Spinless t-V square 8x8, V = 4, n = 0.5

Best values (rules §6)

best variational	-0.1601344	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⟩ (c^†_ic_j + h.c.) + V ∑_⟨ij⟩ n_in_j

Spinless fermions, t = 1, V = 4, 32 particles on 64 sites.

lattice: square
sites: 64
boundary: periodic
Nf: 32
V: 4
energies: 4

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).

Numerically exact (1)

Exact diagonalization, an exact solution, or sign-problem-free QMC where that is established. An exact energy is shown first, before the best of each other class. QMC is exact only within its error bar, which it must state, and reads exact (stochastic); an exact diagonalization outranks it. For a diagonalization resolved by symmetry sector only the ground state's sector is listed; the lowest energies of the other sectors are the instance's spectrum, in its JSON, and are not shown as the exact energy.

E/site	distance to class best	σ	Var(E)	V-score	method	source	code	year	comment
-0.153	n/a	2.6e-3	n/a	n/a	CT-QMC above a variational bound VarBench reference run	Wu et al. (2024)	run script	2024

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
-0.1601344best variational	0	n/a	2.86e-2	4.9e-5	DMRG VarBench reference run	Wu et al. (2024)	run script	2024
-0.1597047	4.3e-4	n/a	n/a	n/a	HF peer reviewed	Humeniuk et al. (2022)		2022
-0.1450	1.5e-2	6.3e-4	n/a	n/a	VMC (arSJVMC) peer reviewed	Humeniuk et al. (2022)	shifushuimenu/autoregressive_Slater-Jastrow	2022

Flagged rows

A flagged row stays listed in rank order but cannot be the best of its class. Rules §10 says how a flag is lifted or upheld.

above a variational bound, CT-QMC, -0.153(3)

Finding. Stored -9.7906 +- 0.166 sits 0.458 ABOVE the variational DMRG row -10.2486 (2.76 sigma) and 0.430 above Hartree-Fock (2.59 sigma), which an unbiased ground-state estimate can only do by fluctuation. The committed h5 is a second run of the same estimator: -10.0373 +- 0.1194, again 1.77 sigma above DMRG; the two runs combined, -9.953 +- 0.097, are 3.05 sigma above. On that h5 the vertex-count estimator (KinE + IntE2, error 20x smaller) gives -10.2491 +- 0.0069, 0.08 sigma from DMRG: the projection and the sign are fine and E0 is about -10.249; it is the single-random-site Wick Energy estimator whose error bar is understated at V = 4 (per-measurement variance 1096, binning reports no autocorrelation where the vertex observables show tau = 137). Below the 3 sigma of RULES.md 9.4, so not a validator issue; held off the record at its stated sigma.

How these numbers were read

Where each number below was read, and how it was checked.

-0.1601344, DMRG

Checked 2026-09-29 · varbench/methods ed31bb0 scripts/tV/square_64_P_32_4/dmrg.sh -> programs/dmrg_itensors/tv_2d.jl, common.jl (ITensors.jl, Manifest pinned)

Script builds the instance (8x8 torus, fermionic PBC through ITensors' Jordan-Wigner handling of 'Fermion' OpSums, Nf = 32 fixed by the QN-conserving MPS, V = 4, V n_i n_j without shift; einf and dof correct). Variational energy of the last sweep at the stated bond dimension, no extrapolation; no output log (config). Provenance: 96a2b26 (2023-02-02); printed as '-10.248601' in arXiv:2210.05871v4 Table 4 (a quote). 0.458 below the flagged LCT-INT row -9.7906(166) and 0.0275 below the HF row -10.2211008 (arXiv:2210.05871, other group); variance 0.0286. Record row. M11 flag (record 0.27% below the runner-up) settled: the runner-up is that HF determinant, a genuine margin for a correlated MPS; the only row that could compete, the LCT-INT, is flagged.

-0.1597047, HF

Checked 2026-09-15 · source text of arXiv:2210.05871 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. -10.221101

Table 4, V/t=4 column, HF row | Same as convention note above.

-0.1450(6), VMC (arSJVMC)

Reported as. -9.28(4)

Table 4, V/t=4 column, arSJVMC row; text notes this falls short of even the HF energy due to training difficulty | Same instance.

Coverage

Newest published number here: 2024.

This instance as data

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