Kitaev's Chain: How to Get Unpaired Majorana Fermions
Electrons can be used as qubits with spin up and spin down states to store information. The problem is, that these qubits are sensitive to electromagnetic and thermal disturbances as they occur in electric circuits. We call such the interaction as decorence. The question is: How can we theoretically build decorenceprotected quantum memory?
Alexei Kitaev came (in his paper from the year 2000) to an idea, not to use spins as quantum memory, but the so called Majoara fermions in a onedimensional fermionic chain, which are much less sensitive to disturbances.
In the following we consider a onedimensional chain of length \(L\). Each site can be either empty \( 0\rangle \) or occupied \( 1\rangle \) by an electron. We assume that the spin orientation is fixed here. With the annihilation operator \(a_j\) we can annihilate an electron at the \(j\)th site and with \(a_j^{\dagger}\) we can create an electron at the \(j\)th site. Here \(j\) goes over the sites: \( j ~=~ 1, ~..., ~ L \).
Two types of errors can occur in the quantum system and thus can modify the system:

Classical error  represented by the Pauli matrix \(\sigma^{\mathrm x}_j \) changes the \(j\)th state from \(0\rangle\) to \(1\rangle\) or vice versa. This error cannot occur using this Kitaev chain because of the charge conservation.

Phase flip error  represented by the Pauli matrix \(\sigma^{\mathrm z}_j = 1  2\, a^{\dagger}_j \, a_j \) (JordanWigner transformation) may change the sign of \(j\)th state:
$$ \begin{align} \sigma^{\mathrm z}_j \, 0\rangle ~&=~ 0\rangle \\
\sigma^{\mathrm z}_j \, 1\rangle ~&=~ 1\rangle \end{align} $$
Majorana fermions
In 1937 Ettore Majorana speculated that there could be a particle that is ist own antiparticle. In the language of second quantization it means that creating and annihilating a particle should be done with the same operator:
So we can simply use either the \(c\) or \(c^{\dagger}\) operator to create and annihilate a Majorana fermion.
We can write the electron annihilation operator \(a_{j}\) by splitting it into a real \(c_{2j1}\) and imaginary part \(c_{2j}^{\dagger}\). Here \(c_{2j1}\) and \(c_{2j}^{\dagger}\) are majorana operators. And then we form the adjoint of \(a_{j}\), which is the creation operator \(a_{j}^\dagger\):
a_j^{\dagger} ~&=~ \frac{1}{2} \left( c_{2j1}  \mathrm{i}\,c_{2j}^{\dagger} \right) \end{align} $$
For each physical (electron) site \(j\) there are always two Majorana sites \(2j1\) and \(2j\). Thus, the Majorana sites occur in pairs and their number is even. We can reformulate Eq. 3
equivalently and express the Majorana operators as superposition of the electron operators:
c_{2j} ~&=~ \mathrm i \left( a_j^{\dagger}  a_j\right) \end{align} $$
As you can see: Majorana operators \(c_{2j1}\) and \(c_{2j}\) act on a single electron site. We can easily see that \(c_{2j1}\) and \(c_{2j}\) are indeed selfadjoint operators: \( c_{2j1} = c_{2j1}^\dagger\) and \(c_{2j} = c_{2j}^\dagger\).
Let us express the number operator \( a^{\dagger}_j \, a_j \) inside the phase error using Majorana representation 3
:
~&=~ 1  2\, \frac{1}{2}(1+\mathrm{i}\,c_{2j1}c_{2j}) \\\\
~&=~ \mathrm{i}\,c_{2j1}c_{2j}
\end{align} $$
If the two Majorana operators in Eq. 5
belonged to different electron sites, the phase error could be made arbitrarily small if we make the chain long enough. Indeed, there is still an interaction between the Majorana sites at the edges of the chain, but it decreases exponentially with the length of the chain.
Next, let's construct a Hamiltonian operator which would give rise to Majorana fermions (more precisely: Majorana Zero Modes):
~&\mu \, \underset{j ~=~1}{\overset{L}{\boxed{+}}}~ \left( a^{\dagger}_j\,a_j  1/2 \right) \\\\
~&+~ \underset{j ~=~1}{\overset{L}{\boxed{+}}}~ \left( \Delta\,a_j\,a_{j+1} ~+~ \Delta^\ast \, a^{\dagger}_{j+1} \, a^{\dagger}_{j} \right)
\end{align} $$

The first summand in
6
, with the hopping amplitude \(w\), describes the hopping of an electron from site \(j\) to \(j+1\) and vice versa. 
The second summand in
6
, with the chemical potential \(\mu\), describes the energy on site \(j\). 
The last summand in
6
describes superconductivity with the energy gap \(\Delta\). The parameter \(\Delta\) specifies the energy gain, when two electrons form a Cooper pair. This part of the Hamiltonian creates or annihilates two adjacent electrons on site \(j\) and \(j+1\).
Next, we use the Majorana operators 4
to rewrite the Hamilton operator 6
:
~&+~ \frac{\mathrm{i} }{2} \, \left(w+\Delta\right)\, \underset{j ~=~1}{\overset{L1}{\boxed{+}}}~ c_{2j}\,c_{2j+1} \\\\
~&+~ \frac{\mathrm{i} }{2} \,\left(\Deltaw\right) \, \underset{j ~=~1}{\overset{L2}{\boxed{+}}}~ c_{2j1}\,c_{2j+2} \end{align} $$
By expressing the Hamiltonian operator with Majorana operators, we will be able to spot isolated Majorana fermions. So we have three parameters \(\mu\), \(w\) and \(\Delta\) that we can change to achieve different phases of the Kitaev chain. The Kitaev chain has two phases: the trivial phase and the topological phase. In the topological phase we will discover isolated Majoranas.

Special case #1: If we set \(\Delta=w=0\) in the Hamilton operator
7
, then it simplifies to:$$ \begin{align} H_1 ~&=~ \frac{\mathrm{i} }{2} \, \mu\, \underset{j ~=~1}{\overset{L}{\boxed{+}}}~ c_{2j1}\,c_{2j} \\\\
~&=~ \frac{\mathrm{i} }{2} \, \mu\, \left( c_{1}\,c_{2} ~+~ c_{3}\,c_{4} ~+~ ... ~+~ c_{2L1}\,c_{2L} \right) \end{align} $$In this case, two Majorana sites from the same physical site are coupled. This is the trivial case because here ALL Majorana sites are paired up (see illustration 2).

Special case #2: If we set superconduting gap and hopping amplitude equal \(\Delta=w \) and set chemical potential \(\mu=0\), then Hamilton operator
7
becomes:$$ \begin{align} H_1 ~&=~ \mathrm{i} \, w\, \underset{j ~=~1}{\overset{L}{\boxed{+}}}~ c_{2j}\,c_{2j+1} \\\\
~&=~ \mathrm{i} \, w\, \left( c_{2}\,c_{3} ~+~ c_{4}\,c_{5} ~+~ ... ~+~ c_{2L2}\,c_{2L1} \right) \end{align} $$By choosing the parameter \(\mu=0\), we have decoupled the two isolated Majorana sites from the rest of the chain. The Operators \(c_1\) and \(c_{2L}\) are not in the Hamiltonian. If we diagonalize the Hamiltonian operator
8
and plot the eigenvalues of these two Majora edge states, we will see that they have energy equal to zero. That is why we call them Majorana Zero Modes. This special case forms the topological phase of the Kitaev chain (see illustration 2).If we plot the probability densities \(\psi_0^2\) and \(\psi_1^2\) of the Majorana Zero Modes as a function of position on the Kitaev chain, we see that they do indeed reside at the edges of the chain.
We have considered a special case where the chemical potential is \( \mu = 0 \). In this topological phase, we have obtained two Majorana Zero Modes. What is special about the Kitaev chain is that this topological phase occurs not only at \(\mu=0\):

Topological phase: As can be seen in illustration 3, the spectrum indicates two degenerate ground states with energies equal to zero, which remain stable up to the ratio \(\mu/w = 2 \). Moreover, these Majorana zero modes have an energy gap to the excited states (shown in gray) that decreases as the chemical potential \(\mu \) increases. This energy gap makes the Majorana Zero Modes stable against phase errors.

Trivial phase: Above the ratio \(\mu/w > 2 \) (i.e. strong pairing of two Majorana sites at the same physical site) the degeneracy of the Majorana Zero Modes is lifted and their energy is no longer zero. In this phase, the Majorana Zero Modes disappear.
In our model 6
, in the superconductor term with \(\Delta\), we pair up electrons from DIFFERENT physical sites \(j\) and \(j+1\). We cannot pair up two electrons from the same site because only one electron can occupy a site in our model. In other words: We have no spin in this model. Such a pairing describes a pwave superconductor.
Of course, we can ask ourselves whether the model 6
for a pwave superconductor is experimentally realizable. After all, it is a onedimensional pwave superconductor with electrons without spin. It is difficult to engineer a pwave superconductor and Kitaev wants it to be without spin degeneracy, which is practically impossible because electrons DO HAVE a spin.
So in a real chain we have two such pairings, one for spinup and one for spindown electrons. However, this leads to the fact that the two Majorana Zero Modes at the edge are now paired up which eliminates the purpose of the Kitaev chain.
10 years after Alexei Kitaev published his paper, the problem of an experimentally impossible spinless pwave superconductor was solved by Lutchyn, Sau and Das Sarma. We could actually engineer an effective pwave superconductor by using a 1d chain placed on a conventional swave superconductor with spinorbit coupling and an external magnetic field. This ingredients will eliminate the spin degeneracy.