Majorana excitations

We consider the following Hamiltonian written in Nambu representation,

\[H =\Psi_k^\dagger \mathcal{H}\Psi_k\]

with a Nambu spinor defined as

\[\Psi_k = \left\lbrack \matrix{c_{k\uparrow} \cr c_{k\downarrow} \cr c_{-k\downarrow}^\dagger \cr -c_{-k\uparrow}} \right\rbrack \]

Where,

\[\mathcal{H} = \left\lbrack \matrix{\epsilon_k & 0 & \Delta & 0 \cr 0 & \epsilon_k & 0 & \Delta \cr \Delta & 0 & -\epsilon_k & 0 \cr 0 & \Delta & 0 & \epsilon_k} \right\rbrack \]

Majorana Fermion

It is a very special type of fermion which is by definition it's own antiparticle(\(\Psi^\dagger = \Psi\)). These particles do not appear naturally in materials, as we only have electrons.

Mathematically, each electron can be written as two Majoranas

\[c = \Psi_\alpha + i\Psi_\beta, c^\dagger = \Psi_\alpha - i\Psi_\beta\]

\[\Psi_\alpha^\dagger = \Psi_\alpha, \Psi_\beta^\dagger = \Psi_\beta\]

Questions

Can we isolate a single Majorana in materials?
Can we have superconductors in nature that show these excitations?

Minimal model for 1D topological superconductor¶

One dimensional spinless p-wave superconductor(Kitaev model)

\[H = \sum_n tc_{n+1}^\dagger c_n + \Delta c_n c_{n+1} + c.c\]

Where, \(c_k = \sum_n e^{ikn} c_n\) and \(\Delta(k) = -\Delta(-k)\) After Fourier transform, we have

\[H = \sum_k \epsilon_k c_k^\dagger c_k + i \Delta[c_{-k}c_k \sin k - c_{-k}^\dagger c_k^\dagger \sin k]\]

Spinless fermions in a 1D chain, (hopping and superconducting order are equal)

\[H = \sum_n c_{n+1}^\dagger c_n + c_n c_{n+1} + h.c\]

can be transformed into (\(\gamma\) are Majorana operators)

\[H = i\sum_n \gamma_{2n}\gamma_{2n+1}\]

\(c_n = \gamma_{2n-1} + i \gamma_{2n}\)

At the ends of the chain there are zero energy excitations in the superconducting state.