Kitaev1D — 1D p-wave Majorana Wire

Kitaev1D(; μ = 0.0, t = 1.0, Δ = 1.0) <: AbstractQAtlasModel

Kitaev (2001) one-dimensional p-wave superconducting wire,

\[H = -\mu \sum_i c_i^{\dagger} c_i - t \sum_i (c_i^{\dagger} c_{i+1} + \text{h.c.}) + \Delta \sum_i (c_i c_{i+1} + \text{h.c.}).\]

μ is the chemical potential, t the hopping, Δ the p-wave pairing. |μ| < 2|t| is the topological phase (Majorana edge modes); |μ| > 2|t| is the trivial phase; |μ| = 2|t| is the gapless critical line.

This is the 1D Majorana wire and is distinct from the 2D KitaevHoneycomb spin model.

The TFIM is a special case (μ = -2h, t = J, Δ = J); the BdG spectrum of Kitaev1D(μ=-2h, t=J, Δ=J) agrees exactly with that of TFIM(J=J, h=h) at OBC.

Currently registered fetches:

fetch(model::Kitaev1D, ::CorrelationLength, ::Infinite) -> Float64

T = 0 correlation length of the infinite Kitaev1D chain, set by the inverse bulk gap,

\[\xi = \frac{1}{\Delta_{\mathrm{gap}}}.\]

Returns Inf on the gapless line |μ| = 2|t| (and on the gapless metal Δ = 0, |μ| < 2|t|). In QAtlas convention ξ is dimensionless (in units of the lattice spacing).

fetch(model::Kitaev1D, ::EdgeModeEnergy, bc::OBC; N::Int) -> Float64

Energy of the lowest-lying boundary mode on an N-site OBC Kitaev1D chain — the smallest non-negative BdG eigenvalue.

In the topological phase (|μ| < 2|t|, Δ ≠ 0) the two end-localised Majorana modes hybridise into a single complex fermion with exponentially-small splitting ~ e^{-N/ξ} where ξ ~ 1/log(2|t|/|μ|) for |μ| ≪ 2|t|.

Numerically equal to fetch(model, MassGap(), OBC(N)); the two methods exist as separate names so call sites can be explicit about which physical interpretation they have in mind.

fetch(model::Kitaev1D, ::Energy{:per_site}, ::Infinite; beta=nothing) -> Float64

Energy per site of the infinite Kitaev1D chain. With no beta (or beta = nothing), the T = 0 ground state

\[\varepsilon_0 = -\frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{E(k)}{2}\, dk, \qquad E(k) = \sqrt{(2t\cos k + \mu)^2 + 4\Delta^2 \sin^2 k};\]

with beta = β > 0, the finite-temperature value

\[\varepsilon(\beta) = -\frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{E(k)}{2}\, \tanh\!\frac{\beta E(k)}{2}\, dk\]

(the factor 1/2 accounts for the BdG particle-hole doubling; β → ∞ recovers ε₀). Adaptive Gauss-Kronrod quadrature.

fetch(model::Kitaev1D, ::ExactSpectrum, bc::OBC; N::Int) -> Vector{Float64}

Return the N non-negative BdG quasiparticle energies of the OBC Kitaev1D chain, sorted ascending.

N is read from bc.N (OBC(N) / OBC(; N)) or, as a legacy fallback, from the N keyword argument.

In the topological phase (|μ| < 2|t|, Δ ≠ 0) the lowest entry is the exponentially-small Majorana edge-mode hybridisation energy ~ e^{-N/ξ}. In the trivial phase the lowest entry approaches the bulk gap min(|2t + μ|, |2t - μ|).

fetch(model::Kitaev1D, ::MassGap, ::Infinite) -> Float64

Bulk single-quasiparticle gap of the infinite Kitaev1D chain,

\[\Delta_{\mathrm{gap}} = \min_k \sqrt{(2t\cos k + \mu)^2 + 4\Delta^2 \sin^2 k}.\]

Closed form (for t ≠ 0, Δ ≠ 0):

• |μ| ≥ 2|t|: minimum at k = 0 or k = π, giving Δ_gap = ||μ| - 2|t||.
• |μ| < 2|t|: stationary point at cos k* = -μ t / (2(t² - Δ²)) when |t| ≠ |Δ| and |cos k*| ≤ 1; otherwise the minimum is at k = 0 or k = π.

Δ = 0 gives the gapless metal whenever |μ| < 2|t|, and the routine returns 0.0 in that case.

MassGap at OBC is provided as the smallest non-negative BdG eigenvalue (numerically equal to EdgeModeEnergy at OBC).

fetch(model::Kitaev1D, ::MassGap, bc::OBC; N::Int) -> Float64

Single-quasiparticle gap of the N-site OBC Kitaev1D chain — the smallest non-negative BdG eigenvalue of the 2N × 2N BdG matrix.

In the topological phase this is the Majorana edge-mode energy ~ e^{-N/ξ} (use EdgeModeEnergy for the same value under a boundary-mode-explicit name). In the trivial phase it converges to the bulk gap as N → ∞.

fetch(model::Kitaev1D, ::TopologicalInvariant, ::Infinite) -> Int

Pfaffian Z₂ invariant of the infinite Kitaev1D chain (Kitaev 2001),

\[\nu = \operatorname{sgn}\bigl[\operatorname{Pf} A(k=0) \cdot \operatorname{Pf} A(k=\pi)\bigr] = \operatorname{sgn}\bigl[(\mu + 2t)(\mu - 2t)\bigr] = \operatorname{sgn}(\mu^2 - 4t^2).\]

Returns -1 in the topological phase (|μ| < 2|t|) and +1 in the trivial phase (|μ| > 2|t|). Throws on the gapless line |μ| = 2|t| (Pfaffian vanishes; invariant ill-defined) and on the gapless metal Δ = 0 with |μ| < 2|t|.

The two 2 × 2 Pfaffians are computed using the generic pfaffian routine in src/core/pfaffian.jl, exercising the same numerical machinery used elsewhere in QAtlas for free-fermion Wick contractions.

fetch(model::Kitaev1D, ::FreeEnergy, ::Infinite; beta) -> Float64

Helmholtz free energy per site f(β) = -β⁻¹ log Z/L of the infinite Kitaev chain (BdG free fermions):

f(β) = -(1/2π) ∫_{-π}^{π} [ E(k)/2 + β⁻¹ ln(1 + e^{-βE(k)}) ] dk.

β → ∞ recovers the ground-state energy per site.

fetch(model::Kitaev1D, ::SpecificHeat, ::Infinite; beta) -> Float64

Specific heat per site of the infinite Kitaev chain, the energy fluctuation–dissipation form for free BdG quasiparticles:

c_v(β) = (β²/8π) ∫_{-π}^{π} E(k)² sech²(βE(k)/2) dk
       = β² ∫_{-π}^{π} (dk/2π) E(k)² f_k (1 - f_k),   f_k = 1/(e^{βE}+1).

fetch(model::Kitaev1D, ::ThermalEntropy, ::Infinite; beta) -> Float64

Entropy per site s(β) = β(ε − f) of the infinite Kitaev chain. Bounded between 0 (T → 0) and ln 2 (T → ∞: two states per spinless mode).