Phase Diagram and Edge States of Surface States of Topological Superconductors

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 pa-per, 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 rea-lizing topological quantum computation.


Introduction
In 1937, Ettore Majorana [1] proposed the existence of a type of fermion, known as Majorana fermion, which is its own antiparticle [2] [3]. Since the Majorana fermions were proposed, people have been looking for them [4] [5] [6]. The most representative one is the p-wave superconductor, which is influenced by the Moore-Read fractional quantum Hall (Pfaffian) state [7]. Because of the nontrivial topology of bulk Chern number, there are chiral Majorana fermion edge modes trapped in the vortices in the 2-dimensional p-wave superconductor [8]. Because of the non-abelian weaving of Majorana fermions, topological quantum computation can be realized based on them. For example, topological quantum computing can be achieved by using a non-abelian topological order containing the Ising The study of angular resolved spectrum shows that, due to the influence of volume superconductivity, a superconducting energy gap similar to the energy band gap will be generated on the topological surface states of a fully gap superconductor with temperature below T C , which is the topological superconducting surface states [21]. Topological surface state superconductivity is proposed by Fu-Kane [5], which is realized by inducing superconductivity in topological surface states of strong topological insulators by s-wave superconductivity proximity effect. In the absence of an external magnetic field, the Majorana zero model can be observed in the gap of FeTe 0.55 Se 0.45 and in the quantum anomalous vortices nucleated at Fe site [22] [23] [24]. Based on the above research progress, we will discuss below the chiral topological superconductor used to generate nonmagnetic two-dimensional time-inversion symmetry breaking in topological surface state superconductivity, and construct a superconducting topological surface state coupling model on the upper and lower surfaces of the film, as shown in Figure 1(a). Or construct a coupled model of two topological surface state superconductors with opposite surfaces as shown in Figure 1

Model Hamiltonian
First, we need to construct the Hamiltonian of the model, which is selected based on Nambu basis: (  )   T   †   †   †   †  1  1  2  2  1  1  2  2   ,  ,  ,  ,  , , , 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) [8] Hamiltonian is: (1) ( ) x h q and x ∆ are expressed as follows: We make a substitution such that σ χ τ σ χ τ µσ χ τ λσ χ τ σ χ τ By comparing Equations (4) and (5), we can get α , The phase diagram between its module value θ and phase Angle φ is shown in Figure 3. The influence of the upper and lower surfaces on the phase diagram is not considered here, so the parameters of the upper and lower surfaces are set to be the same here, and their sizes are set to be the same as the module of the complex number As can be seen from Figure Next, the corresponding phase diagram of 1 h in the θ φ − plane is drawn, as shown in Figure 4(a). Figure 4(a) is also symmetric about the coordinate axis. In the first quadrant of the figure, there are two regions, in which the Chern number of the region I is −1 and the Chern number of the region II is 0. We have given the phase diagram corresponding to 2 h , as shown in Figure 4(b).
The selection of parameters in this diagram is the same as that in the phase diagram drawn by 1 h . There are also two regions in the first quadrant of (b) in Figure 4, where the Chern number of the region I is +1, and the positive stale number of the region II is 0. By comparing Figure 3 and Figure 4, it can be 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.

Discussion on Edge States
Next, edge states and their state density diagrams are discussed. The BdG Hamiltonian can be written as follows in the case of Nambu basis: In the lattice model, Hamiltonian variables are replaced as follows: ( )  Figure 5(a), in this figure the red (blue) line stands for the spectrum of the chiral edge mode localized on the right (left) side of the sample (see Figure 6(a)). From Figure 5(a), it can be found that there is an intersection point within the small range of coordinate (0, 0), as shown in the red line and blue line in the figure, the two lines are the gapless edge states. The red line and the blue line will only travel along the edge, and the direction of the electrons will only travel in a certain direction. The selection of parameters in Figure 5(a) corresponds to the I region in Figure 3, and the Chern number of this region can be calculated as: N = −1, and its absolute value is 1, corresponding to edge states in Figure 5.
Next, the edge states of points corresponding to the region II in Figure 3 are plotted, as shown in Figure 5(b), in this figure the red and green (blue and orange) lines stand for the spectrum of the chiral edge mode localized on the left (right) side of the sample (see Figure 6(b)). By comparing Figure 5(a) and  II Chen number 0 is not 0 + 0 = 0, but rather the result of 1 and −1 in Figure   5(b) there will be two edge, one of the representative 1 h describes the edge of the Chern number of 1, the other is 2 h describes the edge of the corresponding Chern number is 1.
The edge state of the corresponding point in the region III in Figure 3 is shown in Figure 5(c), in this figure the red (blue) line stands for the spectrum of the chiral edge mode localized on the left (right) side of the sample (see Figure   6(c)). Contrast Figure 5(b) and Figure 5(c) can be found that the Chern num-World Journal of Condensed Matter Physics ber of the edge of graph into a group, which means in Figure 5(b) and Figure   5(c) between two groups of parameters will be after a topological phase transition point, it is because the topological transformation happened, so will lead to the edge of the area III in Figure 3 state is only a set, the topological transformation point is Figure 3 part within the solid line.
Finally, the edge state graph of the corresponding point in the region IV in Figure 3 is drawn, as shown in Figure 5(d). As can be seen in Figure 5

Density Distribution of States in Real Space of Wave Function
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 Figure 5  will be localized at both ends, which can indicate that the Chern number of the II region in Figure 3 is the structure of ( ) ( ) For two symmetrical Density of States diagrams, they correspond to the same pattern. be found that when the slope is positive, the density distribution of real space states corresponding to points on the edge states will be localized at opposite edges. This is because Figure 6(a) corresponds to the region with a Chern number of −1 in Figure 3, while Figure 6(c) corresponds to the region with a Chern number of 1 in Figure 3, indicating that the Chern number will affect the number of edge states. Although the positive and negative Chern number will not affect the number of edge states, it will affect the distribution of the energy state density of the wave function.

Conclusion
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