Ansatz for Majorana modes for a quantum walk in a finite wire

The topological phases in a one-dimensional quantum walk can be classified by the coin parameters. By solving for the general exact solutions of bound states in a one-dimensional quantum walk with boundaries specified by different coin parameters, we show that these bound states are Majorana modes with quasienergy $E=0,\pi$. These modes are qualitatively different for different boundary conditions used. For systems with two boundaries, such as finite wires, we find the same exact solution as the single boundary system if the boundary condition is antisymmetric. For a two-boundary system with symmetric boundary conditions in a finite wire, the interaction energy between two Majorana bound states can be computed as a function of the length of the wire and the coin parameter in the wire. Suggestions of observing these modes are provided.

Figure 1

The spatial profile of the coin parameter (top) and topological number (bottom) for the symmetric (a) and antisymmetric (b) configuration. The topological profiles for the symmetric and antisymmetric configuration have one and two jumps, respectively. In general, the symmetric configuration is defined by $\text{sgn}\left({\theta }_3\right)=\text{sgn}\left({\theta }_1\right)=-\text{sgn}\left({\theta }_2\right)$ and the antisymmetric configuration is defined by $\text{sgn}\left({\theta }_3\right)=-\text{sgn}\left({\theta }_1\right)=\text{sgn}\left({\theta }_2\right)$. There are two jumps in the signs of $\theta$ in the symmetric case (a), while there is only one jump in the antisymmetric case.

Figure 2

The probability distribution of boundary modes for three different configurations of a finite system with length $N=10$ and ${\theta }_2=-\pi /4$. The first and the second value in the legend is the coin parameter on the left and right end, respectively.

Figure 3

The trajectory of vectors $\mathbf{n}\left(k\right)$ (black circle), $\mathbf{A}\left(\theta \right)$ (blue long dash semi-circle) in (a) and $\mathbf{B}\left(\theta \right)$ (red short dash semicircle) in (b) on the unit sphere. The vector $\mathbf{A}\left(\theta \right)$ is perpendicular to the black curves and defines the winding direction of the vector $\mathbf{n}\left(k\right)$.