks, SciPy eigsh/ARPACK k=1 SA; commit 716e043); results file read locally, no rerun
Reported as. energy -16.460626753681726 (stored_exact -16.46062675368178, relative_difference 3.24e-15, reproduces_stored true)
|reproduced - stored| = 5.33e-14 (relative 3.24e-15); within 1e-9 relative. Sector N_up = N_dn = 4, dimension 3312400, 171 Lanczos steps (the file states no residual). The stored value is the source's own ED (varbench@2024-10-22), so two codes agree; a third code, E1's node Lanczos (qmbl-runs/qmbl-verify/out/E1-exact-recompute/results/Hubbard__square_16_P_4_8.json), gives -16.46062675368358 (1.85e-12 from the cost run). Also read by VB2-varbench-code-hubbard#51 (scope value: VarBench Hubbard/square_16_P_4_8.md row 'Exact diagonalization'; checks/hubbard-u-labels/ed_results.jsonl job {lattice: square, L: 4, n: 4, U: 8}).
-1.0287892, DMRG
Checked 2026-09-29 · run script programs/dmrg_itensors_hubbard/square_16_P_4_8.jl (varbench/methods@ed31bb0) read in full (author copy MaxSilvester/Hubbard-and-Heisenberg-codes identical in physics); VarBench history (git clone of github.com/varbench/varbench, partial, read with git log --follow; old names normalised); energy compared with the QMBL full-space recompute
Reported as. -16.460626633 | | 1.6764e-7 | DMRG (MaxBondDim ~ 3200) (VarBench upload, J. Maxwell Silvester)
4x4 torus (32 bonds), t = 1, U n_up n_dn with no shift, N_up = N_dn = 4: the instance at U = 8. The energy is the reached DMRG energy at the final maxdim and lies above the exact energy at the script's U, as a variational energy must; variational correct. Temple check (E - E0 <= Var/(E1 - E), E1 = -16.338331380898335 the next level, eigsh lowest): E - E0 = 1.2e-7 vs bound 1.4e-6, consistent. History: uploaded 2022-09-12 (d547055, variance 1.67644e-7, then 1.6764e-7 in 22d4daf). No output log (scope config).