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All instances / Hubbard

Hubbard rectangular 8x48, U = 8, n ≈ 0.9167

No record

—

no variational row (rules §6). Every row is still listed below.

The Hamiltonian

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

t = 1, U = 8, N↑ = N↓ = 176 on 384 sites.

lattice
rectangular-8x48
sites
384
boundary
periodic
Nf
176
U
8
energies
3

Stored as a total energy in the convention of VarBench; quoted here as E/site, which is what the papers report (how the conversion works).

Every published energy

Grouped by what the number is. Ranking happens only inside the first group.

Projected or fixed-node estimates (3)

Variational only within a constraint: fixed-node, constrained-path, GFMC on a trial state. Node- or constraint-dependent, so not cleanly comparable to each other or to the group above, and never the record.

E/sitedistance to recordσVar(E)V-score methodsourceyear
-0.6854 n/a 2.0e-4 n/a n/a CP-AFQMC (trial state: 2/3-filled stripe, λ = 8)
peer reviewed
arXiv:2112.02187 2022
-0.6852 n/a 2.0e-4 n/a n/a CP-AFQMC (trial state: filled stripe, λ = 12)
peer reviewed
arXiv:2112.02187 2022
-0.6821 n/a 2.0e-4 n/a n/a CP-AFQMC (trial state: half-filled stripe, λ = 6)
peer reviewed
arXiv:2112.02187 2022

How these numbers were read

Rows added after the VarBench import record where the number came from.

-0.6854(2) — CP-AFQMC (trial state: 2/3-filled stripe, λ = 8)

Checked 2026-10-07 · source text of arXiv:2112.02187 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. -0.6854(2)

Table II, row 48 x 8, column λ = 8 (2/3-filled stripe state); mixed-estimator constrained-path AFQMC, not a variational bound | Energy per site in units of t, U = 8, hole doping 1/12, 'fully periodic supercells ... with no pinning fields' (Table II caption); 48 x 6 is QMBL's 6 x 48 torus with 132 + 132 electrons.

-0.6852(2) — CP-AFQMC (trial state: filled stripe, λ = 12)

Checked 2026-10-07 · source text of arXiv:2112.02187 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. -0.6852(2)

Table II, row 48 x 8, column λ = 12 (filled stripe state); mixed-estimator constrained-path AFQMC, not a variational bound | Energy per site in units of t, U = 8, hole doping 1/12, 'fully periodic supercells ... with no pinning fields' (Table II caption); 48 x 6 is QMBL's 6 x 48 torus with 132 + 132 electrons.

-0.6821(2) — CP-AFQMC (trial state: half-filled stripe, λ = 6)

Checked 2026-10-07 · source text of arXiv:2112.02187 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. -0.6821(2)

Table II, row 48 x 8, column λ = 6 (half-filled stripe state); mixed-estimator constrained-path AFQMC, not a variational bound | Energy per site in units of t, U = 8, hole doping 1/12, 'fully periodic supercells ... with no pinning fields' (Table II caption); 48 x 6 is QMBL's 6 x 48 torus with 132 + 132 electrons.

Coverage

Newest published number here: 2022. 3 rows added from the 2026-09 literature sweep, on top of the VarBench snapshot of 2024-10-22.

This instance as data

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