The Hamiltonian
H = ∑⟨ij⟩γ [K σγiσγj]
Honeycomb lattice; γ = x, y, z labels the three bond directions and (α, β) the other two components. K = -1, J = 0, Γ = 0, the couplings of the spin-1/2 form with S = σ/2, normalised to |K| = 1 (RULES.md 2.1); written here with Pauli matrices, so the stored energy is 4 times the S-operator energy and E/N in S units is E/(4N).
- lattice
- honeycomb-11x11
- sites
- 242
- boundary
- open
- K
- -1
- J
- 0
- Gamma
- 0
- energies
- 5
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.
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.
Projected or fixed-node estimates (2)
Variational only within a constraint: fixed-node, constrained-path, GFMC on a trial state. Each depends on its node or constraint, so two of them are not cleanly comparable to each other or to strict bounds.
How these numbers were read
Where each number below was read, and how it was checked.
-0.1882584, Exact solution
Checked 2026-10-20 · source text of arXiv:2112.08394 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; first reading and an adversarial second reading (2026-10-20)
Reported as. -0.75303346
Table III, row (11,11), column Exact; version read: arXiv text (sources/2112.08394.txt); the journal version was not reached from this session, to be read on the verification pass (RULES.md 8) | Eq. (25) H = -Jx sum sigma^x sigma^x - Jy sum sigma^y sigma^y - Jz sum sigma^z sigma^z, Pauli matrices, Jx = Jy = Jz = 1: K = -4 in the S = sigma/2 form, 4 times the instance's K = -1 (scale 4). Table III caption 'energy per site E0/N'; footnote 'N = Lx x Ly x 2'. QMBL's flux-free free-Majorana energies of the open 5 x 5, 7 x 7 and 11 x 11 patches give -0.71402401, -0.73416737 and -0.75303346 per site, the Exact column to all printed digits, which fixes the open boundaries the text does not print.
-0.1867058, isoTNS
Checked 2026-10-20 · source text of arXiv:2112.08394 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; first reading and an adversarial second reading (2026-10-20)
Reported as. -0.74682331
Table III, row (11,11), column isoTNS D=6, η=12; version read: arXiv text (sources/2112.08394.txt); the journal version was not reached from this session, to be read on the verification pass (RULES.md 8) | Eq. (25) H = -Jx sum sigma^x sigma^x - Jy sum sigma^y sigma^y - Jz sum sigma^z sigma^z, Pauli matrices, Jx = Jy = Jz = 1: K = -4 in the S = sigma/2 form, 4 times the instance's K = -1 (scale 4). Table III caption 'energy per site E0/N'; footnote 'N = Lx x Ly x 2'. QMBL's flux-free free-Majorana energies of the open 5 x 5, 7 x 7 and 11 x 11 patches give -0.71402401, -0.73416737 and -0.75303346 per site, the Exact column to all printed digits, which fixes the open boundaries the text does not print.
-0.1861212, isoTNS
Checked 2026-10-20 · source text of arXiv:2112.08394 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; first reading and an adversarial second reading (2026-10-20)
Reported as. -0.74448479
Table III, row (11,11), column isoTNS D=4, η=8; version read: arXiv text (sources/2112.08394.txt); the journal version was not reached from this session, to be read on the verification pass (RULES.md 8) | Eq. (25) H = -Jx sum sigma^x sigma^x - Jy sum sigma^y sigma^y - Jz sum sigma^z sigma^z, Pauli matrices, Jx = Jy = Jz = 1: K = -4 in the S = sigma/2 form, 4 times the instance's K = -1 (scale 4). Table III caption 'energy per site E0/N'; footnote 'N = Lx x Ly x 2'. QMBL's flux-free free-Majorana energies of the open 5 x 5, 7 x 7 and 11 x 11 patches give -0.71402401, -0.73416737 and -0.75303346 per site, the Exact column to all printed digits, which fixes the open boundaries the text does not print.
-0.1820213, DMRG
Checked 2026-10-20 · source text of arXiv:2112.08394 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; first reading and an adversarial second reading (2026-10-20)
Reported as. -0.72808531
Table III, row (11,11), column DMRG χ=1024; version read: arXiv text (sources/2112.08394.txt); the journal version was not reached from this session, to be read on the verification pass (RULES.md 8) | Eq. (25) H = -Jx sum sigma^x sigma^x - Jy sum sigma^y sigma^y - Jz sum sigma^z sigma^z, Pauli matrices, Jx = Jy = Jz = 1: K = -4 in the S = sigma/2 form, 4 times the instance's K = -1 (scale 4). Table III caption 'energy per site E0/N'; footnote 'N = Lx x Ly x 2'. QMBL's flux-free free-Majorana energies of the open 5 x 5, 7 x 7 and 11 x 11 patches give -0.71402401, -0.73416737 and -0.75303346 per site, the Exact column to all printed digits, which fixes the open boundaries the text does not print.
-0.1810139, DMRG
Checked 2026-10-20 · source text of arXiv:2112.08394 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; first reading and an adversarial second reading (2026-10-20)
Reported as. -0.72405548
Table III, row (11,11), column DMRG χ=512; version read: arXiv text (sources/2112.08394.txt); the journal version was not reached from this session, to be read on the verification pass (RULES.md 8) | Eq. (25) H = -Jx sum sigma^x sigma^x - Jy sum sigma^y sigma^y - Jz sum sigma^z sigma^z, Pauli matrices, Jx = Jy = Jz = 1: K = -4 in the S = sigma/2 form, 4 times the instance's K = -1 (scale 4). Table III caption 'energy per site E0/N'; footnote 'N = Lx x Ly x 2'. QMBL's flux-free free-Majorana energies of the open 5 x 5, 7 x 7 and 11 x 11 patches give -0.71402401, -0.73416737 and -0.75303346 per site, the Exact column to all printed digits, which fixes the open boundaries the text does not print.