How these numbers were read
Where each number below was read, and how it was checked.
-0.220223, ED
Checked 2026-09-29 · this pass on an AMD EPYC 9654 server: ed_fsym.py (E2 ed_sym.py structure with fermionic signs), sectors of translations x C4v
Reported as. -7.9280262409783155 (true residual 3.84e-10)
H = -t sum_<ij> (c+c + h.c.) + V sum n_i n_j, t = 1, V = 10, 13 spinless fermions on the 6x6 torus (72 bonds). ed_fsym.py (fermionic symmetric basis: translations x C4v, sign sigma(g, rep) in the table; validated on the 4x4 torus, N_f = 5, against the stored values at V = 0.01, 0.1, 1, 10 with all 16 momenta). Ground state in the sectors computed: Gamma B1, -7.9280262409783155 (true residual 3.84e-10, sector dimension 8026392 of C(36,13) = 2.31e9). Sectors computed (Lanczos to 1e-9 with second pass): Gamma {C2:-1,sigma_axis(j->-j):1,sigma_axis(i->-i):-1} -6.91618234; Gamma A1 -7.10991483; Gamma A2 -6.02200005; Gamma B1 -7.92802624; Gamma B2 -5.82664355; k=(1,0) (translations only: all irreps at this k) -7.83501530; k=(1,1) (translations only: all irreps at this k) -7.77429202; k=(2,0) (translations only: all irreps at this k) -7.72881151; k=(2,1) (translations only: all irreps at this k) -7.72050574; k=(2,2) (translations only: all irreps at this k) -7.77135552; k=(3,0) (translations only: all irreps at this k) -7.69167783; k=(3,1) (translations only: all irreps at this k) -7.72663375; k=(3,2) (translations only: all irreps at this k) -7.84092710; M {C2:1,sigma_axis(j->-j):1,sigma_axis(i->-i):1} -7.91872065; M {C2:-1,sigma_axis(j->-j):1,sigma_axis(i->-i):-1} -6.91641124; M {C2:1,sigma_axis(j->-j):-1,sigma_axis(i->-i):-1} -6.02199047. All 10 momentum stars of the 6x6 torus are covered (Gamma and M with every point-group irrep, the other 8 stars with translations only, which holds every irrep at that k), so this is the global ground state; the next level is M {C2:1,sigma_axis(j->-j):1,sigma_axis(i->-i):1} -7.91872065 (+0.00931). |E - stored| = 5.51e-14 (relative 6.95e-15). Second determination: the stored value is VarBench's own ED (lattice-symmetries, a different code and basis); E2's unconverged full-space Ritz value -7.928026237338893 (estimate 1.4e-4) is an upper bound 3.64e-9 above it.
-0.2201239, DMRG
Checked 2026-09-29 · varbench/methods ed31bb0 scripts/tV/square_36_P_13_10/dmrg.sh -> programs/dmrg_itensors/tv_2d.jl, common.jl (ITensors.jl, Manifest pinned)
Reported as. VarBench tV/square_36_P_13_10.md: Energy -7.924461196776301 | Sigma (empty) | Energy Variance 0.12396407883914667 | DOF 13 | Einf 89.14285714285715 | Method "DMRG (maxbonddim = 4096)"
Script builds the instance (6x6 torus, fermionic PBC through ITensors' Jordan-Wigner handling of 'Fermion' OpSums, Nf = 13 fixed by the QN-conserving MPS, V = 10, V n_i n_j without shift; einf and dof correct). Variational energy of the last sweep at the stated bond dimension, no extrapolation; no output log (config). Provenance: 96a2b26 (2023-02-02). 3.57e-3 above the ED -7.928026240978371 (whose sector is questioned in its own entry; the bound holds either way); variance 0.124. Record row.
-0.2008(8), VMC (arSJVMC)
Checked 2026-09-15 · source text of arXiv:2210.05871 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. -7.23(3)
Table 3, V/t=10 column, arSJVMC row | Same instance.
-0.191906, HF
Checked 2026-09-15 · source text of arXiv:2210.05871 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. -6.908615
Table 3, V/t=10 column, HF row | Same as convention note above.
-0.171(2), VMC (SJ + RBM)
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. -6.168047091994236730 | 8.708582883788365536e-02 | var 3.426841426704378790e+01
6 x 6, Nf = 13, V = 10. Science SM S1 Eq. (S4) (journal): H = -t sum (c+_i c_j + h.c.) + V sum n_i n_j, t = 1, Nf fixed (total, spinless) — no V(n-1/2)(n-1/2) shift, matching the stored convention. Method paper Romero, Nys & Carleo, Commun. Phys. 8, 46 (2025) read in its journal version (Europe PMC full-text XML PMC11779646, sources/journal/commphys8-46.epmc.xml, sha256 a2e037eb; Supplementary Information sources/journal/commphys8-46-supp.pdf, sha256 0489aee4): it studies closed-shell L = 4, 8, 10 spectra and L = 4..10 structure factors and prints none of the 6 x 6, Nf = 13 energies (searched both for the stored digits and per-site forms), and its data statement reads 'The data underlying this study are accessible upon reasonable request'; the dataset is the printing source, as the row's reference already says. No duplicate.