How these numbers were read
Where each number below was read, and how it was checked.
-0.736431(9), 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-4x8_32_PO_14_8.jl (varbench/methods@ed31bb0)
Reported as. extrap energy -23.5658 +/- 0.0003
Transcribed as printed and consistent with the division by 4 of the factor-4 upload (unlike the variational energy of the same line, see that row). bound_type extrapolated right; M6: the +/- is the uploader's extrapolation uncertainty, not defined anywhere. 14 up + 14 down on the 4x8 cylinder (PO), U = 8.
-0.7363906, DMRG
Checked 2026-09-29 · VarBench history of Hubbard/rectangular_4x8_PO_14_14_8.md (git log, git show; varbench-history clone), read by VB2 on 2026-09-29; run script programs/dmrg_itensors_hubbard/rectangular-4x8_32_PO_14_8.jl (varbench/methods@ed31bb0)
Reported as. -23.5645 (total)
The stored number is J. M. Silvester's upload of 2022-04-01, transcribed correctly, and the script's configuration matches the instance (U = 8, 8 x 4, periodic along the 4-site direction, 14 + 14). It may have lost a digit: the uploader's factor-4 entry of the day before was -94.257, and -94.257 / 4 = -23.56425, while the same day's division turned the variance (0.24766 -> 0.0155) and the extrapolated energy (-94.263 +/- 0.001 -> -23.5658 +/- 0.0003) exactly, as did his conversions of the same week on other instances (-192.213 -> -48.0533, -127.599 -> -31.8998). -23.5645 lies 2.5e-4 below -23.56425, outside the +/-1.25e-4 that the three decimals of -94.257 allow. Kept as uploaded (ruling E13, 2026-10-02); only the uploader can say which is right. The row holds the instance's variational record either way.
-0.736329, DMRG
Checked 2026-09-15 · source text of arXiv:2510.11710 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. -0.736329(9)
Table I (Appendix E): 'DMRG (8x4) -0.736329(9) NA'; caption: 'results for the 8x4 and 8x8 cylinders and 8x8 torus lattices with U/t=8 and nh=1/8 hole doping' | 8x4 lattice = 'rectangular-4x8', 32 sites; text (Sec. IV.A) states this is an '8x4 lattice with open boundary conditions along the long side and periodic boundary conditions along the short side' i.e. cylinder = catalog boundary PO; nh=1/8 -> Ne=28, Nf=14, matching focus instance directly
-0.73500(3), 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. -0.7350(6) (energy per site, units of t)
Energy verified against the journal; sigma differs (ambiguous entry). Version difference: v2 printed -0.7309(6), the metastable half-filled-stripe state ('S = 0 or small values of S lead to a half-filled stripe configuration of higher energy'), the journal the lower filled-stripe state ('Larger values of S yield a filled-stripe configuration', main Sec. 3 B, p. 6); the stored value follows the journal and the first author's upload (b71872b, 2022-07-16, after v2 of 18 Jun 2022: -23.520 | 0.001 | 0.035(4)). 4 x 8, n = 7/8 = 14 up + 14 down at zero magnetization, U = 8, t = 1, Eq. 1 with no shift. N~ = 16 and alpha = 14 for L = 8 ('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. 0.0445 above the VarBench DMRG row -23.5645; not the record. 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 (the '8x4 cylinder' HFDS row -23.46944 is PRB 113, 245104's own computation).
-0.73342(8), VMC (HFDS-ViT)
Checked 2026-09-14 · arXiv HTML parsed locally, no LLM transcription
Reported as. E_0/M = -0.73342 (+/- 0.00008), sigma^2/M = 0.0225
Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Row "HFDS (8x4)".
-0.7332(6), VMC (JBf-ViT)
Checked 2026-09-14 · arXiv HTML parsed locally, no LLM transcription
Reported as. E_0/M = -0.7332 (+/- 0.0006), sigma^2/M = 0.0235
Read from Table 1 of arXiv:2510.11710 (U/t = 8, n_h = 1/8, energies per site, 131072 samples), parsed from the arXiv HTML. Row "JBf (8x4)".