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

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, characterize 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.