How these numbers were read
Where each number below was read, and how it was checked.
-1.0564725, ED
Checked 2026-09-29 · QMBL ED cost run checks/cost/ed/results/Hubbard--square_16_P_5_10.json (checks/cost/ed/run_ed.py: NetKet 3.22.4 operator, sparse CSR built in blocks, SciPy eigsh/ARPACK k=1 SA; commit 5763525; host eu-g5-035-1); results file read locally, no rerun
Reported as. energy -16.903559984243575 (stored_exact -16.903559984243547, relative_difference 1.68e-15, reproduces_stored true)
|reproduced - stored| = 2.84e-14 (relative 1.68e-15); within 1e-9 relative. Sector N_up = N_dn = 5, dimension 19079424, 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_5_10.json), gives -16.903559984267847 (2.43e-11 from the cost run). Also read by VB2-varbench-code-hubbard#52 (scope value: VarBench Hubbard/square_16_P_5_10.md row 'Exact diagonalization'; checks/hubbard-u-labels/ed_results.jsonl job {lattice: square, L: 4, n: 5, U: 10}; SI Table III, row 'ED', column 'U =10').
-1.0564725, DMRG
Checked 2026-09-29 · run script programs/dmrg_itensors_hubbard/square_16_P_5_10.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. -16.903559816 | | 5.71e-6 | DMRG(MaxBondDim = 7000) (VarBench upload, J. Maxwell Silvester)
4x4 torus (32 bonds), t = 1, U n_up n_dn with no shift, N_up = N_dn = 5: the instance at U = 10. 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 = -15.899067046025197 the next level, lanczos ritz (no reorth)): E - E0 = 1.7e-7 vs bound 5.7e-6, consistent. History: an empty DMRG line (MaxBondDim ~3200) on 2022-09-09, filled 2022-09-12 (f1ef7ab). No output log (scope config).
-1.056468(9), VMC (HFPS)
Checked 2026-09-15 · source text of arXiv:2507.10705 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. -1.056468(9)
TABLE II, row U=10, column 'E/N_site', sigma^2/N_site=6.1e-5 | Matches record_per_site -1.0564724885.
-1.056364(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. -16.90183(2) (total energy, units of t)
4 x 4 torus, 5 + 5, U = 10, N~ = 10, alpha = 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. 0.00173 above the paper's ED -16.9035599(8); not the record (VarBench DMRG -16.903559816 and the HFPS row -16.903488 lie lower). Stored sigma equals the printed bar. Upload 5a046db: -16.90183 | 0.00002 | 0.047(1). 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.
-1.056342(1), 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. -16.90147(2)
Appendix H Table III, U=10, 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#94: sigma 0.0000208 -> 0.00002 (The table prints totals, so the bar is 0.00002 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 4.0%. 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).
-1.056324(1), 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. -16.90119(2)
Appendix H Table III, U=10, 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; qmbl-verify 2026-10-19 SP2-arxiv-2111-10420#93: sigma 0.0000208 -> 0.00002 (The table prints totals, so the bar is 0.00002 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 4.0%. 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).
-1.056263(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. -16.90020(3)
Appendix H Table III, U=10, 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; qmbl-verify 2026-10-19 SP2-arxiv-2111-10420#92: sigma 0.0000304 -> 0.00003 (The table prints totals, so the bar is 0.00003 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).