The Hamiltonian
H = −t ∑⟨ij⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t = 1, U = -8, N↑ = N↓ = 63 on 144 sites.
- lattice
- square
- sites
- 144
- boundary
- periodic
- Nf
- 63
- U
- -8
- energies
- 4
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. On a solved instance the exact energy is the record; otherwise the lowest unflagged energy of any group holds it (rules §6).
Strict variational upper bounds (2)
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.
Projected or fixed-node estimates (2)
Variational only within a constraint: fixed-node, constrained-path, GFMC on a trial state. Each depends on its node or constraint, so two of them are not cleanly comparable to each other or to strict bounds.
How these numbers were read
Where each number below was read, and how it was checked.
-4.0161, Fixed-node GFMC (trial state: ACE)
Checked 2026-10-22 · source text of arXiv:2604.25775 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. -4.0161
Table IV (App. D, attractive Hubbard, t'=0, delta=1/8, PBC); columns 8x8 U=-2 | 8x8 U=-8 | 12x12 U=-2 | 12x12 U=-8; row ACE+GFMC, column 12x12 U=-8; sources/2604.25775.txt line 763; version read: arXiv v1 (sources/2604.25775.txt); APS copy not reached (403, 2026-10-16) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, U = 8, doping 1/8 (Sec. III), periodic unless the column says OBC; energies per site. Pinning fields only in pre-training (App. C).
-4.0159, ACE (16 conv layers)
Checked 2026-10-22 · source text of arXiv:2604.25775 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. -4.0159
Table IV (App. D, attractive Hubbard, t'=0, delta=1/8, PBC); columns 8x8 U=-2 | 8x8 U=-8 | 12x12 U=-2 | 12x12 U=-8; row ACE, column 12x12 U=-8; sources/2604.25775.txt line 762; version read: arXiv v1 (sources/2604.25775.txt); APS copy not reached (403, 2026-10-16) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, U = 8, doping 1/8 (Sec. III), periodic unless the column says OBC; energies per site. Pinning fields only in pre-training (App. C).
-4.0158, Fixed-node GFMC (trial state: SCALE, 1 conv layer)
Checked 2026-10-22 · source text of arXiv:2604.25775 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. -4.0158
Table IV (App. D, attractive Hubbard, t'=0, delta=1/8, PBC); columns 8x8 U=-2 | 8x8 U=-8 | 12x12 U=-2 | 12x12 U=-8; row SCALE+GFMC, column 12x12 U=-8; sources/2604.25775.txt line 761; version read: arXiv v1 (sources/2604.25775.txt); APS copy not reached (403, 2026-10-16) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, U = 8, doping 1/8 (Sec. III), periodic unless the column says OBC; energies per site. Pinning fields only in pre-training (App. C).
-4.0141, SCALE (1 conv layer)
Checked 2026-10-22 · source text of arXiv:2604.25775 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. -4.0141
Table IV (App. D, attractive Hubbard, t'=0, delta=1/8, PBC); columns 8x8 U=-2 | 8x8 U=-8 | 12x12 U=-2 | 12x12 U=-8; row SCALE, column 12x12 U=-8; sources/2604.25775.txt line 760; version read: arXiv v1 (sources/2604.25775.txt); APS copy not reached (403, 2026-10-16) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, U = 8, doping 1/8 (Sec. III), periodic unless the column says OBC; energies per site. Pinning fields only in pre-training (App. C).