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

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.

arXiv comment14 pages, 2 figures

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