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 (2)
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.
-0.4387039, ED
Checked 2026-10-15 · published PDF Phys. Rev. E 98, 033309 (2018), sources/journal/pre98-033309.pdf, sha256 e8e3383f, pypdf layout; arXiv:1804.05028v2 compared (sources/arxiv/1804.05028v2.pdf, its Table I 'g.s. energy' row identical)
Reported as. -21.057 787 063 5 (S.S total, J = 1); sector Gamma.A1.even
H = J sum S_i . S_j, J = 1 (Eq. 44), energies in units of J ('The energy is given in units of the nearest neighbor coupling constant J = 1'), S.S total; stored Pauli total = 4 x printed. Own ED (Lanczos with sublattice coding). Uploaded to VarBench by the first author (1fa8a1c, A. Wietek, 2022-02-11). sigma null for exact (corrected). peer_reviewed true. Cluster: 48-site kagome with full D6 point group (Fig. 5; VarBench's original file name kagome_sqrt48_PP_48, renamed kagome-4x4_48_P). The paper says the value 'has been previously computed [14] with a specialized code and agrees with our results' ([14] = Lauchli, Sudan & Moessner, arXiv:1611.06990): an own recomputation, so this paper is a primary source. No duplicate (the instance's other rows are variational/extrapolated).
-0.4383(2), DMRG
Checked 2026-09-14 · arXiv PDF text extracted locally with pypdf, no LLM transcription
Reported as. -0.4383 per site in S.S units
Table I row "Torus 4 -0.4383(2)". The paper states energies are "extrapolated in the truncation error of single-site DMRG", so this is not a strict variational bound (RULES.md 4). Sits above the exact -0.4387039 on this instance.
-0.4375(2), VMC (GCNN)
Checked 2026-09-14 · arXiv PDF text extracted locally with pypdf, no LLM transcription
Reported as. -0.4375 per site in S.S units
Sec. III: "R = (E_GCNN - E_ED)/E_ED ~ 0.3% with E_ED = -21.057787063 and E_GCNN = -21.00 +/- 0.01" on the 48-site (4x4x3) PBC cluster, chi0 sector. Total energy in S.S units; E/N = -0.4375. The instance's existing exact row is -0.4387039, i.e. -21.057787063/48 to 8 digits, which confirms the cluster and the convention. Sampling here is the ergodic spin-exchange update, so the objection in arXiv:2605.28861 does not apply to this size.
-0.43043(3), 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. | -82.643 | 0.005 | 10.6 | 48 | 0 | VMC with Dirac spin liquid + Jastrow | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/kagome-4x4_48_P/vmc_gutzwiller.sh) |
vmc_gutzwiller inputs: geometry.d 6-site supercell (two kagome cells, lattice constant 2, 60 degrees) x '2 0 0 4' = 4x4 kagome cells = 48 sites on a periodic spin torus (the '1 0' flags only set antiperiodic auxiliary fermions); nelup = neldo = 24 (Sz = 0); jperp = 1, uvv = 0.25 (H = sum S.S); Dirac spin liquid fluxes (wf.in 180-degree phases) + spin Jastrow (iesspin=8); binned error (sigma/sqrt(Var/2^20) = 1.57). Stored = 4N x <H>/N. 1.588 above the instance's ED -84.2311482540 (1.9%). Entered 7b68237 (2022-02-14, F. Ferrari), never changed; linked by Ferrari d186176 (2024-07-30).