Quasi-Majorana modes in the $p$-wave Kitaev chains on a square lattice

TL;DR

This paper investigates the topological properties of a $p$-wave Kitaev chain on a square lattice, revealing the existence of quasi-Majorana modes, Dirac points, and topological edge states characteristic of a weak topological superconductor.

Contribution

It introduces a detailed analysis of quasi-Majorana modes and Dirac points in a $p$-wave Kitaev model on a square lattice, highlighting their topological nature and robustness.

Findings

Identification of topological gapless phases with edge modes

Presence of Dirac points with zero Chern number

Robustness of edge states against disorder

Abstract

The topological characteristics of the $p$-wave Kitaev chains on a square lattice with nearest-neighbor and next-nearest-neighbor inter-chains hopping and pairing are investigated. Besides gapless exact zero-energy modes, this model exhibits topological gapless phase hosting edge modes, which do not reside strictly at zero energy. However, these modes can be distinguished from the bulk states. These states are known as pseudo- or quasi-Majorana Modes (qMMs). The exploration of this system's bulk spectrum and Berry curvature reveals singularities and flux-carrying vortices within its Brillouin zone. These vortices indicate the presence of four-fold Dirac points arising from two-fold degenerate bands. Examining the Hamiltonian under a cylindrical geometry uncovers the edge properties, demonstrating the existence of topological edge modes. These modes are a direct topological consequence of the Dirac semimetal characteristics of the system. The system is analyzed under open boundary conditions to distinguish the multiple MZMs and qMMs. This analysis includes a study of the normalized site-dependent local density of states, which pinpoints the presence of localized edge states. Additionally, numerical evidence confirms the robustness of the edge modes against disorder perturbations. The emergence of topological edge states and Dirac points with zero Chern number indicates that this model is a weak topological superconductor.