Hubbard square 8x8, U = 4, n ≈ 0.7813

Best values (rules §6)

best variational	-1.13048(8)	VAFQMC Levy et al. (2023) · the lowest unflagged variational energy, a strict variational bound
best projected	-1.13253(3)	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^†_iσc_jσ + h.c.) + U ∑_i n_i↑n_i↓

t = 1, U = 4, N_↑ = N_↓ = 25 on 64 sites.

lattice: square
sites: 64
boundary: periodic
Nf: 25
U: 4
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/site	distance to class best	σ	Var(E)	V-score	method	source	code	year
-1.13048best variational	0	8.0e-5	5.00e-1	2.0e-3	VAFQMC peer reviewed	Levy et al. (2023)		2023
-1.130375tied with best variational at 2σ	1.1e-4	5.2e-6	5.90e-1	2.4e-3	VMC (PP + RBM) VarBench reference run	Wu et al. (2024)	run script	2024
-1.12683	3.6e-3	1.4e-5	1.43e+0	5.8e-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/site	distance to class best	σ	Var(E)	V-score	method	source	code	year	comment
-1.13253best projected	0	3.0e-5	n/a	n/a	CP-AFQMC peer reviewed	Levy et al. (2023)		2023
-1.128397	4.1e-3	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.

-1.13253(3), 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. −1.13253(3)

Table I | 8x8=64 sites P; n=0.78125 filling => Nf=25 per spin; U=4; not starred (not deemed effectively exact); catalogue best_per_site for this instance is -1.130375, so this quoted value is lower (a stronger record) than the current catalogue best

-1.13048(8), VAFQMC

Reported as. −1.13048(8)

Table I | same instance as above; text: 'at n=0.78125 we obtain nearly the same energies for U=4 and U=8...to a very sophisticated VMC wave function', i.e. the SU(2)+Lanczos VMC† row | variance reported as 0.5, not stored: convention not verified

-1.130375(5), VMC (PP + RBM)

Checked 2026-09-29 · scripts/Hubbard/square_64_P_25_4/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. -72.344(4) | 0.000331 | 0.59(2) | ... (U=4) (Ne = 50) | alpha = 8, with 1x1 subblattice (overview); VarBench file: '-72.344(4) | 3.31e-4 | 0.59(2) | mVMC ... (U=4), alpha = 8 with 1x1 RBM subspace'

The generator targets exactly these runs (README: 'we focus on the case of Ns=8x8, Ne=50 or 64, U=4 or 8'): 8x8 torus P-P, t = 1 (trans.def writes 1.0, mVMC H_T = -sum t_ij c+c per calham.c), U n_up n_dn, 25 + 25 electrons (Ne total 50, 2Sz 0), SU(2) and K = 0 projections, one Lanczos step, RBM with translation-invariant (1x1) weights (sub_rbm_phidx.py), alpha 8: the instance and method as printed; variational (Lanczos-step energy of a variational state) right. History: first uploaded 2021-10-05 (2ab3a17) as '-1.130374 | 6.65e-6 | 1.13e-4': the energy per site (x 64 = -72.3439), the sigma per electron (x 50 = 3.3e-4) and the variance relative to E^2 (x 72.344^2 = 0.59), i.e. the same state; as totals -72.344(4) | 3.31e-4 | 0.59(2) since 2021-10-21 (16077f2), unchanged. See the separate sigma question.

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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