Charge $2e/3$ Superconductivity and Topological Degeneracies without Localized Zero Modes in Bilayer Fractional Quantum Hall States

It has been recently shown that non-Abelian defects with localized parafermion zero modes can arise in conventional Abelian fractional quantum Hall (FQH) states. Here we propose an alternate route to creating, manipulating, and measuring topologically protected degeneracies in bilayer FQH states coupled to superconductors, without the creation of localized parafermion zero modes. We focus mainly on electron-hole bilayers, with a $\pm 1/3$ Laughlin FQH state in each layer, with boundaries that are proximity coupled to a superconductor. We show that the superconductor induces charge $2e/3$ quasiparticle-pair condensation at each boundary of the FQH state, and that this leads to (i) topologically protected degeneracies that can be measured through charge sensing experiments and (ii) a fractional charge $2e/3$ ac Josephson effect. We demonstrate that an analog of non-Abelian braiding is possible, despite the absence of a localized zero mode. We discuss several practical advantages of this proposal over previous work, and also several generalizations.

Figure 1

(a) Cross section. Bilayer electron-hole FQH system at ${\nu }_{\text{total}}=-1/3+1/3$. Cooper pairs tunnel between the counterpropagating FQH edge modes and the superconductor (red), in the regime $d<{\xi }_{\mathrm{sc}}$, where ${\xi }_{\mathrm{sc}}$ is the superconducting coherence length. (b) 3D view of two regions filled with superconductor. (c) Top view. Read-out of topological states can be performed by measuring the charge, in units of $2e/3$, in the region enclosed by the loop $\alpha$. (d) Top and bottom gates (dark and light gray, respectively) to enhance quasiparticle tunneling along $\alpha$ or $\beta$.

Figure 2

(a) S-FQH-S geometry for fractional Josephson effect. (b)–(d) Evolution of $H(\tau )$ for $\tau =0$, 1, 2, 3. System includes three disconnected boundaries coupled to superconductors (red). Dashed lines indicate loops and lines of interest. Dark gray bars indicate active top gates, light gray bars indicate active bottom gates.