The Hamiltonian
H = −t ∑⟨ij⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t = 1, U = 8, N↑ = N↓ = 72 on 144 sites.
- lattice
- square
- sites
- 144
- boundary
- periodic/antiperiodic
- Nf
- 72
- U
- 8
- energies
- 3
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.
Strict variational upper bounds (2)
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.
How these numbers were read
Where each number below was read, and how it was checked.
-0.5249(2), AFQMC
Checked 2026-09-15 · arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription
Reported as. E = -75.58(3) (total), 12 x 12, U = 8, PBC-APBC
Table IV, "Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit." Row 12 x 12, column U = 8 PBC-APBC. Half filling: N_up = N_dn = 72. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.
-0.5224, VMC (tensor BF)
Checked 2026-09-15 · source text of arXiv:2507.01856 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. -0.5224
Table 1 (ar5iv id S3.T1), row n=1,U=8,12x12,A-PBC, column E_p=1 | same as above
-0.5205, VMC (tensor BF)
Checked 2026-09-15 · source text of arXiv:2507.01856 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. -0.5205
Table 1 (ar5iv id S3.T1), row n=1,U=8,12x12,A-PBC, column E_p=0 | N=144, Nf=72=N/2 (n=1), A-PBC boundary matches instance's PA boundary