The Hamiltonian
H = −t ∑⟨ij⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t = 1, U = 12, N↑ = N↓ = 126 on 288 sites.
- lattice
- rectangular-6x48
- sites
- 288
- boundary
- periodic
- Nf
- 126
- U
- 12
- energies
- 6
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 (4)
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.
-0.63217(2), FN-GFMC (Jastrow + BF)
Checked 2026-09-29 · run script scripts/Hubbard/rectangular-6x48_288_P_126_12/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. -182.065 | 0.005 | (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 = 12 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 = 126, N_dn = 288 - neldo = 126 (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). FN not printed in the papers.
-0.62556(3), VMC (Jastrow + BF)
Checked 2026-09-29 · run script scripts/Hubbard/rectangular-6x48_288_P_126_12/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. -180.16 | 0.01 | 21.5 (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 = 12 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 = 126, N_dn = 288 - neldo = 126 (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: arXiv:2111.04623v4 (SciPost Phys. 12, 180 (2022)) Table 2, U/t = 12, t'/t = 0, Lx = 48: '-0.6255 (λ = 6)' ('The error bar on the energy is always smaller than 10^-4'); stored -180.16/288 = -0.625556 differs by 5.6e-5.
-0.6255, VMC (Jastrow + BF)
Checked 2026-10-07 · source text of arXiv:2111.04623 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. -0.6255
Table 2 (U/t = 12, Lx = 48), row t'/t = 0, column E_stripe (λ = 6); no per-value error bar | Energy per site in units of t on Lx x 6 ladders with 'periodic boundary conditions along both the x and the y directions', hole doping 1/8 (n = 7/8). 'The error bar on the energy is always smaller than 10^-4' is one statement for the whole table, not a per-value sigma.
-0.6227, VMC (Jastrow + BF)
Checked 2026-10-07 · source text of arXiv:2111.04623 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. -0.6227
Table 2 (U/t = 12, Lx = 48), row t'/t = 0, column E_uniform; no per-value error bar | Energy per site in units of t on Lx x 6 ladders with 'periodic boundary conditions along both the x and the y directions', hole doping 1/8 (n = 7/8). 'The error bar on the energy is always smaller than 10^-4' is one statement for the whole table, not a per-value sigma.