Identification of the Majorana edge modes in tight-binding systems based on the Krylov method

Abstract

Low dimensional structures in the non-trivial topological phase can host the in-gap Majorana bound states, identified experimentally as zero-bias peaks in differential conductance. Theoretical methods for studying Majorana modes are mostly based on the bulk-boundary correspondence or exact diagonalization of finite systems via, e.g., Bogoliubov–de Gennes formalism. In this paper, we develop an efficient method for identifying the Majorana in-gap (edge) states via looking for extreme eigenvalues of symmetric matrices. The presented approach is based on the Krylov method and allows for study the spatial profile of the modes as well as the spectrum of the system. The advantage of this method is the calculation cost, which shows linear dependence on the number of lattice sites. The latter problem may be solved for very large clusters of arbitrary shape/geometry. In order to demonstrate the efficiency of our approach, we study two- and three-dimensional clusters described by the Kitaev and Rashba models for which we determine the number of Majorana modes and calculate their spatial structures. Additionally, we discuss the impact of the system size on the physical properties of the topological phase of the magnetic nanoisland deposited on the superconducting surface. In this case, we have shown that the eigenvalues of the in-gap states depend on the length of the system edge.