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All instances / Heisenberg

Heisenberg chain, 20 sites

Exact energy and best values (rules §6)

exact -0.4452193 ED
Wu et al. (2024) · the instance is solved
best variational -0.4452193 DMRG
Wu et al. (2024) · the lowest unflagged variational energy, a strict variational bound

Energies as E/N (S.S). Classes are different claims and are not ranked against each other.

The Hamiltonian

H = ∑⟨ij⟩ σi·σj

Nearest-neighbour Heisenberg antiferromagnet, J = 1, written with Pauli matrices.

lattice
chain
sites
20
boundary
periodic
energies
7

Stored as a total energy; quoted here as E/N (S.S), 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.

E/N (S.S)distance to class bestσVar(E)V-score methodsourcecodeyearcomment
-0.4452193exact 0 n/a 0.00e+0 0.0e+0 ED
VarBench reference run
 Wu et al. (2024) run script 2024

Strict variational upper bounds (6)

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.

E/N (S.S)distance to class bestσVar(E)V-score methodsourcecodeyearcomment
-0.4452193best variational 0 n/a 1.20e-10 1.9e-12 DMRG
VarBench reference run
 Wu et al. (2024) run script 2024
-0.44521887 4.5e-7 5.7e-8 5.34e-5 8.4e-7 VMC (RNN)
VarBench reference run
 Wu et al. (2024) run script 2024
-0.44521861 7.1e-7 9.0e-8 1.28e-4 2.0e-6 VMC (RNN)
VarBench reference run
 Wu et al. (2024) run script 2024
-0.445182 3.8e-5 4.1e-6 4.88e-2 7.7e-4 VMC (Gutzwiller)
VarBench reference run
 Wu et al. (2024) run script 2024
-0.445049 1.7e-4 5.1e-6 1.76e-1 2.8e-3 VMC (RBM)
VarBench reference run
 Wu et al. (2024) run script 2024
-0.444397 8.2e-4 6.1e-6 2.42e-1 3.8e-3 VMC (Jastrow)
VarBench reference run
 Wu et al. (2024) run script 2024

How these numbers were read

Where each number below was read, and how it was checked.

-0.4452193, ED

Checked 2026-09-29 · QMBL ED cost run checks/cost/ed/results/Heisenberg--chain_20_P.json (checks/cost/ed/run_ed.py: NetKet 3.22.4 operator, sparse CSR built in blocks, SciPy eigsh/ARPACK k=1 SA; commit 716e043); results file read locally, no rerun

Reported as. energy -35.61754611950575 (stored_exact -35.61754611950579, relative_difference 1.20e-15, reproduces_stored true)

|reproduced - stored| = 4.26e-14 (relative 1.20e-15); within 1e-9 relative. Sector total Sz = 0, dimension 184756, 101 Lanczos steps (the file states no residual). The stored value is the source's own ED (varbench@2024-10-22), so two codes agree; a third code, E1's node Lanczos (qmbl-runs/qmbl-verify/out/E1-exact-recompute/results/Heisenberg__chain_20_P.json), gives -35.617546119506194 (4.48e-13 from the cost run). Also read by VB3-varbench-code-heisenberg#12 (scope config: vendor/varbench/Heisenberg/chain_20_P.md (snapshot 390a21e), row 'Exact diagonalization'; linked scripts/Heisenberg/chain_20_P/ed_netket.sh).

-0.4452193, 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. | -35.6175461195 | | 1.2E-10 | 20 | 0 | DMRG (max truncation error ~ 1.0E-13) | [code](https://github.com/varbench/methods/blob/main/programs/dmrg_itensors_heisenberg/chain_20_P.jl) |

Program chain_20_P.jl: N = 20, periodic = true (edge [1,20]), Pauli form, Sz = 0, maxdim up to 1000. Best variational bound on the solved instance: 5.8e-12 above the exact -35.61754611950577/579, i.e. equal at its 10 printed decimals. M6 (V-score 1.9e-12 outlier): genuine. History: entered 56d7863 (2021-12-05, J. M. Silvester) as -35.617546119 | 0 | 0 | DMRG; one digit added and label '~ 1.0E-13', variance 0.000029 in be2ca60 (2022-03-09); variance -> 1.2E-10 (318b106/8a65fb2, 2022-03-10); nothing restored. Program linked 2024-08-08 (d944d80).

-0.44521861(9), VMC (RNN)

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. | -35.6174890 | 0.0000072 | 0.0001278 | 20 | 0 | RNN | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/chain_20_P/vmc_rnn.sh) |

vmc_rnn.sh: periodic 20-site ring, Pauli, J = 1, Marshall sign rule (bipartite ring, N even), full Hilbert space with exact autoregressive sampling, FastLSTMNet 2 x 16, Adam 10^5 steps. The stored (sigma, Var) imply 2.5e6 independent samples (ratio 0.652), not the script's 2^20: numbers from the 2021 run (d9e1806, 2021-10-15), script added 2024-07-30 (d54f7c0). 5.7e-5 above the exact energy: a valid bound.

-0.445182(4), 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. | -35.61452 | 0.00033 | 0.0488 | 20 | 0 | VMC with projected fermions + Jastrow | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/chain_20_P/vmc_gutzwiller.sh) |

vmc_gutzwiller.sh + inputs: geometry.d 1-site cell x 20 (ring; the last flag 1 only makes the auxiliary fermions antiperiodic, the spin exchange uses abs(ivicn) and is periodic), nelup = neldo = 10 (Sz = 0), jperp = 1, uvv = 0.25 (H = sum S.S, no constant), BCS-paired projected fermions (iesBCS=3) + spin Jastrow (iesspin=9); read_ene.x binned error. Stored = 4N x <H>/N (Pauli total). sigma/sqrt(Var/2^20) = 1.53 (binned SEM; implied 4.5e5 effective samples). 0.00303 above the exact energy. Entered 37e1c6a (2021-11-25, F. Ferrari), never changed; linked by Ferrari d186176 (2024-07-30).

-0.445049(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. | -35.60392 | 0.00041 | 0.1757833 | 20 | 0 | RBM (alpha = 1) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/chain_20_P/vmc_rbm.sh) |

scripts/Heisenberg/chain_20_P/vmc_rbm.sh: 'python3 vmc.py --ham heis --boundary peri --sign mars --ham_dim 1 --L 20 --zero_mag --net rbm --seed 123 --show_progress' -> 20-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) = 1.001: the stored sigma is the error of that final estimate. 0.0136 above the exact -35.61754611950579. History: entered 10f5c12 (2022-07-11, Dian Wu) as -35.6039 | 0.00041 | 0.175783 | RBM (alpha = 1); af1d48f (2022-09-01, 'Add Jastrow baseline'): energy -35.6039 -> -35.60392; variance 0.175783 -> 0.1757833; 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.

-0.444397(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. | -35.55179 | 0.00049 | 0.2419342 | 20 | 0 | Jastrow baseline | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/chain_20_P/vmc_jastrow.sh) |

scripts/Heisenberg/chain_20_P/vmc_jastrow.sh: 'python3 vmc.py --ham heis --boundary peri --sign mars --ham_dim 1 --L 20 --zero_mag --net jas --seed 123 --show_progress' -> 20-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) = 1.020: the stored sigma is the error of that final estimate. 0.0658 above the exact -35.61754611950579. History: entered af1d48f (2022-09-01, Dian Wu) as -35.55179 | 0.00049 | 0.2419342 | 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.

Coverage

Newest published number here: 2024.

This instance as data

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