Heisenberg pyrochlore 4x4x4 (256 sites)

Best values (rules §6)

best variational	-0.485584(3)	VMC (RVB) Cheng & Li (2026) · 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: pyrochlore-4x4x4
sites: 256
boundary: periodic
energies: 2

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 (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.

E/N (S.S)	distance to class best	σ	Var(E)	V-score	method	source	code	year	comment
-0.485584best variational	0	2.9e-6	7.83e+1	8.1e-2	VMC (RVB) peer reviewed	Cheng & Li (2026)		2026
-0.4831	2.5e-3	1.0e-4	1.09e+2	1.1e-1	VMC (PP)	Astrakhantsev et al. (2021)	uses mVMC	2021

How these numbers were read

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

-0.485584(3), VMC (RVB)

Checked 2026-09-15 · arXiv PDF text extracted locally with pypdf in layout mode, no LLM transcription

Reported as. E0 ~ -0.4855 J/site, approximate, no error bar

Sec. A: "The obtained ground state energy is E0 ~ -0.5118 J/site for the L = 2 cluster and E0 ~ -0.4855 J/site for the L = 4 cluster", periodic boundary conditions on both; the L = 4 cluster carries Nv = 4N^2 = 262144 parameters, so N = 256. Units: the same passage quotes the mVMC energy -0.5162 J/site for L = 2, which is the pyrochlore-2x2x2_32_P record -0.516266 in the S.S per-site convention. Approximate and without an error bar. It is 0.5% below the VarBench mVMC record -0.4830957, although the ansatz has no symmetry projection and sits above mVMC at L = 2 - which says more about that record than about this row. 2026-09-19: the authors measured the fixed state on request, E/N = -0.4855840(29) J with Var(H) = 4.894(22) J^2 (see corrections and error_metrics on this row).

-0.4831(1), VMC (PP)

Checked 2026-10-15 · published PDF Phys. Rev. X 11, 041021 (2021) (open access), sources/journal/prx11-041021.pdf, sha256 e65c26b3, pypdf layout (ð1Þ = (1) in the font); arXiv:2101.08787v2 compared (qmbl/sources/2101.08787.txt Table II identical)

Reported as. -0.4831(1) per site (S.S, J = 1)

H = j1 sum h_lambda(i,j) + j2 sum ..., h_lambda = lambda/2 (S+S- + S-S+) + SzSz (Eqs. 1-2), Heisenberg point lambda = 1, j2 = 0; Table II: 'The ground-state energy per site is listed in the first line of every cell'; 'We consider pyrochlore clusters with periodic boundary conditions'. Geometry 4 x L^3 = 4L^3 sites. Stored Pauli total = 4 x N x E/N. Own computation (mVMC / NQS CNN columns). The paper prints no energy variance: the stored energy_variance is the VarBench upload's only. Published 29 October 2021, peer reviewed. No duplicate.

Coverage

Newest published number here: 2026. Literature last checked 2026-09-15 (table harvest of cached arXiv sources (match_tables.mjs --all), every matched paper read for this instance): 3 papers read, 1 of them with an energy on this page.

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