QMBLthe quantum many-body leaderboard

All instances / Hubbard

Hubbard square 12x12, U = 4, n ≈ 0.8333

Record

-1.0751(1) E/site

HFPS (GCNN; initialised at Δ0 = 0.5, U0 = 3; d-wave superconductor)

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↓ = 60 on 144 sites.

lattice
square
sites
144
boundary
periodic
Nf
60
U
4
energies
6

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 (6)

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
-1.0751record 0 1.0e-4 n/a n/a HFPS (GCNN; initialised at Δ0 = 0.5, U0 = 3; d-wave superconductor)
preprint
 Roth et al. (2025) 2025
-1.07508 <1e-4 4.0e-5 n/a n/a HFPS (GCNN; lowest energy over the initialisations tried)
preprint
 Roth et al. (2025) 2025
-1.0743 8.0e-4 1.0e-4 n/a n/a HFPS (GCNN; initialised at Δ0 = 1.0, U0 = 3; d-wave superconductor)
preprint
 Roth et al. (2025) 2025
-1.0737 1.4e-3 1.0e-4 n/a n/a HFPS (GCNN; initialised at Δ0 = 0.2, U0 = 3; d-wave superconductor)
preprint
 Roth et al. (2025) 2025
-1.0717 3.4e-3 2.0e-4 n/a n/a HFPS (GCNN; initialised at Δ0 = 0.0, U0 = 1; d-wave superconductor)
preprint
 Roth et al. (2025) 2025
-1.0700 5.1e-3 2.0e-4 n/a n/a HFPS (GCNN; initialised at Δ0 = 0.0, U0 = 3; stripe)
preprint
 Roth et al. (2025) 2025

How these numbers were read

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

-1.0751(1), HFPS (GCNN; initialised at Δ0 = 0.5, U0 = 3; d-wave...

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. -1.0751(1)

Table II (144-site 12x12 cluster, delta=1/6, U=4, t'=0; columns Delta_0 | U_0 | E/N | C_p | Phase); row Delta_0=0.5, U_0=3; sources/2511.07566.txt line 373; 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).

-1.07508(4), HFPS (GCNN; lowest energy over the initialisations tried)

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. -1.07508(4)

Table VII (App. D, lowest-energy wavefunction on the 144-site cluster at various U; t'=0, delta=1/6 from Fig. 4 caption and Sec. IV C); row U=4; sources/2511.07566.txt line 1224; 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).

-1.0743(1), HFPS (GCNN; initialised at Δ0 = 1.0, U0 = 3; d-wave...

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. -1.0743(1)

Table II (144-site 12x12 cluster, delta=1/6, U=4, t'=0; columns Delta_0 | U_0 | E/N | C_p | Phase); row Delta_0=1.0, U_0=3; sources/2511.07566.txt line 374; 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).

-1.0737(1), HFPS (GCNN; initialised at Δ0 = 0.2, U0 = 3; d-wave...

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. -1.0737(1)

Table II (144-site 12x12 cluster, delta=1/6, U=4, t'=0; columns Delta_0 | U_0 | E/N | C_p | Phase); row Delta_0=0.2, U_0=3; sources/2511.07566.txt line 372; 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).

-1.0717(2), HFPS (GCNN; initialised at Δ0 = 0.0, U0 = 1; d-wave...

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. -1.0717(2)

Table II (144-site 12x12 cluster, delta=1/6, U=4, t'=0; columns Delta_0 | U_0 | E/N | C_p | Phase); row Delta_0=0.0, U_0=1; sources/2511.07566.txt line 371; 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).

-1.0700(2), HFPS (GCNN; initialised at Δ0 = 0.0, U0 = 3; 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. -1.0700(2)

Table II (144-site 12x12 cluster, delta=1/6, U=4, t'=0; columns Delta_0 | U_0 | E/N | C_p | Phase); row Delta_0=0.0, U_0=3; sources/2511.07566.txt line 375; 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

JSON · source file on GitHub · add a result · open an issue about this instance