Edge state behavior in a Su-Schrieffer-Heeger like model with periodically modulated hopping

TL;DR

This paper explores topological edge and corner states in a generalized SSH model with spatially periodic hopping modulations in one and two dimensions, revealing new in-gap states and the effects of modulation periodicity.

Contribution

It introduces a SSH-like model with periodic hopping modulations in 1D and 2D, uncovering novel in-gap states, zero-energy modes, and corner modes not seen in standard models.

Findings

New in-gap end states appear with longer hopping periodicity in 1D.

Zero energy Majorana modes become more accessible with increased periodicity.

2D model exhibits edge, corner, and satellite modes with complex distribution.

Abstract

Su-Schrieffer-Heeger (SSH) model is one of the simplest models to show topological end/edge states and the existence of Majorana fermions. Here we consider a SSH like model both in one and two dimensions where a nearest neighbor hopping features spatially periodic modulations. In the 1D chain, we witness appearance of new in-gap end states apart from a pair of Majorana zero modes (MZM) when the hopping periodicity go beyond two lattice spacings. The pair of MZMs, that appear in the topological regime, characterise the end modes each existing in either end of the chain. These, however, crossover to both-end end modes for small hopping modulation strength in a finite chain. Contrarily in a 2D SSH model with symmetric hopping that we consider, both non-zero and zero energy topological states appear in a finite square lattice even with a simple staggered hopping, though the zero energy modes disappear in a ribbon configuration. Apart from edge modes, the 2D system also features corner modes as well as modes with satellite peaks distributed non-randomly within the lattice. In both the dimensions, an increase in the periodicity of hopping modulation causes the zero energy Majorana modes to become available for either sign of the modulation. But interestingly with different periodicity for hopping modulations in the two directions, the zero energy modes in a 2D model become rarer and does not appear for all strength and sign of the modulation.