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 (2)
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 (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.
Projected or fixed-node estimates (1)
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.
How these numbers were read
Where each number below was read, and how it was checked.
-0.86036(6), 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. -55.063 | 0.004 (total energy, units of t)
8 x 8 PBC, half filling, U = 4. Prior finding stands: Qin et al. PRB 94, 085103 Table IV prints -55.05(1), so the row is VarBench's own AFQMC (uploaded 2022-03-16), cited to the Science paper. 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.
-0.8602(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 = -55.05(1) (total), 8 x 8, U = 4, PBC
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 8 x 8, column U = 4 PBC. Half filling: N_up = N_dn = 32. 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.
-0.859156(8), VMC (PP + RBM)
Checked 2026-09-29 · scripts/Hubbard/square_64_P_32_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. -54.986(3) | 0.00049 | 0.47(1) | ... (U=4) (Ne = 64) | alpha = 4 (overview; VarBench file '... (U=4), alpha = 4')
8x8 P-P half filling 32 + 32, U = 4, SU(2) + K = 0 projections, one Lanczos step, alpha 4: the instance and the printed method; variational right. History: per-site -0.849567 (earlier run, 2021-10-05), -54.986(3) since 2021-10-25, alpha added 2021-11-14; unchanged. Best variational row on the instance by 0.068 (HB K=1 -54.9184, Néel VMC -54.81638); the stochastic-exact AFQMC rows (-55.063(4), -55.05(1)) lie 0.07 below, as they must. See the sigma question.
-0.8581, VMC (HB)
Checked 2026-09-15 · source text of arXiv:2606.00924 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. −0.8581
Table II caption 'Energies per site for the Hubbard model at half-filling under periodic boundary conditions...'; row '8 x 8', column 'U = 4', sub-column 'K = 1' | Table II states full periodic BC (not PA) at half filling (N_up=N_down=n_sites/2); 8x8 row + U=4 column fix size/coupling; per-site t=1 units confirmed by QMC row matching this instance's exact_per_site (see paper-level convention note).