Line nodes and surface Majorana flat bands in static and kicked p-wave superconducting Harper model

TL;DR

This paper explores how adding p-wave superconducting pairing to static and kicked extended Harper models induces topological phases with Majorana flat bands and controllable Floquet Majorana modes, revealing new topological phenomena in driven systems.

Contribution

It introduces a comprehensive analysis of p-wave pairing effects in static and kicked Harper models, demonstrating the emergence of Majorana flat bands and Floquet Majorana modes with controllable properties.

Findings

Majorana flat bands appear at zero and π quasienergy under open boundary conditions.

System transitions between gapped and gapless phases with point or line nodes.

Generation of two controllable Floquet Majorana modes at quasienergy zero and π.

Abstract

We investigate the effect of introducing nearest-neighbor $p$-wave superconducting pairing to both the static and kicked extended Harper model with two periodic phase parameters acting as artificial dimensions to simulate three-dimensional systems. It is found that in both the static model and the kicked model, by varying the $p$-wave pairing order parameter, the system can switch between a fully gapped phase and a gapless phase with point nodes or line nodes. The topological property of both the static and kicked model is revealed by calculating corresponding topological invariants defined in the one-dimensional lattice dimension. Under open boundary conditions along the physical dimension, Majorana flat bands at energy zero (quasienergy zero and $\pi$) emerge in the static (kicked) model at the two-dimensional surface Brillouin zone. For certain values of pairing order parameter, (Floquet) Su-Schrieffer-Heeger-like edge modes appear in the form of arcs connecting different (Floquet) Majorana flat bands. Finally, we find that in the kicked model, it is possible to generate two controllable Floquet Majorana modes, one at quasienergy zero and the other at quasienergy $\pi$, at the same parameter values.