KitaevHoneycomb — Honeycomb Lattice Kitaev Model

Hamiltonian

H = − Σ{⟨ij⟩ ∈ x-bonds} Kx σˣᵢ σˣⱼ − Σ{⟨ij⟩ ∈ y-bonds} Ky σʸᵢ σʸⱼ − Σ{⟨ij⟩ ∈ z-bonds} Kz σᶻᵢ σᶻⱼ

(σ-convention; eigenvalues ±1)

Phases (Kitaev 2006)

• isotropic gapless (B-phase): each |K_α| ≤ sum of others
• anisotropic gapped (Ax, Ay, A_z phases): one K dominates
• gap: Δ = 2·max(0, |Kmax| − |Kother₁| − |K_other₂|)

Coverage Matrix

*ZZStructureFactor router を参照 (TFIMinfinitedynamics.jl に同様の実装あり; Kitaev 自身の動的 SF は別 issue)。

v0.17 Highlights — Matter-Majorana Free-Fermion Finite-T

Lieb 1994: GS は flux-free sector に固定。flux-free セクター内では matter Majorana が自由フェルミオン hopping を成し、

Z_matter(β) = ∏_k 2 cosh(β λ_k)

(λk は bipartite hopping matrix M の正の SVD 値; λk は full Majorana eigenvalue 規約 — 半固有値ではなく)。

per-site (= per Majorana site, 2 atoms per unit cell) 物理量:

ε(β) = -(1/N_sites) Σ_k λ_k tanh(β λ_k)
f(β) = -(1/(N_sites β)) Σ_k log(2 cosh(β λ_k))
s(β) = (1/N_sites) Σ_k [log(2 cosh(βλ)) − βλ tanh(βλ)]
c_v(β) = (1/N_sites) Σ_k (βλ)² sech²(βλ)

Infinite では Σ_k → ∫_BZ d²θ / (2π)² (factor 2 で 2 atoms/cell)。

Code Examples

m = KitaevHoneycomb(Kx=1.0, Ky=1.0, Kz=1.0)  # isotropic gapless

# T = 0 GS
QAtlas.fetch(m, Energy(:per_site), Infinite())   # Baskaran-Mandal-Shankar 2007 ≈ -0.787
QAtlas.fetch(m, Energy(:per_site), OBC(0); Lx=4, Ly=4)
QAtlas.fetch(m, Energy(:per_site), PBC(0); Lx=4, Ly=4)
QAtlas.fetch(m, MassGap(), Infinite())  # = 0 (gapless)

# T > 0 matter sector
β = 5.0
QAtlas.fetch(m, FreeEnergy(), Infinite(); beta=β)
QAtlas.fetch(m, SpecificHeat(), Infinite(); beta=β)

# Anisotropic gapped phase
m_az = KitaevHoneycomb(Kx=0.5, Ky=0.5, Kz=2.0)
QAtlas.fetch(m_az, MassGap(), Infinite())  # = 2(2 − 1) = 2

Validity Boundary of Matter-Sector Finite-T

flux gap Δ_v ≈ 0.07 |K| (isotropic) — 凍結する温度スケール。

References

• Kitaev 2006 — exactly solved model and beyond
• Lieb 1994 — flux-free ground state theorem
• Baskaran-Mandal-Shankar 2007 — TL ε_gs ≈ −0.787 |K|
• Nasu-Udagawa-Motome 2014 — full flux-sector finite-T (sign-free QMC)
• Loss-Pu 2008 — spin correlators (z-bond local, off-diagonal)

Related

• TFIM — 1D version of "single Majorana species" (BdG hopping); Kitaev's matter sector is the 2D analogue

<!– ATLAS:HUBS:START – auto-generated by docs/atlas/generate.jl. Do not edit by hand; edits between these markers are overwritten on next regen. –>

Verified hubs

In the Verified Atlas, these 2 models register 11 hubs (quantity / BC pair). The badge column shows the R1 assurance level; click a hub link to see the exact verify(...) calls, references, and corroboration mechanism.

<!– ATLAS:HUBS:END –>