The model and phase diagram–
In this section we will consider the simplest and paradigmatic model for topological supercoductor in one-dimension, the Kitaev chain. This was proposed by Kitaev in 2000 [47]. As seen in the section on the Dirac equation, Majorana’s original paper already had a Hamiltonian of discrete, real ‘Majorana’ variables. The mathematical form of the Kitaev chain as a quadratic, fermionic Hamiltonian on a lattice appeared in the works of Lieb, Schulz and Mattis [54] and [55] . This was in the context of spin chain physics, where the spin variables are converted to fermionic operators through a non-local transformation. Kitaev’s insight was in the identification of this Hamiltonian as a model for a topological p-wave superconductor and occurrence of an isolated Majorana zero modes at the edge in the topological phase of the model. Historically, this was set in the context of previous works which discovered Majorana modes as quasiparticles in fractional quantum Hall phases [56].Kitaev’s further insight was the possible use of the MBS for topological quantum computation.
Figure 2.3: Kitaev chain - A lattice model of fermions in one-dimension, with hopping parameter w, Super- conducting gap ∆ and onsite chemical potential µ.
This lattice model consists of non-interacting spinless fermions on each site having nearest neighbor tunneling of strength w, nearest neighbor superconducting pairing of strength ∆, and an on-site chemical
potential µn (Fig. 2.3).Its associated Hamiltonian takes the form H = N −1 X n=1 (−wc†n+1cn+ ∆c†n+1c † n+ h.c) − N X n=1 µn(c†ncn− 1/2), (2.39)
where h.c. denotes Hermitian conjugate. Here, the c†n and cn operators represent the creation and annihila-
tion of electrons on site n, respectively. They obey the fermionic commutation relations {cm, cn} = 0 and
{cm, c†n} = δmn. In the thermodynamic limit (infinite wire) or for a closed chain, one can transform the
Hamiltonian into Fourier space, and the single particle energy spectrum takes the form
Ek= ±
q
(2w cos k + µ)2+ 4∆2sin2
k. (2.40)
The system exhibits multiples phases as shown in the Fig. ??. The spectrum has a finite superconducting gap in all the phases except at the ‘critical lines’ µ/2w = 1 and ∆ = 0(represented as dark lines in the figure). The gap vanishes as one crosses one of the ‘critical lines’ and reopens upon entering another phase. The study of isolated Majorana modes become transparent if the Hamiltonian is transformed to the Majorana basis. Let us introduce 2N Majorana fermion operators ˆan and ˆbn, namely, ˆan= cn+ c†nand ˆbn= i(c†n− cn).
The Majorana operators satisfy the relations ˆa†n = ˆan, ˆb†n = ˆbn and {ˆan, ˆam} = {ˆbn, ˆbm} = 2δmn. In terms
of the Majorana operators, the Hamiltonian is given by
HM = − i 2 N −1 X n=1 (w − ∆)ˆanˆbn+1− (w + ∆)ˆbnˆan+1 − N X n=1 iµn 2 ˆan ˆ bn. (2.41)
Now let us look at the different phases of this model. The phases I and II are topologically non-trivial and, in the thermodynamic limit, have zero energy Majorana modes bound to the ends of the wire, whereas such modes are absent in the topologically trivial phases III and IV. These Majorana modes have finite support at the ends and decay rapidly into the bulk with a decay length proportional to the reciprocal of the bulk gap. One can understand the existence of the Majorana end modes by considering the extreme limit of w = ∆ and µ = 0. The Hamiltonian reduces to H = iwP
nbnan+1. The Majorana operators a1 and
bN are not paired with any other operators in the system and therefore do not appear in the Hamiltonian.
These isolated modes correspond to the zero energy eigenvectors localized at the ends. The existence of these modes is robust even away from this extreme limit and they only disappear with the closing of the
bulk gap.
The ground state of the system in the topological phase is thus doubly degenerate and has two zero energy eigenvalues corresponding to the Majorana modes. These Majorana modes can be combined to form two complex Dirac fermion states, which can be either empty or occupied. Hence, each of the degenerate ground states has a specific fermion parity and the system can be characterized by a related Z2-valued
topological invariant.
Class D: The existence of the zero-energy modes is in fact a manifestation of the symmetry and the topology of the system. As described in one of the previous sections, the anti-unitary symmetries completely classify topological phases that can occur in gapped non-interacting fermionic systems and also provide the topological invariant in different dimensions. In the present case of a spin-less superconductor, there is the particle-hole symmetry as described above but no time-reversal symmetry. This identifies the system in class D and in one dimension, such a system is characterized by a Z2topological invariant. One can calculate the
invariant as a winding number from the Dirac form of the Hamiltonian in k-space [5]. This invariant also counts the number of zero modes in the topological defects such as vortices or the edges of the system.
Dirac equation and the edge mode: Finally let us see the emergence of the Dirac equation in the Ki- taev Hamiltonian and show that the zero energy solution corresponds to the Majorana mode. By Fourier transforming the Hamiltonian in Eq. 2.39,
HBdG= 1 2 X p Ψ†k
−2w cos k − µ 2i|∆| sin k −2i|∆| sin k 2w cos k + µ
Ψk (2.42)
where Ψk= (ckc†−k)T is the Nambu-spinor. When this is expanded near k ≈ 0 and k → i∂x one obtains
HBdG(k) = |∆|(i∂x)σ1− µσ3. (2.43)
This is the Dirac Hamiltonian. To obtain the Majorana modes, consider the solitonic profile of the chemical potential- µ(x): µ(−∞) < 0 and µ(+∞) > 0. The zero energy solution of the BdG Hamiltonian is
γM = N Z dxe iπ/4 √ 2 exp − 1 |∆| Z ∞ 0 µ(x0)dx0 (c(x) − ic†(x)), (2.44)
where N is the normalization. Thus we have seen that the Kitaev chain Hamiltonian is a simple lattice realization of a one-dimensional topological superconductor and harbors Majorana modes at the edges.