The Hamiltonian
H = −t ∑⟨ij⟩ (c†icj + h.c.) + V ∑⟨ij⟩ ninj
Spinless fermions, t = 1, V = 1, 13 particles on 36 sites.
- lattice
- square
- sites
- 36
- boundary
- periodic
- Nf
- 13
- V
- 1
- energies
- 6
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 (5)
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.
-0.6132603, ED
Checked 2026-09-29 · prior recomputation: E2 ed_full.py, output qmbl-runs/qmbl-verify/out/E2-exact-remote/results/tV__square_36_P_13_1.json (evidence cited by the plan: qmbl-verify/out/E2-exact-remote (verified))
Reported as. -22.077372357830814 (Lanczos Ritz value; estimate beta_m|s_m| = 6.24e-10; no second pass, so no true residual); N_f = 13, full space (2.31e9)
Real recomputation (code and output file). |E - stored| = 4.62e-14 (relative 2.09e-15). Second determination: the stored value is the source's own ED (VarBench), a different code; this pass, ed_fsym.py (fermionic symmetric basis, Gamma C4v B1, dimension 8026392): -22.077372357830782 with true residual 4.86e-10, 3.20e-14 from E2 (out/VE-exact-recompute/results/tvsq/tV__square_36_P_13__V_1__k0-0__c4v-irrep2.json). Also read by VB4-tfising-tv#44 (scope config: ed_lattice_symmetries.sh: python3 ed_ls.py --ham tv --boundary peri --ham_dim 2 --L 6 --V 1 --Nf 13; arXiv:2210.05871v4 p. 32 Table 3).
-0.6132263, DMRG
Checked 2026-09-29 · varbench/methods ed31bb0 scripts/tV/square_36_P_13_1/dmrg.sh -> programs/dmrg_itensors/tv_2d.jl, common.jl (ITensors.jl, Manifest pinned)
Reported as. VarBench tV/square_36_P_13_1.md: Energy -22.076147854622903 | Sigma (empty) | Energy Variance 0.01800294336391062 | DOF 13 | Einf 8.914285714285715 | 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 = 1, 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). 1.22e-3 above the ED -22.07737235783086; variance 0.018. Record row when read (the best bound then; the VMC SBJ K=0 row -22.07494(24) is 1.2e-3 higher).
-0.6116(1), 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. -22.01792828440505900 | 3.586025996321709226e-03 | var 5.431811628015638105e-01
6 x 6, Nf = 13, V = 1. 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.
-0.6069(3), 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. -21.85(1)
Table 3, V/t=1 column, arSJVMC row | total/36 = per-site
-0.5925926, 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. -21.333333
Table 3, V/t=1 column, HF row | total/36 = per-site