Hubbard rectangular 4x16, periodic/open, U = 8, n = 0.9375

Best values (rules §6)

best variational	-0.637875	DMRG Wu et al. (2024) · 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^†_iσc_jσ + h.c.) + U ∑_i n_i↑n_i↓

t = 1, U = 8, N_↑ = N_↓ = 30 on 64 sites.

lattice: rectangular-4x16
sites: 64
boundary: periodic/open
Nf: 30
U: 8
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. 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 (1)

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
-0.637875best variational	0	n/a	1.32e-1	3.4e-4	DMRG 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.

-0.637875, DMRG

Checked 2026-09-29 · programs/dmrg_itensors_hubbard/rectangular-4x16_64_PO_30_8.jl (varbench/methods@ed31bb0) read in full; the author's own copy MaxSilvester/Hubbard-and-Heisenberg-codes@b5c28f3 rectangular-4x16_64_PO_30_8.jl has the same Hamiltonian, filling and schedule; VarBench history (git clone of github.com/varbench/varbench, partial, read with git log --follow; old names normalised)

Reported as. -40.824 | | 0.132 | DMRG(MaxLinkDim = 4000, MaxTruncErr~1.5E-05) (VarBench upload by J. Maxwell Silvester)

4x16 cylinder: periodic around the 4-site circumference, open along the 16-site length (PO), t = 1, U n_up n_dn with no shift, N_up = N_dn = 30, conserve_qns: the instance. Maximum bond dimension in the script equals the MaxLinkDim printed. Energy is the reached DMRG energy: variational is right. History: uploaded 2022-12-15 (de78b35), unchanged. No output log in either repository (scope config). Provenance note: in VarBench's copies of the ten rectangular scripts the variance line is evaluated on the initial product state ('E,psi = dmrg(H,ψ,sweeps)' then 'var = inner(H,ψ,H,ψ) - E^2'; ITensors v0.6.16 dmrg optimises psi = copy(psi0), src/lib/ITensorMPS/src/dmrg.jl line 189, so ψ stays the product state), so as committed they would print <ψ0|H^2|ψ0> - E^2 (for the half-filled 32_8 start, a Néel product state, 2 x 124 allowed hops give 248 - 32.923^2 = -836), not the stored positive variance. The author's repository fixed it on 2024-08-05 ('E,ψ = dmrg(...)' plus the var line, e.g. commits e78c997, 4d3e522); the stored variances were uploaded in 2022, before either script existed. Suggest pointing the code link at MaxSilvester/Hubbard-and-Heisenberg-codes or fixing varbench/methods.

Coverage

Newest published number here: 2024. Literature last checked 2026-09-15 (table harvest of cached arXiv sources (match_tables.mjs --all), every matched paper read for this instance): 2 papers read, 0 of them with an energy on this page.

This instance as data

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