Fractional chiral superconductors

Two-dimensional $p_x+ip_y$ topological superconductors host gapless Majorana edge modes, as well as Majorana bound states at the core of $h/2e$ vortices. Here we construct a model realizing the fractional counterpart of this phase: a fractional chiral superconductor. Our model is composed of an array of coupled Rashba wires in the presence of strong interactions, Zeeman field, and proximity coupling to an $s$-wave superconductor. We define the filling factor as $\nu=l_{\text{so}}n/4$, where $n$ is the electronic density and $l_{\text{so}}$ is the spin-orbit length. Focusing on filling $\nu=1/m$, with $m$ being an odd integer, we obtain a tractable model which allows us to study the properties of the bulk and the edge. Using an $\epsilon$-expansion with $m=2+\epsilon$, we show that the bulk Hamiltonian is gapped and that the edge of the sample hosts a chiral $\mathbb{Z}_{2m}$ parafermion theory with central charge $c=\frac{2m-1}{m+1}$. The tunneling density of states associated with this edge theory exhibits an anomalous energy dependence of the form $\omega^{m-1}$. Additionally, we show that $\mathbb{Z}_{2m}$ parafermionic bound states reside at the cores of $h/2e$ vortices. Upon constructing an appropriate Josephson junction in our system, we find that the current-phase relation displays a $4\pi m$ periodicity, reflecting the underlying non-abelian excitations.