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.
Categories: cond-mat.str-el