The Hamiltonian
H = −t ∑⟨ij⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t = 1, U = 2.15443469, N↑ = N↓ = 5 on 16 sites.
Upstream this instance is Hubbard/square_16_P_5_2.1544. The name's 2.1544 cuts off the coupling of this Hamiltonian, 2.15443469 (10^(3/9)). The exact diagonalization, -21.21223577699911, is the ground state there and 3.65e-5 above the ground state at U = 2.1544. Evidence: checks/hubbard-u-labels.
- lattice
- square
- sites
- 16
- boundary
- periodic
- Nf
- 5
- U
- 2.15443469
- energies
- 5
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 (4)
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.
-1.3257647, 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. -21.21223577699904 (ed_check.py) | -21.212235776995612 (ed_lib.mjs) | -21.212235776999105 (skeptic_ed.py)
Reproduced at the grid U the instance now carries, not at the U of VarBench's name (relabels.mjs).
-1.3257435(1), VMC (HFDS)
Checked 2026-09-29 · PNAS 119(32), e2122059119 (2022), SI Appendix PDF (PMC9371695; sources/pnas.2122059119.sapp.PMC9371695.pdf, sha256 c6e62d9c...), pypdf layout mode, re-extracted 2026-09-29 (byte-identical to the earlier extraction); same cell read in arXiv:2111.10420v2 (18 Jun 2022) Appendix H (sources/2111.10420v2.pdf, pypdf layout)
Reported as. -21.211896(2) (total energy, units of t)
Record row (best variational bound; ED holds the instance record, 0.00034 below). 4 x 4 torus, 5 + 5, grid U = 2.15443469, N~ = 10 and alpha = 64 (main Fig. 3 caption B: 'n = 5/8 average site occupancy (first closed shell) and N~ = 10. The rows of the hidden submatrix are given by single-hidden-layer neural networks with α = 64'). Projection: 'Projections to different symmetry subspaces are applied to the converged trial state: rotation of π/2 (C4), translations of momentum K = 0 and the intersection of both subspaces' (SI Sec. 8 B); the stored row is the intersection, as the method string says. A projected state's energy is at least its sector's minimum and so still an upper bound on the ground state: variational is right. Stored sigma equals the printed bar. Upload 5a046db: -21.211896 | 0.000002 | 0.027(4); energy_variance upload-only. Own computation. Venue PNAS, peer reviewed ('Edited by Angel Rubio ... received December 6, 2021; accepted June 28, 2022'; 'This article is a PNAS Direct Submission'); the imported row carries no peer_reviewed field (loader-level, see report). No duplicate.
-1.3257408(2), VMC (HFDS)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -21.211853(3)
Appendix H Table III, U=2.15, row HFSD (alpha=64)+K=0; sources/2111.10420.txt line 1229; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29; qmbl-verify 2026-10-19 SP2-arxiv-2111-10420#99: sigma 0.00000304 -> 0.000003 (The table prints totals, so the bar is 0.000003 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 1.3%. Same fix as the trailing-float-noise ruling: store sigma_per) | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).
-1.3257382(2), VMC (HFDS)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -21.211811(4)
Appendix H Table III, U=2.15, row HFSD (alpha=64)+C4; sources/2111.10420.txt line 1228; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29 | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).
-1.3257132(2), VMC (HFDS)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -21.211411(4)
Appendix H Table III, U=2.15, row HFSD (alpha=64); sources/2111.10420.txt line 1227; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29 | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).