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All instances / Hubbard

Hubbard square 8x8, U = 8, n ≈ 0.7813

Best values (rules §6)

best variational -0.91483(2) VMC (PP + RBM)
Wu et al. (2024) · the lowest unflagged variational energy, a strict variational bound
best projected -0.9252(1) CP-AFQMC
Levy et al. (2023) · 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†iσcjσ + h.c.) + U ∑i ni↑ni↓

t = 1, U = 8, N↑ = N↓ = 25 on 64 sites.

lattice
square
sites
64
boundary
periodic
Nf
25
U
8
energies
5

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 (3)

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
-0.91483best variational 0 1.6e-5 2.58e+0 6.9e-3 VMC (PP + RBM)
VarBench reference run
 Wu et al. (2024) run script 2024
-0.9148tied with best variational at 2σ 2.8e-5 1.0e-4 2.40e+0 6.4e-3 VAFQMC
peer reviewed
 Levy et al. (2023) 2023
-0.91012 4.7e-3 2.2e-5 3.46e+0 9.3e-3 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/sitedistance to class bestσVar(E)V-score methodsourcecodeyearcomment
-0.9252best projected 0 1.0e-4 n/a n/a CP-AFQMC
peer reviewed
 Levy et al. (2023) 2023
-0.913753 1.1e-2 9.4e-6 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.

-0.9252(1), CP-AFQMC

Checked 2026-09-15 · source text of arXiv:2308.08594 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. −0.9252(1)

Table I | 8x8=64 sites P; n=0.78125 => Nf=25 per spin; U=8; catalogue best_per_site for this instance is -0.91482812, so this quoted value is notably lower (stronger) than the current catalogue best

-0.91483(2), VMC (PP + RBM)

Checked 2026-09-29 · scripts/Hubbard/square_64_P_25_8/mVMC/{mVMC.sh, dat_input_makedef.txt}; programs/mVMC_makedef/{make_all.py, sub_trans.py, sub_coulombinter.py, sub_qptransidx.py, sub_modpara_namelist.py, sub_rbm_phidx.py} and mVMC source (programs/mVMC_RBM/src/mVMC/readdef.c, calham.c) read (varbench/methods@ed31bb0); VarBench Hubbard/supplements/Imada_group_overview.md (created ca26677, M. Schmid, 2021-11-14); VarBench history (git clone of github.com/varbench/varbench, partial, read with git log --follow; old names normalised)

Reported as. -58.549(1) | 0.001031 | 2.58(1) | ... (U=8) (Ne = 50) | alpha = 8, with 1x1 subblattice (overview; VarBench file '... (U=8), alpha = 8 with 1x1 RBM subspace')

8x8 P-P, t = 1, U = 8, 25 + 25, SU(2) + K = 0 projections, one Lanczos step, alpha 8 with translation-invariant RBM: the instance and the printed method; variational right; here the parenthetical (1) and Sigma 0.001031 agree. History: per-site -0.914842 / 1.96e-5 (earlier run, 2021-10-05), -58.549(1) since 2021-10-21, unchanged. Record check: VAFQMC (N_l=4) -58.5472(64) is 0.3 sigma above, the two tie at 2 sigma; the mVMC row is lower. No log (scope config).

-0.9148(1), VAFQMC

Checked 2026-09-15 · source text of arXiv:2308.08594 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. −0.9148(1)

Table I | same instance as above | variance reported as 2.4, not stored: convention not verified

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): 7 papers read, 0 of them with an energy on this page.

This instance as data

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