The Hamiltonian
H = −∑⟨ij⟩ σziσzj − h ∑i σxi
Transverse field h = 3. Already in Pauli operators, with no S·S term to rescale, so the per-site energy is E/N.
- lattice
- square
- sites
- 400
- boundary
- open
- h
- 3
- energies
- 7
Stored as a total energy; quoted here as
E/N (Pauli), which is what the papers report
(how the conversion works).
How these numbers were read
Where each number below was read, and how it was checked.
-3.18197, DMRG
Checked 2026-09-29 · arXiv:2204.05693v1 PDF, pypdf layout + plain; LaTeX source; Fig. 3 vector paths digitised from the source figure PDF; VarBench history
Reported as. -3.18197(2) (energy per site, Pauli, J = 1, B = 3.0)
Record row. Number, convention, class and bond dimension agree: the printed value is the energy reached at D = 2^15 (Fig. 3 plots D = 1024-32768, no extrapolation; the D = 2^14 point is -3.1819720), so variational with compute.bond_dimension 32768. Boundary conditions are not stated in the paper; open-open is established indirectly (card C3: the caption's own comparison with open finite-PEPS values at 11 x 11 and 21 x 21 brackets it, and it lies just below VarBench's open-boundary DMRG and MPS-RNN rows on this instance; any periodic direction would lower it by several 1e-3 per site). The sigma question is a separate ambiguous entry. Journal version not read. Not a duplicate: no other row on the instance carries this number.
-3.181580(3), VMC (Jastrow)
Checked 2026-09-29 · varbench/methods ed31bb0 scripts/TFIsing/square_400_O_3/vmc_jastrow.sh -> programs/vmc_netket/vmc.py, args_parser.py (NetKet 3.13.0 per requirements.txt)
Reported as. VarBench TFIsing/square_400_O_3.md: Energy -1272.6319 | Sigma 0.0014 | Energy Variance 2.0295256 | DOF 400 | Einf 0 | Method "Jastrow baseline"
Script builds the instance (open square of 400 sites, h = 3, Pauli TFIM) with nk.models.Jastrow (vmc.py lines 118-124); above the best bound -1272.788 (DMRG on TPU). sigma check: sqrt(Var/2^20) = sqrt(2.0295256/1048576) = 0.00139 vs stored 0.0014 (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 (6e7a021 (2023-01-09; earlier -1272.3418(25) af1d48f, -1272.6255(14) 59e621a)); 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 6e7a021; no value was changed and restored.
-3.181431, DMRG
Checked 2026-09-29 · varbench/methods ed31bb0 scripts/TFIsing/square_400_O_3/dmrg.sh -> programs/dmrg_itensors/ising_2d.jl, common.jl, args.jl (ITensors.jl, Manifest pinned)
Reported as. VarBench TFIsing/square_400_O_3.md: Energy -1272.5724160956197 | Sigma (empty) | Energy Variance 6.2083850603085 | DOF 400 | Einf 0 | Method "DMRG (bond dimension = 1024)"
Script builds the instance (Pauli TFIM +sum ZZ + h sum X on the open 20x20 square, h = 3, equivalent to -sum ZZ - h sum X on this bipartite cluster; no symmetry or magnetisation constraint). Bond dimension = max_B = 1024. 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 3c108e3 (2022-07-11, 'Add DMRG results'); script link attached 2024-07-19 (f88f9ba, cd362c4). Not the record (DMRG on TPU -1272.788 is lower; Jastrow -1272.6319(14) also lower, plausible for an unconverged chi = 1024 MPS on 20x20 with variance 6.2). Provenance note: the MPS-RNN supplement (arXiv:2206.12363v1 and v2, Sec. S7) says 'We have computed all the DMRG results in Fig. 4, except for chi >= 1,024 on the 20x20 lattice', but Fig. 4 of both versions shows only 10x10 AFHM, so the sentence cannot be tied to this row; the TPU paper (arXiv:2204.05693, Fig. 3) prints only the final -3.18197(2) per site, not a chi = 1024 energy with a variance, so nothing indicates this number came from elsewhere than the linked ITensors program.
-3.16681(2), VMC (RBM)
Checked 2026-09-29 · varbench/methods ed31bb0 scripts/TFIsing/square_400_O_3/vmc_rbm.sh -> programs/vmc_netket/vmc.py, args_parser.py (NetKet 3.13.0 per requirements.txt)
Reported as. VarBench TFIsing/square_400_O_3.md: Energy -1266.7258 | Sigma 0.0098 | Energy Variance 98.838660 | DOF 400 | Einf 0 | Method "RBM (alpha = 1)"
Script builds the instance (open square of 400 sites, h = 3, Pauli TFIM) with nk.models.RBM(alpha = 1, log_cosh, float32; vmc.py lines 125-140); above the best bound -1272.788 (DMRG on TPU). sigma check: sqrt(Var/2^20) = sqrt(98.83866/1048576) = 0.00971 vs stored 0.0098 (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 (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 af1d48f; no value was changed and restored.