Coulomb blockade in fractional topological superconductors

We study charge transport through a floating mesoscopic superconductor coupled to counterpropagating fractional quantum Hall edges at filling fraction $\nu =2/3$. We consider a superconducting island with finite charging energy and investigate its effect on transport through the device. We calculate conductance through such a system as a function of temperature and gate voltage applied to the superconducting island. We show that transport is strongly affected by the presence of parafermionic zero modes, leading at zero temperature to a zero-bias conductance quantized in units of $\nu e^2/h$ independent of the applied gate voltage.

Figure 1

Schematic diagram of the device layout. Two counterpropagating fQH edges with the filling fraction $\nu =2/3$ are coupled to a floating mesoscopic $s$-wave superconductor with a charging energy $E_C$. In the appropriate parameter regime (see below), two parafermionic zero modes are localized at $x_1$ and $x_2$. Here the distance $|x_2-x_1|$ should be much larger than superconducting coherence length. The low-energy transport is dominated by the coherent charge $e^{*}$ transmission through the superconducting island.

Figure 2

Schematic plot of the differential conductance $G_1\left({\mathcal{N}}_g\right)$ in the low-temperature $T\ll D_s,\mathrm{\Delta }$ (dash-dot blue line) and high-temperature $\mathrm{\Delta }\ll T\ll D_c,E_C$ (solid red line) limits. Here we assume symmetric contacts $r_1=r_2$.