Braiding Statistics of Vortices in $2+1$d Topological Superconductors from Stacking

TL;DR

This paper investigates the braiding statistics of vortices in 2+1D topological superconductors, using stacking laws and anyon condensation to recover known classifications and braiding coefficients across different classes.

Contribution

It introduces a method to derive vortex braiding coefficients and topological classifications for various superconductor classes via stacking and anyon condensation.

Findings

Recovered the $bZ_{16}$ classification and braiding coefficients for all phases in Class D superconductors.

Derived the $bZ_2$ classification for time-reversal invariant Class DIII superconductors.

Validated the stacking law approach for understanding topological order and vortex braiding.

Abstract

Class D topological superconductors in $2+1$ dimensions are known to have a $\mathbb{Z}_{16}$ classification in the presence of interactions, with $16$ different topological orders underlying the $16$ distinct phases. By applying the fermionic stacking law, which involves anyon condensation, on the effective Hamiltonian describing the topological interaction of vortices in the $p+ip$ superconductor, which generates the $16$ other phases, we recover the braiding coefficients of vortices for all remaining phases as well as the $\mathbb{Z}_{16}$ group law. We also apply this stacking law to the time-reversal invariant Class DIII superconductors (which can themselves be obtained from stacking two Class D superconductors) and recover their $\mathbb{Z}_2$ classification.