The Hamiltonian
H = −∑⟨ij⟩ σziσzj − h ∑i σxi
Transverse field h = 1. Already in Pauli operators, with no S·S term to rescale, so the per-site energy is E/N.
- lattice
- chain
- sites
- 32
- boundary
- periodic
- h
- 1
- energies
- 5
Stored as a total energy; quoted here as
E/N (Pauli), 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 (4)
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.
-1.273751, Exact solution
Checked 2026-09-29 · prior recomputation: E1 free-fermion closed form (checked against ED at N = 16, 18, 20), output qmbl-runs/qmbl-verify/out/E1-exact-recompute/results/TFIsing__chain_32_P_1.json (evidence cited by the plan: qmbl-verify/out/E1-exact-recompute (verified))
Reported as. -40.76003249419223; closed form, even-parity sector
Real recomputation (code and output file). |E - stored| = 0 (relative 0). Second determination: the closed form matches E1's ED at N = 16, 18, 20 (h = 0.5, 1) to < 4e-12; the stored value is VarBench's exact solution. Also read by VB4-tfising-tv#2 (scope value: exact_ising_1d.py: energy = sum_{i<L} -sqrt(1 + Gamma^2 + 2 Gamma cos(pi (2i+1)/L))).
-1.273751, DMRG
Checked 2026-09-29 · varbench/methods ed31bb0 scripts/TFIsing/chain_32_P_1/dmrg.sh -> programs/dmrg_itensors/ising_1d.jl, common.jl, args.jl (ITensors.jl, Manifest pinned)
Reported as. VarBench TFIsing/chain_32_P_1.md: Energy -40.76003249394725 | Sigma (empty) | Energy Variance 2.8e-9 | DOF 32 | Einf 0 | Method "DMRG (bond dimension = 71)"
Script builds the instance (Pauli TFIM +sum ZZ + h sum X on the periodic 32-site chain, h = 1, equivalent to -sum ZZ - h sum X on this bipartite cluster; no symmetry or magnetisation constraint). The label's bond dimension is the one reached under the 1e-12 cutoff with max_B = 128. Energy is the last sweep's, variational, no extrapolation; variance <H^2> - E^2 of the final MPS (ising_*.jl lines 63-65). No output log (config scope). Value since f6237d0 (2022-08-29); script link attached 2024-07-19 (f88f9ba, cd362c4). Record row. 2.45e-10 above the exact solution -40.76003249419223; M6 flag (V-score 5.4e-11) settled: converged state at criticality, not an error.
-1.273676(4), VMC (RBM)
Checked 2026-09-29 · varbench/methods ed31bb0 scripts/TFIsing/chain_32_P_1/vmc_rbm.sh -> programs/vmc_netket/vmc.py, args_parser.py (NetKet 3.13.0 per requirements.txt)
Reported as. VarBench TFIsing/chain_32_P_1.md: Energy -40.75763 | Sigma 1.4e-4 | Energy Variance 0.01991942 | DOF 32 | Einf 0 | Method "RBM (alpha = 1)"
Script builds the instance (periodic chain of 32 sites, h = 1, Pauli TFIM) with nk.models.RBM(alpha = 1, log_cosh, float32; vmc.py lines 125-140); +0.00240 from the exact -40.76003249419223 (above it, as a variational energy must be). sigma check: sqrt(Var/2^20) = sqrt(0.01991942/1048576) = 1.38e-4 vs stored 0.00014 (vmc.py lines 268-270: final estimate with estim_size = 2^20 samples); variance is Var(H) of the total energy. The linked script was attached on 2024-07-30 (fdb4bfb Jastrow / 5047bfd RBM) to a number already in VarBench (10f5c12 (2022-07-11) as -40.7576, digits extended in af1d48f (2022-09-01)); vmc.py uses nk.operator.IsingJax (vmc.py line 78), so it is a later reimplementation of the baseline that documents the ansatz and protocol, not the output of the run that produced this number (no log in the methods repo). Value unchanged since 10f5c12; no value was changed and restored.
-1.27191(2), VMC (Jastrow)
Checked 2026-09-29 · varbench/methods ed31bb0 scripts/TFIsing/chain_32_P_1/vmc_jastrow.sh -> programs/vmc_netket/vmc.py, args_parser.py (NetKet 3.13.0 per requirements.txt)
Reported as. VarBench TFIsing/chain_32_P_1.md: Energy -40.70126 | Sigma 5.3e-4 | Energy Variance 0.2877156 | DOF 32 | Einf 0 | Method "Jastrow baseline"
Script builds the instance (periodic chain of 32 sites, h = 1, Pauli TFIM) with nk.models.Jastrow (vmc.py lines 118-124); +0.0588 from the exact -40.76003249419223 (above it, as a variational energy must be). sigma check: sqrt(Var/2^20) = sqrt(0.2877156/1048576) = 5.24e-4 vs stored 0.00053 (vmc.py lines 268-270: final estimate with estim_size = 2^20 samples); variance is Var(H) of the total energy. The linked script was attached on 2024-07-30 (fdb4bfb Jastrow / 5047bfd RBM) to a number already in VarBench (59e621a (2022-09-13; earlier -40.69845(50) af1d48f)); vmc.py uses nk.operator.IsingJax (vmc.py line 78), so it is a later reimplementation of the baseline that documents the ansatz and protocol, not the output of the run that produced this number (no log in the methods repo). Value unchanged since 59e621a; no value was changed and restored.