The Hamiltonian
H = −t ∑⟨ij⟩ (c†icj + h.c.) + V ∑⟨ij⟩ ninj
Spinless fermions, t = 1, V = 4, 16 particles on 32 sites.
- lattice
- chain
- sites
- 32
- boundary
- periodic
- Nf
- 16
- V
- 4
- energies
- 3
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 (2)
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 (1)
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.
-0.2344426, ED
Checked 2026-09-29 · prior recomputation: E2 ed_full.py, output qmbl-runs/qmbl-verify/out/E2-exact-remote/results/tV__chain_32_P_16_4.json (evidence cited by the plan: qmbl-verify/out/E2-exact-remote (verified))
Reported as. -7.502161670760199 (true residual ||Hx-Ex||/||x|| = 5.63e-10); N_f = 16, full space
Real recomputation (code and output file). |E - stored| = 8.74e-13 (relative 1.16e-13). Second determination: the stored value is the source's own ED (VarBench), a different code. Also read by VB4-tfising-tv#34 (scope config: ed_lattice_symmetries.sh: python3 ed_ls.py --ham tv --boundary peri --ham_dim 1 --L 32 --V 4 --Nf 16; ham.py lines 49-63; vmc.py lines 58-62 (SpinOrbitalFermions(32, n_fermions=16))).
-0.2344426, DMRG
Checked 2026-09-29 · varbench/methods ed31bb0 scripts/tV/chain_32_P_16_4/dmrg.sh -> programs/dmrg_itensors/tv_1d.jl, common.jl (ITensors.jl, Manifest pinned)
Reported as. VarBench tV/chain_32_P_16_4.md: Energy -7.50216163113578 | Sigma (empty) | Energy Variance (empty) | DOF 16 | Einf 30.96774193548387 | Method "DMRG (maxbonddim = 200)"
Script builds the instance (32-site ring, fermionic PBC through ITensors' Jordan-Wigner handling of 'Fermion' OpSums, Nf = 16 fixed by the QN-conserving MPS, V = 4, 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: Qi Yang's upload 75839cf (2022-12-01), before the ITensors script was linked (2024-07-19). 3.96e-8 above the ED -7.5021616707610725. Record row when read (the best bound then).
-0.2340(6), CT-QMC
Checked 2026-09-18 · varbench/methods 3edb6a9 scripts/tV/chain_32_P_16_4/lct_int_inputs/{params.in,test.out.h5}, read with h5py; the shift from programs/SpinlesstV-LCT-INT/README.md and PRB 91, 235151 Eq. 3
Reported as. Energy/mean/value = -1.2340126507640288, error 0.0006419694839270352 (per site)
chain periodic (BCmodifier empty), N_f = N/2, V as the instance; Sign = 1 exactly