# Topological strings linking with quasi-particle exchange in superconducting Dirac semimetals

## TL;DR

This paper classifies vortices in topological superconductors based on Dirac semimetals, linking their properties to quasi-particle exchanges and fractional Josephson effects, revealing a $$ topological classification with interaction-dependent fractionalization.

## Contribution

It introduces a topological invariant for vortex linking in Dirac semimetal-based superconductors, connecting linking processes to fractional Josephson effects and quasi-particle exchanges, with implications for strongly interacting systems.

## Key findings

- Vortices are classified by a pair of numbers $(N_\u03a6,N)$.
- Linked vortices exchange Majorana zero-modes or $e/2$ quasi-particles.
- Strong interactions allow for additional fractionalization and exotic quasi-particle exchanges.

## Abstract

We demonstrate a topological classification of vortices in three dimensional time-reversal invariant topological superconductors based on superconducting Dirac semimetals with an s-wave superconducting order parameter by means of a pair of numbers $(N_\Phi,N)$, accounting how many units $N_\Phi$ of magnetic fluxes $hc/4e$ and how many $N$ chiral Majorana modes the vortex carries. From these quantities, we introduce a topological invariant which further classifies the properties of such vortices under linking processes. While such processes are known to be related to instanton processes in a field theoretic description, we demonstrate here that they are, in fact, also equivalent to the fractional Josephson effect on junctions based at the edges of quantum spin Hall systems. This allows one to consider microscopically the effects of interactions in the linking problem. We therefore demonstrate that associated to links between vortices, one has the exchange of quasi-particles, either Majorana zero-modes or $e/2$ quasi-particles, which allows for a topological classification of vortices in these systems, seen to be $\mathbb{Z}_8$ classified. While $N_\Phi$ and $N$ are shown to be both even or odd in the weakly-interacting limit, in the strongly interacting scenario one loosens this constraint. In this case, one may have further fractionalization possibilities for the vortices, whose excitations are described by $SO(3)_3$-like conformal field theories with quasi-particle exchanges of more exotic types.