Fractional topological superconductors with fractionalized Majorana fermions

In this paper, we introduce a two-dimensional fractional topological superconductor (FTSC) as a strongly correlated topological state which can be achieved by inducing superconductivity into an Abelian fractional quantum Hall state, through the proximity effect. When the proximity coupling is weak, the FTSC has the same topological order as its parent state and is thus Abelian. However, upon increasing the proximity coupling, the bulk gap of such an Abelian FTSC closes and reopens resulting in a new topological order: a non-Abelian FTSC. Using several arguments we will conjecture that the conformal field theory (CFT) that describes the edge state of the non-Abelian FTSC is $U(1)/Z_2$ orbifold theory and use this to write down the ground-state wave function. Further, we predict FTSC based on the Laughlin state at $\nu=1/m$ filling to host fractionalized Majorana zero modes bound to superconducting vortices. These zero modes are non-Abelian quasiparticles which is evident in their quantum dimension of $d_m=\sqrt{2m}$. Using the multi-quasi-particle wave function based on the edge CFT, we derive the projective braid matrix for the zero modes. Finally, the connection between the non-Abelian FTSCs and the $Z_{2m}$ rotor model with a similar topological order is illustrated.