How these numbers were read
Where each number below was read, and how it was checked.
-0.75114(6), 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/rectangular-4x16_64_PO_28_8.jl (varbench/methods@ed31bb0) read in full; the author's own copy MaxSilvester/Hubbard-and-Heisenberg-codes@b5c28f3 rectangular-4x16_64_PO_28_8.jl has the same Hamiltonian, filling and schedule
Reported as. extrap energy -48.073 +/- 0.004
Transcribed as printed; bound_type extrapolated right. M6 answer: the +/- is the uploader's extrapolation uncertainty, no code produces it and its definition is not stated (see the chain rows). Instance and method config as for the variational row.
-0.7508328, DMRG
Checked 2026-09-29 · programs/dmrg_itensors_hubbard/rectangular-4x16_64_PO_28_8.jl (varbench/methods@ed31bb0) read in full; the author's own copy MaxSilvester/Hubbard-and-Heisenberg-codes@b5c28f3 rectangular-4x16_64_PO_28_8.jl has the same Hamiltonian, filling and schedule; VarBench history (git clone of github.com/varbench/varbench, partial, read with git log --follow; old names normalised)
Reported as. -48.0533 | | 0.143 | DMRG(MaxLinkDim = 10000, MaxTruncErr~2.0E-05, extrap energy -48.073 +/- 0.004) (VarBench upload by J. Maxwell Silvester)
4x16 cylinder: periodic around the 4-site circumference, open along the 16-site length (PO), t = 1, U n_up n_dn with no shift, N_up = N_dn = 28, conserve_qns: the instance. Maximum bond dimension in the script equals the MaxLinkDim printed. Energy is the reached DMRG energy: variational is right. History: first uploaded 2022-03-29/31 as -192.0004 (var 2.256), then -192.213 (var 2.29, extrap -192.295 +/- 0.016); replaced 2022-04-01 (e2254f3) by -48.0533, var 0.143, extrap -48.073 +/- 0.004 = the old numbers divided by 4 (energy: -192.213/4 = -48.05325) and 16 (variance: 2.29/16 = 0.1431); the same factor-4 episode hit the 4x8 and 8x8 PO uploads that week. M11 (0.56% below the runner-up): the runner-up is the lambda = 8-pinned HFDS row -47.782 of PNAS 119, e2122059119; a DMRG state below it on a 4-leg cylinder is expected. No output log in either repository (scope config). Provenance note: in VarBench's copies of the ten rectangular scripts the variance line is evaluated on the initial product state ('E,psi = dmrg(H,ψ,sweeps)' then 'var = inner(H,ψ,H,ψ) - E^2'; ITensors v0.6.16 dmrg optimises psi = copy(psi0), src/lib/ITensorMPS/src/dmrg.jl line 189, so ψ stays the product state), so as committed they would print <ψ0|H^2|ψ0> - E^2 (for the half-filled 32_8 start, a Néel product state, 2 x 124 allowed hops give 248 - 32.923^2 = -836), not the stored positive variance. The author's repository fixed it on 2024-08-05 ('E,ψ = dmrg(...)' plus the var line, e.g. commits e78c997, 4d3e522); the stored variances were uploaded in 2022, before either script existed. Suggest pointing the code link at MaxSilvester/Hubbard-and-Heisenberg-codes or fixing varbench/methods.
-0.74659(8), VMC (HFDS)
Checked 2026-09-29 · PNAS 119(32), e2122059119 (2022), SI Appendix PDF (PMC9371695; sources/pnas.2122059119.sapp.PMC9371695.pdf, sha256 c6e62d9c...), pypdf layout mode, re-extracted 2026-09-29 (byte-identical to the earlier extraction); same cell read in arXiv:2111.10420v2 (18 Jun 2022) Appendix H (sources/2111.10420v2.pdf, pypdf layout)
Reported as. -0.7466(8) (energy per site, units of t)
Energy verified; sigma differs (ambiguous entry). Boundary: 'Periodic boundary conditions are considered along the short side of the rectangle, while both open and periodic boundary conditions are taken along the long side ... labelled as PBC-PBC and PBC-OBC' (SI Sec. 8 C): PBC-OBC = PO. 4 x 16, n = 7/8 = 28 up + 28 down at zero magnetization, U = 8, t = 1, Eq. 1 with no shift. N~ = 16 and alpha = 6 for L = 16 ('A good trade-off ... is achieved by the addition of N~ = 16 hidden fermions for all system sizes. We also consider a two-layer fully connected neural network of hidden-unit density α = {60, 14, 6} for the L = {4, 8, 16} sizes', main Sec. 3 B, p. 6), lambda = 8 soft mean-field constraint on phi_v ('To guide the wave-function ansatz toward the λ = 8 stripe order, we add a soft mean-field constraint', p. 6): method string and detail right. Variational VMC bound, own computation. 'At L = 16, the best variational energy found by tuning S lies between the DMRG energies that correspond to the half-filled and filled stripes' (p. 7). 0.271 above the VarBench DMRG row -48.0533; not the record. Upload 8df3719 (jrm874, 2022-07-16): -47.782 | 0.005 | 0.47(7), never changed; energy_variance upload-only. M6/M8 settled from the source: a sampled VMC energy ('General expectation values and gradients of the objective function are computed using Markov chain Monte Carlo sampling', main text Sec. 2 C, p. 5) whose stored sigma is the uploader's Monte Carlo error; units.mjs counts the row deterministic only because 'Soft mean-field constraint' matches /mean[- ]field/ (units.mjs line 20), which changes nothing while sigma is stored (isSampled is consulted only when sigma is null). Venue PNAS, peer reviewed ('Edited by Angel Rubio ... received December 6, 2021; accepted June 28, 2022'; 'This article is a PNAS Direct Submission'); the imported row carries no peer_reviewed field (loader-level, see report). No duplicate (other rows: VarBench DMRG, reached and extrapolated).