Hubbard rectangular 6x48, U = 8, n ≈ 0.8333

Best values (rules §6)

best variational	-0.82075(3)	VMC (Jastrow + BF) Wu et al. (2024) · the lowest unflagged variational energy, a strict variational bound
best projected	-0.82575(2)	FN-GFMC (Jastrow + BF) Wu et al. (2024) · the lowest unflagged projected energy, a projected estimate, which depends on its node or constraint

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

The Hamiltonian

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

t = 1, U = 8, N_↑ = N_↓ = 120 on 288 sites.

lattice: rectangular-6x48
sites: 288
boundary: periodic
Nf: 120
U: 8
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. 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 (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.

E/site	distance to class best	σ	Var(E)	V-score	method	source	code	year	comment
-0.82075best variational	0	3.1e-5	1.73e+1	1.0e-2	VMC (Jastrow + BF) VarBench reference run	Wu et al. (2024)	run script	2024
-0.81909	1.7e-3	1.4e-5	2.22e+1	1.3e-2	VMC (Jastrow + BF) VarBench reference run	Wu et al. (2024)	run script	2024

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.

E/site	distance to class best	σ	Var(E)	V-score	method	source	code	year	comment
-0.82575best projected	0	2.1e-5	n/a	n/a	FN-GFMC (Jastrow + BF) VarBench reference run	Wu et al. (2024)	run script	2024
-0.82339	2.4e-3	1.4e-5	n/a	n/a	FN-GFMC (Jastrow + BF) VarBench reference run	Wu et al. (2024)	run script	2024

How these numbers were read

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

-0.82575(2), FN-GFMC (Jastrow + BF)

Checked 2026-09-29 · run script scripts/Hubbard/rectangular-6x48_288_P_120_8/FN-stripes/fn_hubbard.sh with datprep.d and datasfn.d, programs/vmc_hubbard/{Prep_stripes_wf.f, Main_stripes_wf.f, Energy.f90, Variance.f} read (varbench/methods@ed31bb0); VarBench history (git clone of github.com/varbench/varbench, partial, read with git log --follow; old names normalised)

Reported as. -237.816 | 0.006 | (VarBench upload by L. F. Tocchio)

48 x 6 torus (the compiled nx = 48, nh = 288; ny = nh/nx = 6), periodic in both directions, t = 1, no t', U = 8 with U n_up n_dn (no constant shift: the diagonal energy is U per doubly occupied site, Main_*_wf.f 'diag(k)=diag(k)-U'), N_up = nelup = 120, N_dn = 288 - neldo = 120 (particle-hole transformed down spins, as the code states for iopt >= 0): the instance. Stripe wavelength 6 as printed. Lattice fixed-node GFMC (gamma = 0) on the VMC state: projected is right; its trial energy etry agrees with the result per site. The M3 flag (FN below the best variational bound) is expected: FN improves on its own trial state. No output log in the repository (scope config). The FN value is not printed in either paper (arXiv:1905.02658v2 mentions only 'preliminary calculations, based on Green’s Function Monte Carlo ... with the Fixed-Node approximation').

-0.82075(3), VMC (Jastrow + BF)

Checked 2026-09-29 · run script scripts/Hubbard/rectangular-6x48_288_P_120_8/VMC-stripes/vmc_hubbard.sh with datprep.d and datasvmc.d, programs/vmc_hubbard/{Prep_stripes_wf.f, Main_stripes_wf.f, Energy.f90, Variance.f} read (varbench/methods@ed31bb0); VarBench history (git clone of github.com/varbench/varbench, partial, read with git log --follow; old names normalised)

Reported as. -236.376 | 0.009 | 17.3 (VarBench upload by L. F. Tocchio)

48 x 6 torus (the compiled nx = 48, nh = 288; ny = nh/nx = 6), periodic in both directions, t = 1, no t', U = 8 with U n_up n_dn (no constant shift: the diagonal energy is U per doubly occupied site, Main_*_wf.f 'diag(k)=diag(k)-U'), N_up = nelup = 120, N_dn = 288 - neldo = 120 (particle-hole transformed down spins, as the code states for iopt >= 0): the instance. Stripe wavelength 6 as printed. Jastrow + backflow VMC: variational is right. The M11 margins over the uniform state are the physics of the stripe state, not an error. No output log in the repository (scope config). Published corroboration (same group, same method, not the same number): arXiv:1905.02658v2 (SciPost Phys. 7, 021 (2019)) Table 1, p = 6, U/t = 8: 'E/t (Best stripe) -0.8207(1) (λ = 6)', averaged over 6 x 48, 72, 96 ('Error-bars are estimated in order to take into account the weak size dependence'); stored -0.82075 per site is 0.5 of that bar away.

Coverage

Newest published number here: 2024.

This instance as data

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