# Chiral to helical Majorana fermion transition in a $p$-wave superconductor

## TL;DR

This paper demonstrates a topological transition between chiral and helical Majorana edge modes in a superconductor, enabled by an additional symmetry that alters the topological invariant, with implications for quantum computing.

## Contribution

It introduces a model showing a direct transition between two topologically distinct Majorana phases via an extra symmetry, expanding understanding of topological superconductors.

## Key findings

- A simple analytical model illustrating the transition between chiral and helical Majorana modes.
- Identification of an additional symmetry that changes the topological classification.
- Explicit calculation of bulk invariants and edge states for the phases.

## Abstract

Chiral and helical Majorana edge modes are two archetypal gapless excitations of two-dimensional topological superconductors. They belong to superconductors from two different Altland-Zirnbauer symmetry classes characterized by $\mathbb{Z}$ and $\mathbb{Z}_2$ topological invariant respectively. It seems improbable to tune a pair of co-propagating chiral edge modes to counter-propagate without symmetry breaking. Here we show that such a direct topological transition is in fact possible, provided the system possesses an additional symmetry $\mathcal{O}$ which changes the bulk topological invariant to $\mathbb{Z}\oplus \mathbb{Z}$ type. A simple model describing the proximity structure of a Chern insulator and a $p_x$-wave superconductor is proposed and solved analytically to illustrate the transition between two topologically nontrivial phases. The weak pairing phase has two chiral Majorana edge modes, while the strong pairing phase is characterized by $\mathcal{O}$-graded Chern number and hosts a pair of counter-propagating Majorana fermions. The bulk topological invariants and edge theory are worked out in detail. Implications of these results to topological quantum computing based on Majorana fermions are discussed.