Topological states in normal and superconducting $p$-wave chains

TL;DR

This paper explores a 1D two-band fermionic model with antisymmetric hybridization, revealing its topological properties and similarities to p-wave superconductors, including Majorana-like edge states, through analytical and numerical methods.

Contribution

It demonstrates that a normal $sp$-chain can map onto a p-wave superconductor, showing topological features and Majorana-like states with conventional fermionic excitations.

Findings

Zero energy modes exist in the topological phase.

Edge states exhibit topological robustness and similar localization behavior.

The excitations are conventional fermions with protected topological properties.

Abstract

We study a two-band model of fermions in a 1d chain with an antisymmetric hybridization that breaks inversion symmetry. We find that for certain values of its parameters, the $sp$-chain maps formally into a $p$-wave superconducting chain, the archetypical 1d system exhibiting Majorana fermions. The eigenspectra, including the existence of zero energy modes in the topological phase, agree for both models. The end states too share several similarities in both models, such as the behavior of the localization length, the non-trivial topological index and robustness to disorder. However, we show by mapping the $s$- and $p$- fermions to two copies of Majoranas, that the excitations in the ends of a finite $sp$ chain are indeed conventional fermions though endowed with protected topological properties. Our results are obtained by a scattering approach in a semi-infinite chain with an edge defect treated within the $T$-matrix approximation. We augment the analytical results with exact numerical diagonalization that allow us to extend our results to arbitrary parameters and also to disordered systems.