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.
-0.4431678, Exact solution
Checked 2026-10-27 · bethe_xxx.py: Bethe equations in logarithmic form for the M = 100 real roots of the singlet ground state, Newton's method in double precision, then mpmath at 70 and 90 digits
Reported as. -88.633553252241620261082983305428356490257688921703 (S.S units, 50 digits; results.txt)
Pauli total = 4 x -88.633553252241620261082983305428356490257688921703 = -354.53421300896648104433193322171342596103075568681. The same code reproduces the 20-site periodic chain's ED row to all its 16 digits. Agrees to a relative 1e-49 with the value of the bethe library (https://github.com/Uni20-dev/bethe, arbitrary-precision reals), sent to QMBL on 2026-10-27. The DMRG row, -354.534213002, lies 7.0e-9 above it.
-0.4431678, DMRG
Checked 2026-09-29 · VarBench snapshot 390a21e (vendor/varbench) and full upstream history (varbench-history, git log --follow on the file, every version); varbench/methods ed31bb0 run script and program read as text; no output log or result file exists in the methods repo for any Heisenberg row
Reported as. | -354.534213002 | | 2.559E-8 | 200 | 0 | DMRG (max truncation error ~ 2.8E-12) | [code](https://github.com/varbench/methods/blob/main/programs/dmrg_itensors_heisenberg/chain_200_P.jl) |
Program chain_200_P.jl: N = 200, periodic = true adds edge [1,200] (lines 26, 99-100), Pauli form (lines 66-68), Neel start with conserve_qns (Sz = 0), maxdim up to 1000. Independent value check: the Bethe-ansatz ground state of the 200-site ring (work/bethe.mjs) is -354.53421300896656; the stored -354.534213002 lies 6.97e-9 above it: a valid bound, consistent with the stored variance 2.559e-8. M6 (V-score 4.1e-11 outlier): genuine. M11: now read and checked against the exact energy. History: energy unchanged since 35b794c (2021-12-07, J. M. Silvester; label '~ 3.0E-12' -> '~ 2.8E-12' in 9735224); variance 0 -> 0.000080 -> 0.000320 -> 0.000640 -> 2.559E-8 within 2022-03-10 (b6309db), apparently while settling units; nothing restored. Program linked 2024-08-08 (d944d80).
-0.443097(2), VMC (Gutzwiller)
Checked 2026-09-29 · VarBench snapshot 390a21e (vendor/varbench) and full upstream history (varbench-history, git log --follow on the file, every version); varbench/methods ed31bb0 run script and program read as text; no output log or result file exists in the methods repo for any Heisenberg row
Reported as. | -354.4774 | 0.0013 | 0.595 | 200 | 0 | VMC with projected fermions + Jastrow | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/chain_200_P/vmc_gutzwiller.sh) |
vmc_gutzwiller.sh + inputs: geometry.d 1-site cell x 200 (ring; the last flag 1 only makes the auxiliary fermions antiperiodic, the spin exchange uses abs(ivicn) and is periodic), nelup = neldo = 100 (Sz = 0), jperp = 1, uvv = 0.25 (H = sum S.S, no constant), BCS-paired projected fermions (iesBCS=3) + long-range spin Jastrow (iesspin=99), ngen = 5e5; read_ene.x binned error (bins of 10^4, first 4 dropped). Stored = 4N x <H>/N (Pauli total). sigma/sqrt(Var/2^20) = 1.73 (binned SEM; implied 3.5e5 effective samples). 0.0568 above the Bethe-ansatz ground state -354.53421300896656: a valid bound. Entered 61acf55 (2022-02-15, F. Ferrari), never changed; linked by Ferrari d186176 (2024-07-30).
-0.439176(5), VMC (RBM)
Checked 2026-09-29 · VarBench snapshot 390a21e (vendor/varbench) and full upstream history (varbench-history, git log --follow on the file, every version); varbench/methods ed31bb0 run script and program read as text; no output log or result file exists in the methods repo for any Heisenberg row
Reported as. | -351.3410 | 0.0037 | 14.31015 | 200 | 0 | RBM (alpha = 1) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/chain_200_P/vmc_rbm.sh) |
scripts/Heisenberg/chain_200_P/vmc_rbm.sh: 'python3 vmc.py --ham heis --boundary peri --sign mars --ham_dim 1 --L 200 --zero_mag --net rbm --seed 123 --show_progress' -> 200-site ring (nk.graph.Grid, vmc.py line 51); NetKet Heisenberg J = 1 in Pauli form with Marshall sign rule (a sublattice rotation on a bipartite lattice, spectrum unchanged); Sz = 0 (total_sz=0, MetropolisExchange); real float32 RBM, alpha = 1 (nk.models.RBM, log_cosh); 10^4 SR steps; final estimate over 2^20 samples (vmc.py lines 268-269). sigma/sqrt(Var/2^20) = 1.002: the stored sigma is the error of that final estimate. 3.1932 above the Bethe-ansatz ground state -354.53421300896656 (work/bethe.mjs). History: entered 10f5c12 (2022-07-11, Dian Wu) as -351.341 | 0.0037 | 14.3102 | RBM (alpha = 1); af1d48f (2022-09-01, 'Add Jastrow baseline'): energy -351.341 -> -351.3410; variance 14.3102 -> 14.31015; linked 5047bfd (2024-07-30). Program files postdate the value (scripts added to varbench/methods 2024-07-30); they document the settings after the run.
-0.433459(6), VMC (Jastrow)
Checked 2026-09-29 · VarBench snapshot 390a21e (vendor/varbench) and full upstream history (varbench-history, git log --follow on the file, every version); varbench/methods ed31bb0 run script and program read as text; no output log or result file exists in the methods repo for any Heisenberg row
Reported as. | -346.7672 | 0.0045 | 20.28494 | 200 | 0 | Jastrow baseline | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/chain_200_P/vmc_jastrow.sh) |
scripts/Heisenberg/chain_200_P/vmc_jastrow.sh: 'python3 vmc.py --ham heis --boundary peri --sign mars --ham_dim 1 --L 200 --zero_mag --net jas --seed 123 --show_progress' -> 200-site ring (nk.graph.Grid, vmc.py line 51); NetKet Heisenberg J = 1 in Pauli form with Marshall sign rule (a sublattice rotation on a bipartite lattice, spectrum unchanged); Sz = 0 (total_sz=0, MetropolisExchange); real float32 Jastrow (nk.models.Jastrow); 10^4 SR steps; final estimate over 2^20 samples (vmc.py lines 268-269). sigma/sqrt(Var/2^20) = 1.023: the stored sigma is the error of that final estimate. 7.7670 above the Bethe-ansatz ground state -354.53421300896656 (work/bethe.mjs). History: entered af1d48f (2022-09-01, Dian Wu) as -346.7672 | 0.0045 | 20.28494 | Jastrow baseline; linked fdb4bfb (2024-07-30). Program files postdate the value (scripts added to varbench/methods 2024-07-30); they document the settings after the run.