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-3x4
- sites
- 24
- boundary
- periodic
- 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 (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.
How these numbers were read
Where each number below was read, and how it was checked.
-0.19887(2), VMC (PP)
Checked 2026-10-20 · source text of arXiv:1411.5198 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. −9.546±0.001
Table II, 3×4 unit cells, (Wx,Wy) = (+,−), column VMC; version read: arXiv text (sources/1411.5198.txt); the journal version was not reached from this session, to be read on the verification pass (RULES.md 8) | Eq. (28) H_KH = 2 alpha H_K + (1 - alpha) J_H sum S.S at alpha = 1 (the text before Table I: 'the Kitaev model, where alpha = 1'), with H_K = -J sum S^g S^g (Eq. 27), S^g = c^dag sigma^g c / 2, J = 1: H = -2 sum S^g S^g, K = -2 in the S = sigma/2 form, twice the instance's K = -1 (scale 2). Total energies of the cluster: the per-site values -0.3970, -0.3977, -0.3926 the text compares with the thermodynamic -0.39365 are these totals over N. QMBL's flux-free free-Majorana energies of the 3 x 4 and 4 x 4 tori reproduce -9.54589 and -12.56409 (SWEEP.md, 2026-10-20).
-0.1988727, ED
Checked 2026-10-20 · source text of arXiv:1411.5198 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. −9.54589
Table I, row Exact, column 3×4 unit cells; version read: arXiv text (sources/1411.5198.txt); the journal version was not reached from this session, to be read on the verification pass (RULES.md 8) | Eq. (28) H_KH = 2 alpha H_K + (1 - alpha) J_H sum S.S at alpha = 1 (the text before Table I: 'the Kitaev model, where alpha = 1'), with H_K = -J sum S^g S^g (Eq. 27), S^g = c^dag sigma^g c / 2, J = 1: H = -2 sum S^g S^g, K = -2 in the S = sigma/2 form, twice the instance's K = -1 (scale 2). Total energies of the cluster: the per-site values -0.3970, -0.3977, -0.3926 the text compares with the thermodynamic -0.39365 are these totals over N. QMBL's flux-free free-Majorana energies of the 3 x 4 and 4 x 4 tori reproduce -9.54589 and -12.56409 (SWEEP.md, 2026-10-20).
-0.19883(8), VMC (PP)
Checked 2026-10-20 · source text of arXiv:1411.5198 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. −9.544 ± 0.004
Table I, row |ψKL⟩, column 3×4 unit cells; version read: arXiv text (sources/1411.5198.txt); the journal version was not reached from this session, to be read on the verification pass (RULES.md 8) | Eq. (28) H_KH = 2 alpha H_K + (1 - alpha) J_H sum S.S at alpha = 1 (the text before Table I: 'the Kitaev model, where alpha = 1'), with H_K = -J sum S^g S^g (Eq. 27), S^g = c^dag sigma^g c / 2, J = 1: H = -2 sum S^g S^g, K = -2 in the S = sigma/2 form, twice the instance's K = -1 (scale 2). Total energies of the cluster: the per-site values -0.3970, -0.3977, -0.3926 the text compares with the thermodynamic -0.39365 are these totals over N. QMBL's flux-free free-Majorana energies of the 3 x 4 and 4 x 4 tori reproduce -9.54589 and -12.56409 (SWEEP.md, 2026-10-20).
-0.19881(8), VMC (PP)
Checked 2026-10-20 · source text of arXiv:1411.5198 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. −9.543 ± 0.004
Table I, row |ψMP⟩, column 3×4 unit cells; version read: arXiv text (sources/1411.5198.txt); the journal version was not reached from this session, to be read on the verification pass (RULES.md 8) | Eq. (28) H_KH = 2 alpha H_K + (1 - alpha) J_H sum S.S at alpha = 1 (the text before Table I: 'the Kitaev model, where alpha = 1'), with H_K = -J sum S^g S^g (Eq. 27), S^g = c^dag sigma^g c / 2, J = 1: H = -2 sum S^g S^g, K = -2 in the S = sigma/2 form, twice the instance's K = -1 (scale 2). Total energies of the cluster: the per-site values -0.3970, -0.3977, -0.3926 the text compares with the thermodynamic -0.39365 are these totals over N. QMBL's flux-free free-Majorana energies of the 3 x 4 and 4 x 4 tori reproduce -9.54589 and -12.56409 (SWEEP.md, 2026-10-20).
-0.1879(4), VMC (PP)
Checked 2026-10-20 · source text of arXiv:1411.5198 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. −9.02 ± 0.02
Table I, row |ψpair⟩, column 3×4 unit cells; version read: arXiv text (sources/1411.5198.txt); the journal version was not reached from this session, to be read on the verification pass (RULES.md 8) | Eq. (28) H_KH = 2 alpha H_K + (1 - alpha) J_H sum S.S at alpha = 1 (the text before Table I: 'the Kitaev model, where alpha = 1'), with H_K = -J sum S^g S^g (Eq. 27), S^g = c^dag sigma^g c / 2, J = 1: H = -2 sum S^g S^g, K = -2 in the S = sigma/2 form, twice the instance's K = -1 (scale 2). Total energies of the cluster: the per-site values -0.3970, -0.3977, -0.3926 the text compares with the thermodynamic -0.39365 are these totals over N. QMBL's flux-free free-Majorana energies of the 3 x 4 and 4 x 4 tori reproduce -9.54589 and -12.56409 (SWEEP.md, 2026-10-20).