The Hamiltonian
H = −t ∑⟨ij⟩ (c†icj + h.c.) + V ∑⟨ij⟩ ninj
Spinless fermions, t = 1, V = 1, 5 particles on 16 sites.
- lattice
- square
- sites
- 16
- boundary
- periodic
- Nf
- 5
- V
- 1
- energies
- 7
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 (6)
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.
-0.6400407, ED
Checked 2026-09-29 · QMBL ED cost run checks/cost/ed/results/tV--square_16_P_5_1.json (checks/cost/ed/run_ed.py: NetKet 3.22.4 operator, sparse CSR built in blocks, SciPy eigsh/ARPACK k=1 SA; commit 716e043); results file read locally, no rerun
Reported as. energy -10.240650492312088 (stored_exact -10.240650492312078, relative_difference 1.04e-15, reproduces_stored true)
|reproduced - stored| = 1.07e-14 (relative 1.04e-15); within 1e-9 relative. Sector N = 5, dimension 4368, 61 Lanczos steps (the file states no residual). The stored value is the source's own ED (varbench@2024-10-22), so two codes agree; a third code, E1's node Lanczos (qmbl-runs/qmbl-verify/out/E1-exact-recompute/results/tV__square_16_P_5_1.json), gives -10.2406504923121 (1.07e-14 from the cost run). Also read by VB4-tfising-tv#38 (scope config: ed_netket.sh: python3 ed.py --ham tv --boundary peri --ham_dim 2 --L 4 --V 1 --Nf 5; ed.py lines 7-10; ham.py lines 49-63; arXiv:2210.05871v4 p. 32 Table 2, row 'Lanczos ED (QuSpin)').
-0.6400407, DMRG
Checked 2026-09-29 · varbench/methods ed31bb0 scripts/tV/square_16_P_5_1/dmrg.sh -> programs/dmrg_itensors/tv_2d.jl, common.jl (ITensors.jl, Manifest pinned)
Reported as. VarBench tV/square_16_P_5_1.md: Energy -10.240650492312072 | Sigma (empty) | Energy Variance 5.684341886080802e-14 | DOF 5 | Einf 2.666666666666667 | Method "DMRG (maxbonddim = 74)"
Script builds the instance (4x4 torus, fermionic PBC through ITensors' Jordan-Wigner handling of 'Fermion' OpSums, Nf = 5 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); 74 = sum_n min(C(8,n), C(8,5-n)), the exact MPS bond dimension at the central cut. 6e-15 above the ED -10.240650492312078 (exact MPS). Record row. M6 flag (V-score 1.7e-15) settled: exact representation, not an error.
-0.63987(6), 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. -10.238(1)
Table 2, V/t=1 column, arSJVMC row | Same instance.
-0.625, 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. -10.000000
Table 2, V/t=1 column, HF row | Same as convention note above.