The Hamiltonian
H = −t ∑⟨ij⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t = 1, U = 8, N↑ = N↓ = 32 on 64 sites.
- lattice
- square
- sites
- 64
- boundary
- periodic/open
- Nf
- 32
- U
- 8
- 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.
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.
-0.5005(2), AFQMC
Checked 2026-10-15 · published PDF Science 386, 296 (2024), sources/journal/science386-296.pdf, sha256 80f8b792, and Supplementary Materials, sources/journal/science386-296-sm.pdf, sha256 4539b5df (node fetch: Cloudflare 403; fetched in the automation browser after the article page loaded); arXiv:2302.04919v2 compared; the number itself from the VarBench dataset v1.1.0 = ref. [63] of the paper (Zenodo 10.5281/zenodo.13377360, the GitHub release of tag v1.1.0, commit 0064b70; the Zenodo API timed out from this host, so the tag in varbench-history was read)
Reported as. -32.029 | 0.015 (total energy, units of t)
8 x 8 half filling, U = 8, PO boundary (periodic one direction, open the other; bipartite, so sign-free at half filling). Science SM S3 E 1-2 (journal version, same text as arXiv v2 Suppl. S3 E): 'For sign-problem-free systems, e.g., the repulsive Hubbard model (U > 0) at half-filling or the spin-balanced attractive Hubbard model (U < 0) [135, 147], numerically exact results are given by AFQMC using a generalized Metropolis algorithm with force bias [30, 148]'; 'For all systems computed in this work, we have either extrapolated Delta tau to zero or set it to a fixed value (typically Delta tau = 0.01) and verified that the Trotter error is well within our statistical error.' Hamiltonian (S5) H = -t sum c+c + U sum n_up n_dn, no shift, t = 1, N_up = N_dn. Neither the Science main text nor the SM prints any of these energies (searched both PDFs for the stored digits); the dataset is the printing source. Exact (stochastic) with sigma = AFQMC error bar; peer-reviewed venue. No duplicate on the instance.
-0.5000(3), DMRG
Checked 2026-09-29 · VarBench method string + VarBench history (git clone of github.com/varbench/varbench, partial, read with git log --follow; old names normalised) + run script programs/dmrg_itensors_hubbard/square_64_PO_32_8.jl (varbench/methods@ed31bb0)
Reported as. Extrap Energy -32.0027 +/- 0.0206
Transcribed as printed; bound_type extrapolated right; the +/- (M6) is the uploader's extrapolation uncertainty, undefined. 1.0 combined sigma above the AFQMC -32.029(15) on the same instance.
-0.4984344, DMRG
Checked 2026-09-29 · run script programs/dmrg_itensors_hubbard/square_64_PO_32_8.jl (varbench/methods@ed31bb0) read in full; VarBench history (git clone of github.com/varbench/varbench, partial, read with git log --follow; old names normalised)
Reported as. -31.8998 | | 1.107 | DMRG(MaxLinkDim=10000, MaxTruncErr ~ 3.4E-5, Extrap Energy -32.0027 +/- 0.0206) (VarBench upload a6b91d3, Silvester 2022-04-01)
8x8 cylinder, periodic in one direction, open in the other (PO), half filling 32 + 32, U = 8, t = 1: the instance. History: -31.995 +/- 0.01745 at 'max bond dimension ~10500/10900' (2022-01-14), the variance 1.107 added 2022-03-29, the factor-4 upload -127.599 / var 17.717 / extrap -128.011 +/- 0.082 (744e2a5, 2022-03-31), then -31.8998 (= -127.599/4 = -31.89975) with var 1.107 (= 17.717/16) on 2022-04-01; the 1.107 'restore' seen in the history is this division, not a silent reversion. 0.13 above the stochastic-exact AFQMC -32.029(15) on the instance: consistent. No log (scope config).