# Topologically nontrivial Andreev bound states

## TL;DR

This paper predicts the existence of topologically nontrivial Andreev bound states in one-dimensional superconductors, characterized by a new particle-hole Chern number, distinct from Majorana states, due to a synthetic two-dimensional topological space.

## Contribution

The authors introduce a novel topological invariant, the particle-hole Chern number, to describe nontrivial Andreev bound states in 1D superconductors, expanding the understanding of topological phases.

## Key findings

- Topologically nontrivial Andreev bound states can coexist with Majorana states.
- These states are characterized by a particle-hole Chern number in a synthetic 2D space.
- Nontrivial Andreev states have distinct spectral signatures and are topologically different from Majorana states.

## Abstract

Andreev bound states are low energy excitations appearing below the particle-hole gap of superconductors, and are expected to be topologically trivial. Here, we report the theoretical prediction of topologically $nontrivial$ Andreev bound states in one-dimensional superconductors. These states correspond to another topological invariant defined in a synthetic two-dimensional space, the particle-hole Chern number, which we construct in analogy to the spin Chern number in quantum spin Hall systems. Nontrivial Andreev bound states have distinct features and are topologically nonequivalent to Majorana bound states. Yet, they can coexist in the same system, have similar spectral signatures, and materialize with the concomitant opening of the particle-hole gap. The coexistence of Majorana and nontrivial Andreev bound state is the direct consequence of "double dimensionality", i.e., the dimensional embedding of the one-dimensional system in a synthetic two-dimensional space, which allows the definition of two distinct topological invariants ($\mathbb{Z}_2$ and $\mathbb{Z}$) in different dimensionalities.