Kitaev Chain

One-Sentence Description

The Kitaev chain is a one-dimensional spinless p-wave superconducting mean-field model with a topological phase supporting Majorana zero modes at open boundaries.

Physical Setup

The model places spinless fermionic modes on a one-dimensional chain and allows nearest-neighbor hopping, chemical potential, and nearest-neighbor pairing. It is a minimal model of topological superconductivity, not a literal microscopic model of ordinary s-wave superconductors.

Because the Hamiltonian is a mean-field Bogoliubov-de Gennes model, particle number is not conserved, although fermion parity is.

The Jordan–Wigner Transformation shows how the open transverse-field Ising chain becomes a nearest-neighbor Majorana chain. This model card keeps the fermionic Hamiltonian and topological-phase interpretation as its canonical focus.

Hamiltonian

With lattice spacing set to one and real parameters, a common convention is

$$H = -\mu\sum_j c_j^\dagger c_j - t\sum_j \left( c_j^\dagger c_{j+1} + c_{j+1}^\dagger c_j \right) + \Delta\sum_j \left( c_j c_{j+1} + c_{j+1}^\dagger c_j^\dagger \right).$$

In the Nambu basis $(c_k,c_{-k}^\dagger)$, the Bloch Bogoliubov-de Gennes Hamiltonian can be written as

$$H_{\mathrm{BdG}}(k) = \left( -\mu-2t\cos k \right)\tau_z + 2\Delta\sin k\,\tau_y,$$

up to Nambu-basis sign conventions. Its quasiparticle spectrum is

$$E_\pm(k) = \pm \sqrt{ \left(\mu+2t\cos k\right)^2 + 4\Delta^2\sin^2 k }.$$

Topological Regime

For $t\ne0$ and $\Delta\ne0$, the bulk gap closes at

$$\mu=\pm2t.$$

In the standard convention with real nonzero $\Delta$, the topological phase occurs for

$$\lvert\mu\rvert<2\lvert t\rvert.$$

An open chain in this phase has exponentially localized Majorana zero modes at its ends in the ideal model. Finite chains split the zero-mode degeneracy by an exponentially small amount when the end modes overlap.

What It Teaches

The Kitaev chain is the simplest setting where superconducting pairing, particle-hole redundancy, fermion parity, boundary modes, and a bulk topological distinction can be studied explicitly.

It also teaches a useful caution: Majorana zero modes are model properties under stated gap, symmetry, parity, and boundary assumptions. Experimental claims require additional device modeling.

Canonical Links

• BCS Model
• SSH Model
• Topological Invariants
• Topological Superconductors for the class-D Pfaffian invariant, symmetry classification, Majorana protection, and experimental evidence beyond this model card.
• Fermionic Anticommutation Relations
• Many-Particle Hamiltonians

Common Mistakes

• Treating the spinless p-wave chain as an ordinary number-conserving Hamiltonian.
• Forgetting that the end modes require an open chain; periodic chains have no ends.
• Quoting $\lvert\mu\rvert<2\lvert t\rvert$ without stating the Hamiltonian convention.
• Treating finite-chain zero modes as exactly degenerate without checking overlap and parity constraints.
• Interpreting the model as a complete experimental proposal without proximity, disorder, interactions, and measurement details.

Quick Check

Why must the Kitaev-chain phase transition occur through a bulk gap closing in this model?

Solution

The topological invariant cannot change under a smooth deformation that keeps the bulk gap open and preserves the model assumptions. In the displayed spectrum, the gap closes when $\sin k=0$ and $\mu+2t\cos k=0$, giving $\mu=\pm2t$.

References

• A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires,” Physics-Uspekhi 44, 131-136, 2001.
• J. Alicea, “New directions in the pursuit of Majorana fermions in solid state systems,” Reports on Progress in Physics 75, 076501, 2012.
• C. W. J. Beenakker, “Search for Majorana fermions in superconductors,” Annual Review of Condensed Matter Physics 4, 113-136, 2013.