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All instances / Hubbard

Hubbard square 4x4, U = 2.15443469, n = 0.5

Record

-1.15033255(5) E/site

ED

Moreno et al. (2022) · exact energy: the instance is solved, and this is the state of the art on it (rules §6)

The Hamiltonian

H = −t ∑⟨ij⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓

t = 1, U = 2.15443469, N↑ = N↓ = 4 on 16 sites.

lattice
square
sites
16
boundary
periodic
Nf
4
U
2.15443469
energies
9

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. Where the instance is not solved, the lowest unflagged energy of any group is the record.

Strict variational upper bounds (8)

Where the instance is not solved, the record is the lowest unflagged energy in this group or among the projected and extrapolated ones, with or without an error bar; where it is solved, the exact energy is the record and the lowest of these is the closest challenger.

E/sitedistance to recordσVar(E)V-score methodsourceyear
-1.148627 1.7e-3 3.1e-6 n/a n/a HFDS (unrestricted hidden-spin constraint function + RBM in the augmented space)
peer reviewed
 Moreno et al. (2022) 2022
-1.148387 1.9e-3 4.4e-6 n/a n/a RBM (unrestricted Slater determinant × RBM)
peer reviewed
 Moreno et al. (2022) 2022
-1.146052 4.3e-3 4.4e-6 n/a n/a HFDS (unrestricted hidden-spin constraint function)
peer reviewed
 Moreno et al. (2022) 2022
-1.145216 5.1e-3 5.0e-6 n/a n/a HFDS (Gutzwiller-inspired constraint function)
peer reviewed
 Moreno et al. (2022) 2022
-1.14483 5.5e-3 1.3e-5 n/a n/a Slater-Jastrow (unrestricted Slater determinant + Jastrow factor)
peer reviewed
 Moreno et al. (2022) 2022
-1.1441 6.2e-3 1.9e-4 n/a n/a Slater-Jastrow (unrestricted Slater determinant + Gutzwiller factor)
peer reviewed
 Moreno et al. (2022) 2022
-1.129 2.2e-2 3.7e-3 n/a n/a Hartree-Fock (unrestricted Slater determinant)
peer reviewed
 Moreno et al. (2022) 2022
-1.11433 3.6e-2 1.3e-5 n/a n/a HFDS (Jastrow-inspired constraint function)
peer reviewed
 Moreno et al. (2022) 2022

Numerically exact (1)

Exact diagonalization, an exact solution, or sign-problem-free QMC where that is established: the answer, not a claim about it, and therefore the record wherever one exists. QMC is exact only within its statistical error bar, which it must state, and its rows read exact (stochastic); an exact diagonalization outranks it. A diagonalization resolved by symmetry sector is listed for the ground state's sector only; the lowest energies of the other sectors are the instance's spectrum, in its JSON, and hold nothing.

E/sitedistance to recordσVar(E)V-score methodsourceyear
-1.15033255record 0 5.0e-8 n/a n/a ED
peer reviewed
 Moreno et al. (2022) 2022

How these numbers were read

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

-1.15033255(5) — ED

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. -18.4053208(8)

Appendix H Table II, U=2.15, row ED; sources/2111.10420.txt line 1193; 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).

-1.148627(3) — HFDS (unrestricted hidden-spin constraint function + RBM in...

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. -18.37803(5)

Appendix H Table II, U=2.15, 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 | 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).

-1.148387(4) — RBM (unrestricted Slater determinant × 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. -18.37420(7)

Appendix H Table II, U=2.15, 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 | 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).

-1.146052(4) — HFDS (unrestricted hidden-spin constraint function)

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. -18.33683(7)

Appendix H Table II, U=2.15, row Unrestricted hidden spin; sources/2111.10420.txt line 1199; 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).

-1.145216(5) — HFDS (Gutzwiller-inspired constraint function)

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. -18.32346(8)

Appendix H Table II, U=2.15, 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 | 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).

-1.14483(1) — Slater-Jastrow (unrestricted Slater determinant + Jastrow...

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. -18.3173(2)

Appendix H Table II, U=2.15, 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 | 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).

-1.1441(2) — Slater-Jastrow (unrestricted Slater determinant +...

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. -18.306(3)

Appendix H Table II, U=2.15, 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 | 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).

-1.129(4) — Hartree-Fock (unrestricted Slater determinant)

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. -18.06(6)

Appendix H Table II, U=2.15, 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).

-1.11433(1) — HFDS (Jastrow-inspired constraint function)

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. -17.8293(2)

Appendix H Table II, U=2.15, 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.

This instance as data

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