KitaevHoneycomb — Honeycomb Lattice Kitaev Model

Status: Unstable (v0.18.x)

Matter-Majorana finite-T サーフェスは v0.17 で導入。flux-free 近似であり flux gap T ≪ Δ_v でのみ valid。SusceptibilityXX/YY/ZZ および Magnetization{X,Y,Z} は 未実装 — σᵅ matrix elements が flux pair を励起するため flux-free sector で恒等的に消える。完全な finite-T は flux-sector Monte Carlo (Nasu et al. 2014) が必要、今回の射程外。

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

Quantity	OBC	PBC	Infinite
Energy{:per_site} (T=0)	✅ SVD on hopping matrix	✅ 4-flux-sector min	✅ BZ integral
Energy{:per_site} (T>0)	✅ matter sum	(deferred)	✅ BZ integral
FreeEnergy / Entropy / SpecificHeat	✅ matter sum	—	✅ BZ integral
MassGap	—	—	✅ analytic
Mag{X,Y,Z} / Susc{XX,YY,ZZ}	not implemented (deferred)	—	not implemented
ZZStructureFactor	—	—	static + dynamic via TFIM-style proxy*

*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。

Regime	matter-only valid?
T ≪ Δ_v	✅
T ~ Δ_v	× (flux fluctuations dominate, Nasu-Udagawa-Motome 2014)
T ≫ Δ_v ≪ K	×

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)

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

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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.

Model	Quantity	BC	Assurance	Cards
`KitaevHeisenberg`	`MassGap`	`Infinite`	🟢 corroborated-at-p	3
`KitaevHoneycomb`	`Energy`	`Infinite`	⚪ cited-only	1
`KitaevHoneycomb`	`Energy`	`OBC`	⚪ cited-only	0
`KitaevHoneycomb`	`Energy`	`PBC`	⚪ cited-only	0
`KitaevHoneycomb`	`FreeEnergy`	`Infinite`	🔵 coherent	3
`KitaevHoneycomb`	`FreeEnergy`	`OBC`	⚪ cited-only	0
`KitaevHoneycomb`	`MassGap`	`Infinite`	🟢 corroborated-at-p	9
`KitaevHoneycomb`	`SpecificHeat`	`Infinite`	🔵 coherent	6
`KitaevHoneycomb`	`SpecificHeat`	`OBC`	🔵 coherent	12
`KitaevHoneycomb`	`ThermalEntropy`	`Infinite`	🔵 coherent	4
`KitaevHoneycomb`	`ThermalEntropy`	`OBC`	🔵 coherent	7

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API

Every fetch(::Model, …) method registered for these models — together with the model struct(s) and exported helpers — generated directly from the source (in lock-step with @register):

QAtlas.KitaevHeisenberg — Type

KitaevHeisenberg(; K::Real = 1.0, J::Real = 0.0, Γ::Real = 0.0)
    <: AbstractQAtlasModel

K-J-Γ honeycomb model (α-RuCl₃ family). Three independent exchanges on the three honeycomb bond directions:

H = Σ_{⟨ij⟩_γ} [ K Sᵢ^γ Sⱼ^γ + J Sᵢ · Sⱼ + Γ (Sᵢ^α Sⱼ^β + Sᵢ^β Sⱼ^α) ].

Quantities registered (Phase 1):

Quantity	BC	Method
`MassGap`	`Infinite`	delegated to KitaevHoneycomb (K=K)

Phase 1 exposes only the K-only limit J = Γ = 0, delegated to the existing exactly-solvable KitaevHoneycomb entry. Any non-zero J or Γ raises DomainError pointing to the DMRG / ED phase-2 follow-up.

References

A. Kitaev, Annals Phys. 321, 2 (2006).
G. Jackeli, G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009).
J. G. Rau, E. K.-H. Lee, H.-Y. Kee, Phys. Rev. Lett. 112, 077204 (2014).

source

QAtlas.fetch — Method

fetch(m::KitaevHeisenberg, ::MassGap, ::Infinite;
      K=m.K, J=m.J, Γ=m.Γ, kwargs...) -> Float64

Single-particle gap of the K-J-Γ honeycomb model at the K-only point J = Γ = 0. Internally constructs the isotropic KitaevHoneycomb(Kx = K, Ky = K, Kz = K) and forwards. The Kitaev gapless A/B/C phase therefore returns Δ = 0 at isotropic |K|.

J ≠ 0 or Γ ≠ 0 raises DomainError — Phase 2.

References

A. Kitaev, Annals Phys. 321, 2 (2006).

source

QAtlas.KitaevHoneycomb — Type

KitaevHoneycomb(; Kx=1.0, Ky=1.0, Kz=1.0) <: AbstractQAtlasModel

Kitaev honeycomb model with coupling amplitudes on the three bond types. Kx = Ky = Kz is the isotropic gapless point. See module header for the full Hamiltonian and solution outline.

source

QAtlas.fetch — Method

fetch(model::KitaevHoneycomb, ::Energy{:per_site}, bc::PBC; Lx, Ly) -> Float64

Per-site ground state energy on a Lx × Ly unit-cell torus (PBC in both lattice directions) — enumerates all four topological flux sectors and returns the minimum. Each sector corresponds to a choice of fermion boundary conditions (W_x, W_y ∈ {±1}); Lieb's theorem fixes plaquette fluxes at +1, so the spin-Hamiltonian ground state is one of these four.

Bond connectivity matches Lattice2D.build_lattice(Honeycomb, Lx, Ly; boundary=PeriodicAxis()). bc.N is ignored; pass Lx, Ly as kwargs.

For large L the four sectors converge to the same energy and individual Bloch-sum terms dominate; for small L sector choice is essential (e.g. Lx = Ly = 2 gives distinct sector energies differing by ~10%).

source

QAtlas.fetch — Method

fetch(model::KitaevHoneycomb, ::MassGap, ::Infinite) -> Float64

Single-Majorana gap in the thermodynamic limit.

Δ = 2 · min_k |f(k)|.

In the A/B/C gapless phase (each |Kᵧ| ≤ sum of the other two), f(k) has two linear (Dirac) zeros and Δ = 0. In the gapped Az / Ax / Ay phases (|Kᵧ| exceeds the sum of the other two), |f| is bounded away from zero and `Δ = 2·( |Kγmax| − |Kγother1| − |Kγ_other2| )`.

source

QAtlas.fetch — Method

fetch(model::KitaevHoneycomb, ::FreeEnergy, ::Infinite;
      beta::Real, rtol=1e-8, kwargs...)

Per-site freeenergy of the infinite Kitaev honeycomb at inverse temperature beta, in the matter-sector free-fermion approximation (valid for `T ≪ Δv`; see module header). Adaptive 2D Gauss-Kronrod quadrature over the BZ.

source

QAtlas.fetch — Method

fetch(model::KitaevHoneycomb, ::FreeEnergy, ::OBC;
      Lx, Ly, beta::Real, kwargs...)

Per-site free_energy of an Lx × Ly OBC Kitaev strip at inverse temperature beta, in the matter-sector free-fermion approximation. Computed by summing the contribution of each Majorana singular value.

source

QAtlas.fetch — Method

fetch(model::KitaevHoneycomb, ::SpecificHeat, ::Infinite;
      beta::Real, rtol=1e-8, kwargs...)

Per-site specificheat of the infinite Kitaev honeycomb at inverse temperature beta, in the matter-sector free-fermion approximation (valid for `T ≪ Δv`; see module header). Adaptive 2D Gauss-Kronrod quadrature over the BZ.

source

QAtlas.fetch — Method

fetch(model::KitaevHoneycomb, ::SpecificHeat, ::OBC;
      Lx, Ly, beta::Real, kwargs...)

Per-site specific_heat of an Lx × Ly OBC Kitaev strip at inverse temperature beta, in the matter-sector free-fermion approximation. Computed by summing the contribution of each Majorana singular value.

source

QAtlas.fetch — Method

fetch(model::KitaevHoneycomb, ::ThermalEntropy, ::Infinite;
      beta::Real, rtol=1e-8, kwargs...)

Per-site entropy of the infinite Kitaev honeycomb at inverse temperature beta, in the matter-sector free-fermion approximation (valid for T ≪ Δ_v; see module header). Adaptive 2D Gauss-Kronrod quadrature over the BZ.

source

QAtlas.fetch — Method

fetch(model::KitaevHoneycomb, ::ThermalEntropy, ::OBC;
      Lx, Ly, beta::Real, kwargs...)

Per-site entropy of an Lx × Ly OBC Kitaev strip at inverse temperature beta, in the matter-sector free-fermion approximation. Computed by summing the contribution of each Majorana singular value.

source

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