Hubbard square 24x24, U = 8, n = 0.875, t12 (t2 = -0.2)

Record

-0.74397(1) E/site

SBP (transfer-learned from 20x20; symmetric: unspecified)

Rende et al. (2026) · lowest unflagged energy published, a strict variational bound (rules §6)

The Hamiltonian

H = −t₁ ∑_⟨ij⟩,σ (c^†_iσc_jσ + h.c.) −t₂ ∑_{⟨⟨ij⟩⟩,σ} (c^†_iσc_jσ + h.c.) + U ∑_i n_i↑n_i↓

t₁ = 1, t₂ = -0.2, U = 8, N_↑ = N_↓ = 252 on 576 sites.

This instance is a t12 variant: a next-nearest-neighbour hopping t₂, written as in VarBench's Eq. (S6), so t₂ is the t′ of the sources' H = −t ∑⟨ij⟩ − t′ ∑⟨⟨ij⟩⟩ + U ∑ n_i↑n_i↓. Couplings from the Hamiltonian each source prints (see the rows).

lattice: square
sites: 576
boundary: periodic
Nf: 252
U: 8
t1: 1
t2: -0.2
energies: 1

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 (1)

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/site	distance to record	σ	Var(E)	V-score	method	source	year
-0.74397record	0	1.0e-5	n/a	n/a	SBP (transfer-learned from 20x20; symmetric: unspecified) preprint	Rende et al. (2026)	2026

How these numbers were read

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

-0.74397(1) — SBP (transfer-learned from 20x20; symmetric: unspecified)

Checked 2026-10-16 · source text of arXiv:2608.12465 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. -0.74397(1)

TABLE I (Ground-state variational energies and variances per site, PBC, t'=-0.2, U=8, delta=1/8), 24x24 / SBP; sources/2608.12465.txt line 300; version read: arXiv v1 (sources/2608.12465.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t' = -0.2, U = 8, doping 1/8, L x L periodic; Table I: energies and variances per site.

Coverage

Newest published number here: 2026.

This instance as data

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