# Braiding Flux-Tubes in Topological Quantum and Classical Lattice Models from Class-D

## TL;DR

This paper demonstrates how magnetic flux-tubes in a 2D class D superconductor lattice can host non-abelian zero modes that braid non-trivially, with potential applications in classical meta-materials.

## Contribution

It introduces a topological braiding mechanism for flux-tube zero modes in lattice models, linking quantum topological properties to classical systems.

## Key findings

- Zero-energy modes are stabilized by magnetic flux-tubes.
- Adiabatic displacement of flux-tubes results in topological, non-abelian braiding.
- Explicit numerical simulations confirm the topological nature of the braiding.

## Abstract

We use magnetic flux-tubes to stabilize zero-energy modes in a lattice realization of a 2-dimensional superconductor from class D of classification table of topological condensed matter systems. The zero modes are exchanged by slowly displacing the flux-tubes and an application of the adiabatic theorem demonstrates the geometric nature of the resulting unitary time-evolution operators. Furthermore, an explicit numerical evaluation reveals that the evolutions are in fact topological, hence supplying a representation of the braid group, which turns out to be non-abelian. This physical representation is further formalized using single-strand planar diagrams. Lastly, we discuss how these predictions can be implemented with and observed in classical meta-materials and how the standard Majorana representation of the braid group can be generated by measuring derived physical observables.