Record
-0.7406
E/site
Fixed-node GFMC (trial state: SCALE, 1 conv layer, horizontal half-filled stripe λ = 4)
Gu et al. (2026) · lowest unflagged energy published, a projected estimate
(rules §6)
The Hamiltonian
H = −t1 ∑⟨ij⟩,σ (c†iσcjσ + h.c.) −t2 ∑⟨⟨ij⟩⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t1 = 1, t2 = -0.2, U = 8, N↑ = N↓ = 224 on 512 sites.
This instance is a t12 variant with a next-nearest-neighbour hopping t2, written as in VarBench's Eq. (S6). t2 is the t′ of the sources' H = −t ∑⟨ij⟩ − t′ ∑⟨⟨ij⟩⟩ + U ∑ ni↑ni↓. The couplings are those each source prints (see the rows).
- lattice
- rectangular-16x32
- sites
- 512
- boundary
- periodic
- Nf
- 224
- U
- 8
- t1
- 1
- t2
- -0.2
- 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.
-0.7406, 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. -0.7406
Table II (32x16 Hubbard, PBC, delta=1/8; U=8 per Sec. III C): columns t'=0 filled stripe (lambda=8) | t'=-0.2 half-filled stripe (lambda=4); row SCALE+GFMC, horizontal stripe, column t'=-0.2; sources/2604.25775.txt line 573; 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).
-0.7397, Fixed-node GFMC (trial state: SCALE, 1 conv layer, vertical...
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. -0.7397
Table II (32x16 Hubbard, PBC, delta=1/8; U=8 per Sec. III C): columns t'=0 filled stripe (lambda=8) | t'=-0.2 half-filled stripe (lambda=4); row SCALE+GFMC, vertical stripe, column t'=-0.2; sources/2604.25775.txt line 574; 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).
-0.7371, SCALE (1 conv layer, horizontal half-filled stripe λ = 4)
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. -0.7371
Table II (32x16 Hubbard, PBC, delta=1/8; U=8 per Sec. III C): columns t'=0 filled stripe (lambda=8) | t'=-0.2 half-filled stripe (lambda=4); row SCALE, horizontal stripe, column t'=-0.2; sources/2604.25775.txt line 571; 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).
-0.7359, SCALE (1 conv layer, vertical half-filled stripe λ = 4)
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. -0.7359
Table II (32x16 Hubbard, PBC, delta=1/8; U=8 per Sec. III C): columns t'=0 filled stripe (lambda=8) | t'=-0.2 half-filled stripe (lambda=4); row SCALE, vertical stripe, column t'=-0.2; sources/2604.25775.txt line 572; 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).