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 (1)
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.
-0.66951133(3), SSE QMC
Checked 2026-09-15 · arXiv HTML parsed locally, one value per table cell; no LLM transcription
Reported as. e0 = -0.66951133(3) per spin (S.S), L = 32, periodic
Table 1, "Results for different system sizes computed at inverse temperature beta/L = 64 for L > 30 and averaged over beta/L = 32 and beta/L = 64 simulations for L <= 30", column e0, row L = 32. Pauli total = 4 N e0. The L = 6 value agrees with the exact-diagonalization row on square_36_P to 1.6e-8.
-0.6694877(4), VMC (2D RNN)
Checked 2026-09-15 · source text of arXiv:2502.17144 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -0.6694877(4)
Table III, row L=32, column 'RNN, zero-variance E/N' | Same row's quoted QMC[76] value (-0.66951136(4)) matches instance exact_per_site -0.66951133 to displayed precision.
-0.66930(1), VMC (2D RNN)
Checked 2026-09-16 · authors' data release read locally (pickle via numpy), values copied by key, no transcription; the file is committed as sources/2502.17144-repo-final_energy_data.json and reproduces the four values Table V prints
Reported as. -0.6692956686019897 +/- 0.00001276756579333374 per site (S.S), Var(E/N) = 0.0000016301073628710583
github.com/mschuylermoss/HeisenbergRNN, commit 29bf62ac28 (2025-02-24, 'reproduce figures'), Square/final_plotting/final_data/final_energy_data_plotting.pkl: ['periodic']['rate=0.25']['scale=4'], L = 32; the lowest of the ten training schedules (s = 4, r = 0.25). Plotted as a point in Fig. 3(a), not printed in the paper; Table III prints the zero-variance extrapolation over the s >= 1 runs at this L, carried on this instance as an `extrapolated` row. Variance is Var(H)/N^2 in S.S units (see script header); total = var x N^2 x 16.