The Hamiltonian
H = −t ∑⟨ij⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t = 1, U = 1, N↑ = N↓ = 4 on 14 sites.
- lattice
- chain
- sites
- 14
- boundary
- periodic
- Nf
- 4
- U
- 1
- energies
- 3
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.
-0.9116302(1), DMRG
Checked 2026-09-29 · VarBench method string (Silvester upload, 2022-09-09) + run script programs/dmrg_itensors_hubbard/chain_14_P_4_1.jl (varbench/methods@ed31bb0) read in full + QMBL full-space recompute
Reported as. Extrap Energy = -12.762823 +/- 2.e-6 (first uploaded as -12.7628238567016 +/- 2.3e-6, commit c3fe8ad, rounded the same day)
Transcribed correctly from the uploader's method string; bound_type extrapolated is right. Answer to the M6 flag: the sigma is the uploader's +/- on the extrapolation, not a statistical error; neither run script contains extrapolation code (they print E and var after the last sweep), and neither the extrapolation variable nor the meaning of +/- is stated anywhere (the VarBench paper, arXiv:2302.04919v2 S3.B.1, says only 'protocols for extrapolating the truncation error to zero can give greatly improved energy estimates and approximate errors on those estimates' and that VarBench itself 'forgo[es] extrapolations'). method_detail 'truncation error -> 0' is therefore an inference. Against the exact energy the value is within its +/-.
-0.9116302, ED
Checked 2026-09-29 · QMBL ED cost run checks/cost/ed/results/Hubbard--chain_14_P_4_1.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 -12.762822785869522 (stored_exact -12.762822785869506, relative_difference 1.25e-15, reproduces_stored true)
|reproduced - stored| = 1.60e-14 (relative 1.25e-15); within 1e-9 relative. Sector N_up = N_dn = 4, dimension 1002001, 161 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__chain_14_P_4_1.json), gives -12.762822785871055 (1.53e-12 from the cost run). Also read by VB2-varbench-code-hubbard#1 (scope value: VarBench Hubbard/chain_14_P_4_1.md row 'Exact diagonalization'; checks/hubbard-u-labels/ed_results.jsonl job {lattice: chain, L: 14, n: 4, U: 1}).
-0.9116293, DMRG
Checked 2026-09-29 · run script read in full: programs/dmrg_itensors_hubbard/chain_14_P_4_1.jl (varbench/methods@ed31bb0); same physics in MaxSilvester/Hubbard-and-Heisenberg-codes@b5c28f3 hubbard_chain_14_P_4_1.jl; 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. -12.76281 | | 4.643e-6 | DMRG(MaxBondDim = 1550, Extrap Energy = -12.762823 +/- 2.e-6) (VarBench upload c3fe8ad/de21ea7/8712df3 (Silvester, 2022-09-09; the extrapolation text was edited from -12.7628238567016 +/- 2.3e-6 to -12.762823 +/- 2.e-6, the energy never changed))
H = -t sum (Cdag C + h.c.) + U Nupdn with t = 1 on a 14-site ring with the [1,N] bond (ITensors OpSum, automatic fermion signs, so fermionic PBC), N_up = N_dn = 4: the instance. The energy is the reached variational energy after the last sweep at maxdim 1550 ('~1500' in the string), above the exact energy by 1.3e-5, consistent with its variance: Temple check (E - E0 <= Var/(E1 - E), E1 = -12.709836203145427 the next level, eigsh lowest): E - E0 = 1.3e-5 vs bound 8.8e-5, consistent. bound_type variational is right; the extrapolated value in the method string is a separate row (answers the M8/M6 flags; the tiny V-score is genuine 1D convergence). No output log exists in either repository, so the number itself is not located independently (scope config).