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All instances / Hubbard

Hubbard square 16x16, periodic/antiperiodic, U = 8, n ≈ 0.8359, t12 (t2 = -0.25), UV1V2

Best values (rules §6)

best variational 1.25918(3) VMC (PP + RBM)
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 = −t1 ∑⟨ij⟩,σ (c†iσcjσ + h.c.) −t2 ∑⟨⟨ij⟩⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓ + V1 ∑⟨ij⟩ ninj + V2 ∑⟨⟨ij⟩⟩ ninj

t1 = 1, t2 = -0.25, U = 8, V1 = 1, V2 = 0.5, N↑ = N↓ = 107 on 256 sites.

This instance is the t12 variant with the UV1V2 interactions: a next-nearest-neighbour hopping t2, and density-density terms on nearest (V1) and next-nearest neighbours (V2), each bond once. The densities are the total densities ni = ni↑ + ni↓, as mVMC, the program behind its row, defines them; Eq. (S6) prints them between equal spins. Couplings from VarBench's notes on these runs and Eq. (S6) of its supplement.

lattice
square
sites
256
boundary
periodic/antiperiodic
Nf
107
U
8
t1
1
t2
-0.25
V1
1
V2
0.5
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/sitedistance to class bestσVar(E)V-score methodsourcecodeyearcomment
1.25918best variational 0 2.7e-5 4.47e+1 2.9e-2 VMC (PP + RBM)
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.

1.25918(3), VMC (PP + RBM)

Checked 2026-10-07 · VarBench upload history (varbench-history partial clone at 390a21e: 175fe34, fe5d5a4, ca26677), supplements/Imada_group_overview.md, varbench/methods full history (mVMC.sh, make_all.py, sub_trans.py, sub_qptransidx.py) and mVMC readdef.c / StdFace sources

Reported as. | 322.35(6) | 0.00696 | 44.71(6) |

Upload (VarBench 175fe34, M. Schmid, 2021-10-28; supplements/Imada_group_overview.md): t1 = 1, t2 = -0.25, U = 8, boundary AP, Ne = 214; the energy, variance, dof and einf match it and are unchanged since 2021. Open: no public file shows whether the run put the antiperiodic sign on the next-nearest bonds crossing the boundary (and the signed translations of the K = 0 projection); only the group's 2021 trans.def and qptransidx.def would. The linked mVMC.sh is a 2024 reconstruction that writes a periodic Hamiltonian and aborts on PA instances (APFlag 1, three-column qptransidx.def). The printed '(4)' and the Sigma column disagree. The record stands; nothing points to an error (qmbl-verify 2026-10-07, ruling 6a ).

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): 1 paper read, 0 of them with an energy on this page.

This instance as data

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