Hubbard square 4x4, U = 7.74263683, n = 0.5

Exact energy and best values (rules §6)

exact	-1.0318222	ED Moreno et al. (2022) · the instance is solved
best variational	-1.031562(6)	VMC (HFDS) Moreno et al. (2022) · the lowest unflagged variational energy, a strict variational bound

Energies as E/site. Classes are different claims and are not ranked against each other.

The Hamiltonian

H = −t ∑_⟨ij⟩,σ (c^†_iσc_jσ + h.c.) + U ∑_i n_i↑n_i↓

t = 1, U = 7.74263683, N_↑ = N_↓ = 4 on 16 sites.

Upstream this instance is Hubbard/square_16_P_4_7.74264. The name's 7.74264 rounds the coupling of this Hamiltonian, 7.74263683 (10^(8/9)). The exact diagonalization, -16.509154825265142, is the ground state there; at U = 7.74264 the ground state is 6.1e-7 higher, so the stored exact energy sat below the ground state of the instance as named. Evidence: checks/hubbard-u-labels.

lattice: square
sites: 16
boundary: periodic
Nf: 4
U: 7.74263683
energies: 15

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.

E/site	distance to class best	σ	Var(E)	V-score	method	source	code	year	comment
-1.0318222exact	0	n/a	n/a	n/a	ED	Moreno et al. (2022)		2022

Strict variational upper bounds (14)

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.

E/site	distance to class best	σ	Var(E)	V-score	method	source	code	year
-1.031562best variational	0	6.3e-6	5.50e-2	7.5e-4	VMC (HFDS)	Moreno et al. (2022)	uses NetKet	2022
-1.0314tied with best variational at 2σ	1.9e-4	1.9e-4	n/a	n/a	VMC (HFDS) peer reviewed	Moreno et al. (2022)	uses NetKet	2022
-1.03100	5.6e-4	1.9e-5	n/a	n/a	VMC (HFDS) peer reviewed	Moreno et al. (2022)	uses NetKet	2022
-1.03099	5.7e-4	2.5e-5	n/a	n/a	VMC (HFDS) peer reviewed	Moreno et al. (2022)	uses NetKet	2022
-1.03058	9.8e-4	1.9e-5	n/a	n/a	VMC (HFDS) peer reviewed	Moreno et al. (2022)	uses NetKet	2022
-1.03014	1.4e-3	3.7e-5	n/a	n/a	VMC (HFDS) peer reviewed	Moreno et al. (2022)	uses NetKet	2022
-1.02622	5.3e-3	2.5e-5	n/a	n/a	VMC (HFDS) peer reviewed	Moreno et al. (2022)	uses NetKet	2022
-1.02498	6.6e-3	1.3e-5	n/a	n/a	VMC (HFDS) peer reviewed	Moreno et al. (2022)	uses NetKet	2022
-1.02254	9.0e-3	1.3e-5	n/a	n/a	VMC (RBM) peer reviewed	Moreno et al. (2022)		2022
-1.01832	1.3e-2	3.1e-5	n/a	n/a	VMC (SJ) peer reviewed	Moreno et al. (2022)		2022
-1.014969	1.7e-2	6.3e-6	n/a	n/a	VMC (HFDS) peer reviewed	Moreno et al. (2022)	uses NetKet	2022
-1.01369	1.8e-2	6.3e-5	n/a	n/a	VMC (SJ) peer reviewed	Moreno et al. (2022)		2022
-0.897	1.3e-1	5.0e-3	n/a	n/a	HF peer reviewed	Moreno et al. (2022)		2022
-0.88283	1.5e-1	2.5e-5	n/a	n/a	VMC (HFDS) peer reviewed	Moreno et al. (2022)	uses NetKet	2022

How these numbers were read

Where each number below was read, and how it was checked.

-1.0318222, 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. -16.50915482526524 (ed_check.py) | -16.509154825269405 (ed_lib.mjs)

Reproduced at the grid U the instance now carries, not at the U of VarBench's name (relabels.mjs).

-1.031562(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. -16.505(1) (total energy, units of t)

Energy verified; sigma differs from the printed bar (ambiguous entry). 4 x 4 torus, 4 + 4, U '7.75' = grid 7.74263683, N~ = 8, alpha = 32, variational (0.00415 above ED), own computation; not the record (VarBench DMRG -16.5091541 lies lower). Upload 7f646d3: -16.505 | 0.0001 | 0.055(1); 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.

-1.0314(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. -16.502(3)

Appendix H Table I, U=7.75, column HFDS alpha=16; sources/2111.10420.txt line 1178; 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#81: sigma 0.00304 -> 0.003 (The table prints totals, so the bar is 0.003 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_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).

-1.03100(2), VMC (HFDS)

Reported as. -16.4960(3)

Appendix H Table I, U=7.75, column HFDS alpha=8; sources/2111.10420.txt line 1178; 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#80: sigma 0.000304 -> 0.0003 (The table prints totals, so the bar is 0.0003 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).

-1.03099(3), VMC (HFDS)

Reported as. -16.4959(4)

Appendix H Table I, U=7.75, column HFDS alpha=4; sources/2111.10420.txt line 1178; 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).

-1.03058(2), VMC (HFDS)

Reported as. -16.4893(3)

Appendix H Table I, U=7.75, column HFDS alpha=2; sources/2111.10420.txt line 1178; 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#78: sigma 0.000304 -> 0.0003 (The table prints totals, so the bar is 0.0003 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).

-1.03014(4), VMC (HFDS)

Reported as. -16.4823(6)

Appendix H Table I, U=7.75, column HFDS alpha=1; sources/2111.10420.txt line 1178; 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#77: 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).

-1.02622(3), VMC (HFDS)

Reported as. -16.4196(4)

Appendix H Table I, U=7.75, column HFDS alpha=0.5; sources/2111.10420.txt line 1178; 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).

-1.02498(1), VMC (HFDS)

Reported as. -16.3997(2)

Appendix H Table II, U=7.75, 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#90: 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).

-1.02254(1), VMC (RBM)

Reported as. -16.3607(2)

Appendix H Table II, U=7.75, 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#87: 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).

-1.01832(3), VMC (SJ)

Reported as. -16.2932(5)

Appendix H Table II, U=7.75, 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#86: 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).

-1.014969(6), VMC (HFDS)

Reported as. -16.2395(1)

Appendix H Table II, U=7.75, 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#88: 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).

-1.01369(6), VMC (SJ)

Reported as. -16.219(1)

Appendix H Table II, U=7.75, 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#85: sigma 0.001008 -> 0.001 (The table prints totals, so the bar is 0.001 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_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).

-0.897(5), HF

Reported as. -14.36(8)

Appendix H Table II, U=7.75, 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 | 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).

-0.88283(3), VMC (HFDS)

Reported as. -14.1253(4)

Appendix H Table II, U=7.75, 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 | 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).

Coverage

Newest published number here: 2022. Literature last checked 2026-09-15 (citation screen of Robledo Moreno et al. 2022 (PNAS, HFDS): citing works from OpenAlex and Semantic Scholar (scripts/cited_by.mjs)): 29 papers read, 0 of them with an energy on this page.

This instance as data

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