Superconducting Correlations Out of Repulsive Interactions on a Fractional Quantum Hall Edge

We consider a fractional quantum Hall bilayer system with an interface between quantum Hall states of filling fractions $({\nu }_{\mathrm{top}},{\nu }_{\text{bottom}})=(1,1)$ and $(1/3,2)$, motivated by a recent approach to engineering artificial edges [Y. Ronen et al., Nat. Phys. 14, 411 (2018)]. We show that random tunneling and strong repulsive interactions within one of the layers will drive the system to a stable fixed point with two counterpropagating charge modes which have attractive interactions. As a result, slowly decaying correlations on the edge become predominantly superconducting. We discuss the resulting observable effects and derive general requirements for electron attraction in Abelian quantum Hall states. The broader interest in fractional quantum Hall edge with quasi-long-range superconducting order lies in the prospects of hosting exotic anyonic boundary excitations, which may serve as a platform for topological quantum computation.

Figure 1

Pictorial description of the main result. On the left, we show the configuration of the bare edge modes. The configuration of edge modes can be experimentally realized in a bilayer structure; see Fig. 3. There is strong Coulomb interaction, $U_{1/3,-1}$, as well as random tunneling, $\xi (x)$, between the modes ${\varphi }_{1/3},{\varphi }_{-1}$. The third mode ${\varphi }_{-1}$ is weakly coupled with repulsive interaction to the mode ${\varphi }_{1/3}$. As temperature is lowered the random tunneling renormalizes the strongly coupled modes to a fixed point described by a neutral mode ${\varphi }_n$ and a charge mode ${\varphi }_{-2/3}$, shown on the right. Remarkably, the interaction between the charge modes ${\varphi }_{-2/3}$ and ${\varphi }_1$ can be attractive while the neutral mode is decoupled due to the disorder.

Figure 2

Flow of the interaction parameters $U_{-1,1},U_{1/3,1}$ that couple the mode ${\varphi }_1$ to the other modes. The bare interactions in the blue region, $3U_{-1,1}-U_{1/3,1}<0$, flow to a superconducting phase. There is a line of stable fixed points $U_{-1,1}=U_{1/3,1}$ [$U_{n1}=0$] (black). Near this line, the tree-level RG is accurate and flow is along lines of constant $3U_{-1,1}-U_{1/3,1}$ (dark blue line). Far from the line $U_{-1,1}=U_{1/3,1}$ one needs to account for terms beyond tree-level RG. The operator $3U_{-1,1}-U_{1/3,1}$ is marginally irrelevant and no longer remains constant during the flow. The flow diagram is calculated at the fine-tuned point where $U_{n,-2/3}=0$. We took $v_1=v_{-1}=2v_{1/3}$.

Figure 3

Experimental probes to measure attractive interactions in (a) shot noise and (b) Coulomb blockaded Hall droplet. This configuration can be experimentally realized in a bilayer structure at an interface of bulk fillings $({\nu }_{\mathrm{top}},{\nu }_{\text{bottom}})=(1,1)$ and $(1/3,2)$. We have drawn the bare modes at the interface in the same order as in Fig. 1 left. These modes are renormalized to an effective neutral and charge modes ($e$ and $2e/3$); see Fig. 1 right. The dashed edge mode lives on the bottom layer and solid ones on the top. We have not drawn the $\nu =1$ mode of the bottom layer that encircles the entire system since it is not relevant to the physics at the interface.