How these numbers were read
Where each number below was read, and how it was checked.
-1.0089505, ED
Checked 2026-09-29 · QMBL ED cost run checks/cost/ed/results/Hubbard--square_16_P_4_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 716e043); results file read locally, no rerun
Reported as. energy -16.14320817020251 (stored_exact -16.143208170202413, relative_difference 5.94e-15, reproduces_stored true)
|reproduced - stored| = 9.59e-14 (relative 5.94e-15); within 1e-9 relative. Sector N_up = N_dn = 4, dimension 3312400, 191 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_10.json), gives -16.143208170203927 (1.42e-12 from the cost run). Also read by VB2-varbench-code-hubbard#47 (scope value: VarBench Hubbard/square_16_P_4_10.md row 'Exact diagonalization'; checks/hubbard-u-labels/ed_results.jsonl job {lattice: square, L: 4, n: 4, U: 10}; SI Table I, row U = 10, column 'ED energy').
-1.0089505, DMRG
Checked 2026-09-29 · run script programs/dmrg_itensors_hubbard/square_16_P_4_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. -16.14320807 | | 1.81e-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 = 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 = -16.009837208902866 the next level, eigsh lowest): E - E0 = 1.0e-7 vs bound 1.4e-6, consistent. History: uploaded 2022-09-09 (7781a97) as 'MaxBondDim ~3200'; dropped by jrm874's 'Upload ED energies' overwrite (802857d, 2022-09-12) and re-added with identical numbers the same day (fc5be7f): a merge overwrite, not a changed value. No output log (scope config).
-1.008644(6), 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. -16.1383(9) (total energy, units of t)
Energy verified; sigma differs from the printed bar (ambiguous entry). 4 x 4 torus, 4 + 4, U = 10, N~ = 8, alpha = 32; variational, 0.0049 above the paper's ED -16.1432081(7) (same table row) and the stored ED -16.143208170202413; not the record (VarBench DMRG -16.14320807 lies lower). Upload 7f646d3: -16.1383 | 0.0001 | 0.078(3); energy_variance upload-only. 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.
-1.00842(3), 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. -16.1348(5)
Appendix H Table I, U=10, column HFDS alpha=16; sources/2111.10420.txt line 1179; 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#42: sigma 0.000496 -> 0.0005 (The table prints totals, so the bar is 0.0005 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 0.8%. Same fix as the trailing-float-noise ruling: store sigma_per_s) | 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).
-1.00831(3), 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. -16.1329(4)
Appendix H Table I, U=10, column HFDS alpha=8; sources/2111.10420.txt line 1179; 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).
-1.00753(3), 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. -16.1204(5)
Appendix H Table I, U=10, column HFDS alpha=4; sources/2111.10420.txt line 1179; 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#40: sigma 0.000496 -> 0.0005 (The table prints totals, so the bar is 0.0005 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 0.8%. Same fix as the trailing-float-noise ruling: store sigma_per_s) | 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).
-1.00689(5), 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. -16.1103(8)
Appendix H Table I, U=10, column HFDS alpha=2; sources/2111.10420.txt line 1179; 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).
-1.00622(3), 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. -16.0995(5)
Appendix H Table I, U=10, column HFDS alpha=1; sources/2111.10420.txt line 1179; 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#38: sigma 0.000496 -> 0.0005 (The table prints totals, so the bar is 0.0005 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 0.8%. Same fix as the trailing-float-noise ruling: store sigma_per_s) | 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).
-1.002856(6), 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. -16.0457(1)
Appendix H Table I, U=10, column HFDS alpha=0.5; sources/2111.10420.txt line 1179; 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#37: sigma 0.0001008 -> 0.0001 (The table prints totals, so the bar is 0.0001 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 0.8%. Same fix as the trailing-float-noise ruling: store sigma_per_s) | 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).
-0.99867(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. -15.9788(2)
Appendix H Table II, U=10, row Unrestricted hid. spin(RBM); sources/2111.10420.txt line 1202; 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#51: sigma 0.000208 -> 0.0002 (The table prints totals, so the bar is 0.0002 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_s) | 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).
-0.99635(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. -15.9416(2)
Appendix H Table II, U=10, row Gutzwiller inspired CF(RBM); sources/2111.10420.txt line 1201; 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#50: sigma 0.000208 -> 0.0002 (The table prints totals, so the bar is 0.0002 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_s) | 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).
-0.99587(1), VMC (RBM)
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. -15.9339(2)
Appendix H Table II, U=10, row USD (RBM); sources/2111.10420.txt line 1197; 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#47: sigma 0.000208 -> 0.0002 (The table prints totals, so the bar is 0.0002 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_s) | 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).
-0.98721(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. -15.7953(2)
Appendix H Table II, U=10, row Gutzwiller inspired CF; sources/2111.10420.txt line 1198; 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#48: sigma 0.000208 -> 0.0002 (The table prints totals, so the bar is 0.0002 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_s) | 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).
-0.98402(4), VMC (SJ)
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. -15.7443(6)
Appendix H Table II, U=10, row USD (Jastrow); sources/2111.10420.txt line 1196; 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#46: sigma 0.000608 -> 0.0006 (The table prints totals, so the bar is 0.0006 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_s) | 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).
-0.9803(1), VMC (SJ)
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. -15.685(2)
Appendix H Table II, U=10, row USD (Gutzwiller); sources/2111.10420.txt line 1195; 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#45: sigma 0.00208 -> 0.002 (The table prints totals, so the bar is 0.002 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_si) | 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).
-0.859(4), HF
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. -13.74(7)
Appendix H Table II, U=10, row USD; sources/2111.10420.txt line 1194; 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#44: sigma 0.0704 -> 0.07 (The table prints totals, so the bar is 0.07 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 0.6%. Same fix as the trailing-float-noise ruling: store sigma_per_sit) | 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).
-0.82623(3), 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. -13.2197(5)
Appendix H Table II, U=10, row Jastrow inspired CF; sources/2111.10420.txt line 1200; 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#49: sigma 0.000496 -> 0.0005 (The table prints totals, so the bar is 0.0005 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 0.8%. Same fix as the trailing-float-noise ruling: store sigma_per_s) | 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).