Kitaev1D — 1D p-wave Majorana Wire

Overview

The 1D Kitaev (2001) chain is the canonical exactly-solvable model of a 1D topological superconductor. It is a free-fermion Bogoliubov-de Gennes problem with a Z_2 Pfaffian invariant: the topological phase hosts two Majorana zero modes at the ends of an open chain.

Distinct from KitaevHoneycomb. This model is the one-dimensional spinless p-wave wire (Kitaev, Phys.-Usp. 44, 131, 2001). The KitaevHoneycomb model is a two-dimensional anisotropic spin model on the honeycomb lattice (Kitaev, Ann. Phys. 321, 2, 2006). They share a name only by historical accident.

Hamiltonian

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

with c_i spinless fermions, chemical potential μ, hopping t, and p-wave pairing Δ. Defaults: μ = 0, t = 1, Δ = 1.

PBC dispersion (closed form):

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

Phase diagram (Δ ≠ 0, t ≠ 0):

Topological invariant (Kitaev 2001, Pfaffian sign):

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

evaluated on the 2 × 2 Majorana Bloch matrix A(k) at the two time-reversal-invariant momenta.

OBC edge zero modes: in the topological phase the two Majorana ends hybridise into a single complex fermion with hybridisation energy E_edge(N) ~ exp(-N/ξ) where ξ ~ 1/log(2|t|/|μ|) for |μ| ≪ 2|t|. At the sweet spot μ = 0, t = Δ the two Majoranas decouple exactly and E_edge vanishes for all N ≥ 2.

TFIM correspondence

The transverse-field Ising model is a special case of Kitaev1D under the identification

\[\mu = -2h, \qquad t = J, \qquad \Delta = J.\]

The OBC BdG spectra of Kitaev1D(μ=-2h, t=J, Δ=J) and TFIM(J=J, h=h) agree exactly (verified by test/standalone/test_kitaev1d.jl). The helper _kitaev1d_bdg_spectrum is a strict generalisation of _tfim_bdg_spectrum; choosing Δ = J and μ = -2h reproduces TFIM's A and B blocks element-wise.

Coverage Matrix

Usage

using QAtlas

# Topological phase (μ = 0, sweet spot)
m = Kitaev1D(; μ=0.0, t=1.0, Δ=1.0)
fetch(m, TopologicalInvariant(), Infinite())   # -1
fetch(m, MassGap(), Infinite())                # 2.0  (bulk gap)
fetch(m, EdgeModeEnergy(), OBC(40))            # ~1e-15 (sweet-spot zero modes)

# Trivial phase
m_triv = Kitaev1D(; μ=3.0, t=1.0, Δ=1.0)
fetch(m_triv, TopologicalInvariant(), Infinite())   # +1
fetch(m_triv, MassGap(), Infinite())                # 1.0

# TFIM cross-check
J, h = 1.0, 0.7
m_tfim = Kitaev1D(; μ=-2h, t=J, Δ=J)
fetch(m_tfim, ExactSpectrum(), OBC(20))    # matches _tfim_bdg_spectrum(20, J, h)

References

• A. Yu. Kitaev, "Unpaired Majorana fermions in quantum wires", Phys.-Usp. 44, 131 (2001).
• J. Alicea, "New directions for the pursuit of Majorana fermions in solid state systems", Rep. Prog. Phys. 75, 076501 (2012).
• J. K. Asbóth, L. Oroszlány, A. Pályi, A Short Course on Topological Insulators, Lect. Notes Phys. 919 (2016) — Pfaffian invariant.

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Verified hubs

In the Verified Atlas, this model registers 6 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.

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