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All instances / Hubbard

Hubbard rectangular 8x16, U = 8, n = 0.875

Best values (rules §6)

best variational -0.7568 VMC (tensor BF)
Liang (2025) · the lowest unflagged variational energy, a strict variational bound

Energies as E/site. Classes are different claims and are not ranked against each other.

The Hamiltonian

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

t = 1, U = 8, N↑ = N↓ = 56 on 128 sites.

lattice
rectangular-8x16
sites
128
boundary
periodic
Nf
56
U
8
energies
3

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. The exact energy comes first where there is one; in every other group the lowest unflagged energy is its best value, and each row's distance is to that best (rules §6).

Strict variational upper bounds (3)

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 class bestσVar(E)V-score methodsourcecodeyearcomment
-0.7568best variational 0 n/a n/a n/a VMC (tensor BF)
peer reviewed
 Liang (2025) 2025
-0.7568tied with best variational at 2σ <1e-4 n/a n/a n/a VMC (transformer BF)
peer reviewed
 Gu et al. (2026) bytedance/laqx 2026
-0.7529 3.9e-3 n/a n/a n/a VMC (tensor BF)
peer reviewed
 Liang (2025) 2025

How these numbers were read

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

-0.7568, VMC (tensor BF)

Checked 2026-09-15 · source text of arXiv:2507.01856 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; parsed by column from the primary table

Reported as. -0.7568

Table V of arXiv:2507.01856 (pypdf layout text, line 399), 8x16, t' = 0, NN backflow, column Ep = 1 | n = 0.875, U = 8 (Table V caption); PBC: the 8x16 Ep = 1 value equals Table II's 8x16 PBC row, and Figs. 4 and 8 state the 8x32 lattice is under PBC

-0.7568, VMC (transformer BF)

Checked 2026-10-22 · source text of arXiv:2507.02644 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. −0.7568

Supplementary Table 5, row 16 × 8, t' = 0; dl/gu-si.txt line 189; no error bar | Energy per site in units of t. PBC per the Supplementary Table 5 caption ('Raw data for the energies of Hubbard model under PBC'); U = 8 and hole doping 1/8 per the main text (Fig. 4 caption: 'U = 8 and hole doping δ = 1/8 under PBC'; the t' = -0.2 study: 'Lx × 8 with Lx ranging from 8 to 32, with t' = -0.2 and δ = 1/8'). H has -t' sum_<<ij>>, so t2 = t'. The 16 x 4 and 16 x 16 values at t' = 0 equal the arXiv text's -0.76298 and -0.75630 to the printed digits. No error bar printed. Version read: journal Supplementary Information (41467_2026_74028_MOESM1_ESM.pdf); the arXiv text has no raw-data table.

-0.7529, VMC (tensor BF)

Checked 2026-09-15 · source text of arXiv:2507.01856 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; parsed by column from the primary table

Reported as. -0.7529

Table V of arXiv:2507.01856 (pypdf layout text, line 399), 8x16, t' = 0, NN backflow, column Ep = 0 | n = 0.875, U = 8 (Table V caption); PBC: the 8x16 Ep = 1 value equals Table II's 8x16 PBC row, and Figs. 4 and 8 state the 8x32 lattice is under PBC

Coverage

Newest published number here: 2026.

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