The Hamiltonian
H = −t ∑⟨ij⟩ (c†icj + h.c.) + V ∑⟨ij⟩ ninj
Spinless fermions, t = 1, V = 1, 32 particles on 64 sites.
- lattice
- square
- sites
- 64
- boundary
- periodic
- Nf
- 32
- V
- 1
- energies
- 4
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 (3)
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.
-0.4607(4), CT-QMC
Checked 2026-09-18 · varbench/methods 3edb6a9 scripts/tV/square_64_P_32_1/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 = -0.9606551057511292, error 0.0003595724232837266 (per site)
chain periodic (BCmodifier empty), N_f = N/2, V as the instance; Sign = 1 exactly
-0.4590635, DMRG
Checked 2026-09-29 · varbench/methods ed31bb0 scripts/tV/square_64_P_32_1/dmrg.sh -> programs/dmrg_itensors/tv_2d.jl, common.jl (ITensors.jl, Manifest pinned)
Reported as. VarBench tV/square_64_P_32_1.md: Energy -29.38006427648373 | Sigma (empty) | Energy Variance 0.6929288757939429 | DOF 32 | Einf 31.49206349206349 | Method "DMRG (maxbonddim = 4096)"
Script builds the instance (8x8 torus, fermionic PBC through ITensors' Jordan-Wigner handling of 'Fermion' OpSums, Nf = 32 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); printed as 'DMRG (chi_max = 4096) -29.380064' in arXiv:2210.05871v4 Table 4 (new in v4, 2023-03-14, a quote). 0.102 above the exact (stochastic) LCT-INT row -29.4819(230) (4.4 of its sigma), as an unconverged 8x8 MPS should be; variance 0.693. Record row when read (the best bound then).
-0.4561552, 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. -29.193934
Table 4 (System L=8,Np=32), V/t=1 column, HF row | Total energy / N=64 matches catalogue Nf=32 (Np=32) tV/square_64 instances; no catalogue exact_per_site exists at this size, cross-checked instead via the DMRG V/t=4 row matching catalogue best_per_site exactly.
-0.4497(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. -28.78(2)
Table 4, V/t=1 column, arSJVMC row | Same instance.