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All instances / Hubbard

Hubbard rectangular 4x24, U = 4, n ≈ 0.8333

Record

-1.07872(3) E/site

HFPS (GCNN; filled stripe)

Roth et al. (2025) · lowest unflagged energy published, a strict variational bound (rules §6)

The Hamiltonian

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

t = 1, U = 4, N↑ = N↓ = 40 on 96 sites.

lattice
rectangular-4x24
sites
96
boundary
periodic
Nf
40
U
4
energies
2

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.

E/sitedistance to recordσVar(E)V-score methodsourceyear
-1.07872record 0 3.0e-5 n/a n/a HFPS (GCNN; filled stripe)
preprint
Roth et al. (2025) 2025
-1.07779 9.3e-4 5.0e-5 n/a n/a HFPS (GCNN; d-wave superconductor)
preprint
Roth et al. (2025) 2025

How these numbers were read

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

-1.07872(3), HFPS (GCNN; filled stripe)

Checked 2026-10-22 · source text of arXiv:2511.07566 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. -1.07872(3)

Table III (24x4 torus, U=4; columns Filled Stripe | Superconductor); sources/2511.07566.txt line 773; version read: arXiv v1 (sources/2511.07566.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = U sum n_up n_dn - sum t_xx' c+c, t_xx' = t' on next-nearest bonds, periodic; Table V: 16x16, U = 8, doping 1/8, energy per site. The d-wave field Delta_0 enters only the mean-field Hamiltonian of the initial orbitals (Eq. 5, App. D).

-1.07779(5), HFPS (GCNN; d-wave superconductor)

Checked 2026-10-22 · source text of arXiv:2511.07566 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. -1.07779(5)

Table III (24x4 torus, U=4; columns Filled Stripe | Superconductor); sources/2511.07566.txt line 773; version read: arXiv v1 (sources/2511.07566.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = U sum n_up n_dn - sum t_xx' c+c, t_xx' = t' on next-nearest bonds, periodic; Table V: 16x16, U = 8, doping 1/8, energy per site. The d-wave field Delta_0 enters only the mean-field Hamiltonian of the initial orbitals (Eq. 5, App. D).

Coverage

Newest published number here: 2025.

This instance as data

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