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Majorana fermions in two-dimensional systems satisfy non-Abelian statistics. They are possible to exist in topological superconductors as quasi particles, which is of great significance for topological quantum computing. In this paper, we study a new promising system of superconducting topological surface state topological insulator thin films. We also study the phase diagrams of the model by plotting the Majorana edge states and the density of states in different regions of the phase diagram. Due to the mirror symmetry of the topological surface states, the Hamiltonian can be block diagonalized into two spin-triplet p-wave superconductors, which are also confirmed by the phase diagrams. The chiral Majorana edge modes may provide a new route for realizing topological quantum computation.

In 1937, Ettore Majorana [_{2}RuO_{4} [_{2}RuO_{4} is more of a stay in the theoretical discussion, and no definite conclusion has been reached [

Under normal states, iron-based materials with small Fermi levels are considered topological metals with slightly doped topological surface states, which can be observed by the angular resolved spectrum of spin polarization [_{C}, which is the topological superconducting surface states [_{0.55}Se_{0.45} and in the quantum anomalous vortices nucleated at Fe site [

First, we need to construct the Hamiltonian of the model, which is selected based on Nambu basis: Ψ N = ( ψ 1 ↑ q , ψ 1 ↓ q , ψ 2 ↑ q , ψ 2 ↓ q , ψ 1 ↑ − q † , ψ 1 ↓ − q † , ψ 2 ↑ − q † , ψ 2 ↓ − q † ) T , 1 and 2 represent the surface states of the upper and lower surfaces respectively, and these two indexes represent the pseudo spin states in the spin-orbit coupling respectively, and q index represents the momentum, the Fermi velocity is set as 1. With the basis vector selected, the corresponding Bogoliubov-de Gennes (BdG) [

H ( q ) = ( h x ( q ) Δ x † Δ x h x * ( − q ) ) (1)

h x ( q ) and Δ x are expressed as follows:

h x ( q ) = ( λ − μ q 1 − i q 2 t 0 q 1 + i q 2 − λ − μ 0 t t 0 λ − μ − q 1 + i q 2 0 t − q 1 − i q 2 − λ − μ ) (2)

Δ x = ( 0 Δ 1 0 − Δ s − Δ t 1 − i Δ t 2 − Δ 1 0 Δ s − Δ t 1 − i Δ t 2 0 0 − Δ s + Δ t 1 + i Δ t 2 0 − Δ 2 Δ s + Δ t 1 + i Δ t 2 0 Δ 2 0 ) (3)

We make a substitution such that α = Δ 1 + Δ 2 2 and β = Δ 1 − Δ 2 2 , so that Δ 1 and Δ 2 can be represented correspondently in terms of α and β as Δ 1 = α + β and Δ 2 = α − β . After the above substitution, the Hamiltonian of our model will be as follows:

H 1 = q 1 σ x χ z τ 0 + q 2 σ y χ z τ z − μ σ 0 χ 0 τ z + λ σ z χ x τ z + t σ 0 χ x τ z − Δ s σ y χ x τ y + α 2 σ y χ z τ y + β 2 σ y χ 0 τ y − Δ t 1 σ x χ y τ y + Δ t 2 σ x χ y τ x (4)

Define a mirror symmetry M − = − i σ z χ x τ 0 , then change the above Hamiltonian H 1 into M − H 1 M − T under the action of mirror symmetry M − , and the result after change is as follows:

M − H 1 M − T = − q 1 σ x χ z τ 0 − q 2 σ y χ z τ z + μ σ 0 χ 0 τ z + λ σ z χ x τ z − t σ 0 χ x τ z − Δ s σ y χ x τ y − α 2 σ y χ z τ y + β 2 σ y χ 0 τ y + Δ t 1 σ x χ y τ y − Δ t 2 σ x χ y τ x (5)

By comparing Equations (4) and (5), we can get α , Δ t 1 and Δ t 2 are odd symmetry of the mirror image about M − , while the term β is even symmetry of the mirror image about M − . When Δ t 1 , t, Δ 1 and Δ 2 are all non-zero, the time inversion symmetry of Hamiltonian described in Equation (1) will spontaneously break, and only when the time inversion symmetry is broken, will there be a non-zero Chern number.

are set to be the same as the module of the complex number Δ t 1 + i Δ t 2 , that is Δ 1 = Δ 2 = θ . As can be seen from

U − = 1 2 ( 0 0 0 0 1 0 1 0 0 1 0 − 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 − 1 1 0 − 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 − 1 0 0 1 0 1 0 0 0 0 ) (6)

If the unitary operator U − is applied to the Hamiltonian H, then the Hamiltonian becomes a block diagonal matrix under the action of the unitary operator by considering the case of Δ 1 = Δ 2 = Δ = θ :

U − H U − − 1 = ( h 1 0 0 h 2 ) (7)

h 1 = ( − t − λ + μ − θ − θ e − i ϕ 0 q 1 − i q 2 − θ − θ e i ϕ − t − λ − μ q 1 − i q 2 0 0 q 1 + i q 2 t + λ − μ θ + θ e i ϕ q 1 + i q 2 0 θ + θ e − i ϕ t + λ + μ ) (8)

h 2 = ( − t + λ − μ θ − θ e i ϕ 0 q 1 + i q 2 θ − θ e − i ϕ − t + λ + μ q 1 + i q 2 0 0 q 1 − i q 2 t − λ + μ − θ + θ e − i ϕ q 1 − i q 2 0 − θ + θ e i ϕ t − λ − μ ) (9)

Next, the corresponding phase diagram of h 1 in the θ − ϕ plane is drawn, as shown in

found that the phase diagram described by the original Hamiltonian is exactly equal to the superposition of the two-phase diagrams described by the block matrix obtained after the unitary transformation.

Next, edge states and their state density diagrams are discussed. The BdG Hamiltonian can be written as follows in the case of Nambu basis:

H B d G = ∑ q Ψ N † H ( q ) Ψ N . (10)

In the lattice model, Hamiltonian variables are replaced as follows: t → t 0 + t 1 ( q 1 2 + q 2 2 ) , q 2 → a − 1 sin ( q 2 a ) and q 2 2 → 2 ( 1 − a − 2 cos ( q 2 a ) ) Where a represents the lattice constant, which can be taken as 1, and the direction q 2 is the open edge condition. First, the edge states of each region in _{1}-E plane are drawn. The edge state graph of the corresponding point in the region I in

Next, the edge states of points corresponding to the region II in

The edge state of the corresponding point in the region III in

Finally, the edge state graph of the corresponding point in the region IV in

In order to further explain the distribution of wave function in real space in each group of edge state graphs, we draw density distribution of states with different parameters selected, and select the number of lattice points in the y direction as y = 50 in each graph drawing.

In the edge state graph corresponding to

points on edge states of different colors. We can see that for the density of states of the points on the edge states with the same positive and negative slope, they will be localized at both ends, which can indicate that the Chern number of the II region in

In this paper, we study the phase diagram, edge state and density distribution diagram of the topological surface state superconducting in detail. It is found that the phase diagram of Hamiltonian H ( q ) is the same as the superposition of the phase diagram among the block matrices obtained by the Hamiltonian after the unitary transformation of U − . In nontrivial topological phase, the corresponding phase diagrams are drawn by discussing the lattice model, and the edge states of different regions in the phase diagrams are also given, the difference between the corresponding wave function in the real space distribution state density diagram, thus further proving that Majorana zero model of the number is the same with the body of Chern number, body-edge correspondence. In the edge state, the electron propagates in a certain direction and does not backscatter. Because the Majorana zero model can well solve the problems caused by quantum decoherence, and the Majorana fermions obey non-abelian statistics, the encoding problem of quantum information can be solved by using the Majorana fermions, which will have great application value in the development of topological quantum computing.

We thank Xi Luo for helpful discussions. This work is supported by NNSF of China with No. 11804223.

The author declares no conflicts of interest regarding the publication of this paper.

Zhao, W.H. (2021) Phase Diagram and Edge States of Surface States of Topological Superconductors. World Journal of Condensed Matter Physics, 11, 65-76. https://doi.org/10.4236/wjcmp.2021.113005

In a two-dimensional system, it is assumed that a point k l in the discrete Brillouin region can be expressed as [

k l = ( k j 1 , k j 2 ) (1)

k j μ = 2 π j μ q μ N μ , ( j μ = 0 , ⋯ , N μ − 1 ) (2)

| n ( k l ) 〉 is periodic on the lattice. First define the Chern number of the nth band as:

c ˜ n ≡ 1 2 π i ∑ l F ˜ 12 ( k l ) (3)

For the wave function of the nth band, a connection variable can be defined:

U μ ( k l ) ≡ 〈 n ( k l ) | n ( k l + μ ^ ) 〉 / N μ ( k l ) (4)

where N μ ( k l ) ≡ | 〈 n ( k l ) | n ( k l + μ ^ ) 〉 | . Lattice strength is defined as

F ˜ 12 ( k l ) ≡ ln U 1 ( k l ) U 2 ( k l + 1 ^ ) U 1 ( k l + 2 ^ ) − 1 U 2 ( k l ) − 1 (5)

Since c ˜ n is gauge invariant, the gauge potential energy is selected as:

A ˜ μ ( k l ) = ln U μ ( k l ) , − π < 1 i A ˜ μ ( k l ) ≤ π (6)

which is periodic on the lattice: A ˜ μ ( k l + N μ μ ^ ) = A ˜ μ ( k l ) . Under the selection of the gauge potential, F ˜ 12 ( k l ) becomes:

F ˜ 12 ( k l ) = Δ 1 A ˜ 2 ( k l ) − Δ 2 A ˜ 1 ( k l ) + 2 π i n 12 ( k l ) (7)

where Δ μ can be expressed as: Δ μ f ( k l ) = f ( k l + μ ^ ) − f ( k l ) and n 12 ( k l ) is an integer valued field. So we can get the lattice Chern number:

c ˜ n = ∑ l n 12 ( k l ) (8)