Heisenberg triangular 16x16, open

Best values (rules §6)

best variational	-0.51668(9)	VMC (2D RNN) 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⟩ σ_i·σ_j

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

lattice: triangular
sites: 256
boundary: open
energies: 4

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).

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.

E/N (S.S)	distance to class best	σ	Var(E)	V-score	method	source	code	year
-0.51668best variational	0	8.6e-5	7.51e+1	6.9e-2	VMC (2D RNN)	Wu et al. (2024)	run script	2024
-0.5079697	8.7e-3	n/a	1.60e+2	1.5e-1	DMRG VarBench reference run	Wu et al. (2024)	run script	2024
-0.47747	3.9e-2	1.8e-5	3.40e+2	3.6e-1	VMC (RBM) VarBench reference run	Wu et al. (2024)	run script	2024
-0.47581	4.1e-2	1.5e-5	2.37e+2	2.6e-1	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.

-0.5079697, 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. | -520.160985 | | 159.85312 | 256 | 0 | DMRG (Bond dimension = 2000) | [code](https://github.com/varbench/methods/blob/main/programs/dmrg_itensor_cpp/dmrg_triangular_heisenberg_16x16.cc) |

Program dmrg_triangular_heisenberg_16x16.cc (M. Hibat-Allah, ITensor C++): 16x16 triangularLattice, open, H = sum S.S, Sz = 0, schedule ending at 2000, cutoff 1e-10: final energy, variational. Stored = 4 x -130.04024625. arXiv:2207.14314 used D = 2000 at 16x16. Variance 159.85312 has no runnable source (non-compiling lines 82-83, and dmrg.sh cd's into the non-existent programs/dmrg_itensors_c++). History: -129.31321 (S.S, 36d51f4, 2022-02-15) -> x4 -517.25284 (878bf7a) -> -520.160985 with variance (3c96eaa, 2023-02-01, Hibat-Allah). 8.919 above the gated-RNN record. Linked 9369b7d (2024-08-13).

-0.47747(2), VMC (RBM)

Reported as. | -488.934 | 0.018 | 340.191 | 256 | 0 | RBM (alpha = 1) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/triangular_256_O/vmc_rbm.sh) |

scripts/Heisenberg/triangular_256_O/vmc_rbm.sh: 'python3 vmc.py --ham heis_tri --boundary open --sign mars --ham_dim 2 --L 16 --zero_mag --net rbm --seed 123 --show_progress' -> 16x16 triangular lattice (ColoredJ1J2 back_diag=False: square grid + (i,j)-(i+1,j+1) diagonals), open; NetKet Heisenberg J = 1 in Pauli form with Marshall rotation on the square bonds only (sign_rule=[True, False], a unitary: the diagonals join same-sublattice sites); 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) = 0.999: the stored sigma is the error of that final estimate. 40.1460 above the 2D gated RNN record -529.080(88). M2/M6 flags (if any): real float32 parameters make the amplitudes positive, so on this frustrated lattice the ansatz cannot carry the ground-state signs; no constant shift is involved and the energy is a valid bound. History: entered 10f5c12 (2022-07-11, Dian Wu) as -488.258 | 0.018 | 344.140 | RBM (alpha = 1); af1d48f (2022-09-01, 'Add Jastrow baseline'): energy -488.258 -> -488.840; variance 344.140 -> 340.525; 59e621a (2022-09-13, 'Update Jastrow and RBM baselines'): energy -488.840 -> -488.934; variance 340.525 -> 340.191; linked 5047bfd (2024-07-30). The first value change sits in af1d48f, titled 'Add Jastrow baseline'. Program files postdate the value (scripts added to varbench/methods 2024-07-30); they document the settings after the run.

-0.47581(1), VMC (Jastrow)

Reported as. | -487.225 | 0.015 | 237.059 | 256 | 0 | Jastrow baseline | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/triangular_256_O/vmc_jastrow.sh) |

scripts/Heisenberg/triangular_256_O/vmc_jastrow.sh: 'python3 vmc.py --ham heis_tri --boundary open --sign mars --ham_dim 2 --L 16 --zero_mag --net jas --seed 123 --show_progress' -> 16x16 triangular lattice (ColoredJ1J2 back_diag=False: square grid + (i,j)-(i+1,j+1) diagonals), open; NetKet Heisenberg J = 1 in Pauli form with Marshall rotation on the square bonds only (sign_rule=[True, False], a unitary: the diagonals join same-sublattice sites); 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) = 0.998: the stored sigma is the error of that final estimate. 41.8550 above the 2D gated RNN record -529.080(88). M2/M6 flags (if any): real float32 parameters make the amplitudes positive, so on this frustrated lattice the ansatz cannot carry the ground-state signs; no constant shift is involved and the energy is a valid bound. History: entered 59e621a (2022-09-13, Dian Wu) as -487.225 | 0.015 | 237.059 | 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. Literature last checked 2026-09-15 (table harvest of cached arXiv sources (match_tables.mjs --all), every matched paper read for this instance): 1 paper read, 0 of them with an energy on this page.

This instance as data

JSON · source file on GitHub · add a result · open an issue about this instance