Symmetry Protected Topological Phases and Majorana Mode in One-dimensional Quantum Walk with Boundary

TL;DR

This paper classifies topological phases in 1D quantum walks, identifies Majorana bound states at boundaries, and analyzes their properties and interactions, offering insights into boundary-dependent modes and potential experimental observations.

Contribution

It provides exact solutions for Majorana bound states in 1D quantum walks with boundaries, revealing boundary condition effects and interaction energies.

Findings

Majorana modes exist at boundaries with quasi-energy 0 and π

Boundary conditions influence the properties of bound states

Interaction energy between Majorana modes can be computed

Abstract

The topological phases in one-dimensional quantum walk can be classified by the coin parameters. By solving for the general exact solutions of bound states in one-dimensional quantum walk with boundaries specified by different coin parameters, we show that these bound states are Majorana modes with quasi-energy $E=0,\pi$. These modes are qualitatively different for different boundary conditions used. For two-boundary system with symmetric boundary conditions, the interaction energy between two Majorana bound states can be computed, as in the case of a finite wire. Suggestion of observing these modes are provided.