Quantized Transport of $ν= 2/3$ Fractional Quantum Hall Edge with Disordered Superconducting Proximity

TL;DR

This work reveals an infinite family of disorder-stabilized edge phases, SC$_N$, in the ν=2/3 fractional quantum Hall edge proximitized by a superconductor. By analyzing disordered SC couplings through the scaling dimensions $\Delta_{q,p}(g)$ and Pell-type solutions $p_N,q_N$, the authors show RG flows to fixed points where upstream and downstream modes decouple, yielding quantized downstream transport with $R_d=\frac{h}{2q_N^2 e^2}$ when $N\neq0$. They further compute nonlinear corrections to this quantized transport from RG-irrelevant Cooper-pair tunneling and from near-edge vortex tunneling, with universal exponents $\alpha$ controlled by the SC$_N$ fixed point; finite-temperature and spatial variations are also addressed. The theory extends to bilayer Halperin-(112) ν=2/3 states and potentially to fractional Chern insulators, providing a robust, disorder-enabled signature beyond the conventional Hall response and enriching the landscape of edge-state physics under superconducting proximity.

Abstract

Quantum Hall edge states in proximity to a superconductor (SC) usually acquire a non-quantized electron-to-hole conversion probability in transport, due to non-universal SC couplings and disorders. With counter-propagating modes, we show that the situation can be the opposite in the $ν=2/3$ fractional quantum Hall (FQH) edge states with SC proximity, where disordered SC-couplings can reconstruct the edge states into an infinite set of stable phases with quantized electron-to-hole conversion probability along a long edge. Each phase is dominated by a disordered SC-coupling that tunnels $\pm |q_N|$ Cooper pairs, which can take values $|q_N|=1, 4, 15$, etc. We predict that this gives rise to a quantized downstream resistance $R_d = h/(2q^2_Ne^2)$ in an FQH-SC junction, serving as a quantized electrical transport signature beyond the Hall conductance. Higher-order nonlinear transport due to irrelevant Cooper pair tunneling or vortex dissipation is further studied, which becomes dominant when the edge is in a normal phase. Our results apply to both the single-layer state (as a particle-hole conjugate of $ν=1/3$) and the bilayer Halperin-(112) state, revealing a rich landscape of disorder-stabilized phases in FQH edge states with SC proximity, and may as well apply to fractional Chern insulators recently observed at the same filling.

Figures (5)

• Figure 1: The junction of $\nu=2/3$ FQH edge with SC proximity. A current $I$ is injected at lead 1 and flows out passing the SC at lead 3, while lead 2 is floating. This measures the downstream resistance $R_d\equiv V_{23}/I$ in \ref{['eq: Rd', 'eq: Rd-non']}, where $V_{ij}=V_i-V_j$, and $V_j$ is the voltage of lead $j$ ($j=1,2,3$). Note that the SC is at voltage $V_3$.
• Figure 2: The black curves show the scaling dimension $\Delta_{q_N,p_N}$ of vertex operator $\mathcal{O}_{q_N,p_N}$ (label $(q_N,p_N)$ above each curve) with respect to parameter $g$ defined in Eq. \ref{['eq: def_g']}, which reaches the minimum $\Delta_{q_N,p_N}^{\min}=1$ at $g_N = g_0^{2N+1}$, with $g_0=(2+\sqrt{3})^2$. Each colored interval indicates a phase SC$_N$ (with SC$_0$ being the KFP phase), in which $\mathcal{O}_{q_N,p_N}$ is relevant and $g$ flows to the RG fixed point $g_N$.
• Figure 3: The lower edge of \ref{['fig:setup']} which constitutes a KFP-SC$_N$-KFP junction, with the SC-proximitized edge interval $0<x<L$ in the SC$_N$ phase, and the rest of the edge in the KFP phase. (a) The $N\neq 0$ case, where the SC proximity is strong and results in a quantized downstream resistance $R_d$ in \ref{['eq: Rd']}. (b) The $N=0$ case, where the SC proximity is inefficient and leads to a nonlinear $R_d$ in \ref{['eq: Rd-non']}.
• Figure 1.1: Scaling dimension $\Delta_{q,p}$ versus the edge-mode interaction parameter $g$. Each curve is labeled by its $(q,p)$, and we take $p>0$ to avoid redundancies. A black curve corresponds to $\mathcal{O}_{\pm q_N, \pm p_N}$, which is the possibly relevant disorder operator that induces the SC$_N$ fixed point. A blue curve corresponds to $\mathcal{O}_{\pm p_N, \pm 3q_N}$, which is the least irrelevant operator that dominates the nonlinear transport at the SC$_N$ fixed point (assuming no vortex tunneling). Red curves are shown for reference, which correspond to the next least irrelevant operator $\mathcal{O}_{\pm 2q_N, \pm 2p_N}$.
• Figure 3.1: Proposed experimental setup for studying quantized transport of the superconducting proximitized $\nu=2/3$ quantum Hall edges. (a) The three-terminal setup considered in the main text where the downstream resistance $R_d \equiv (V_2-V_3)/I$ is shown to be quantized when the proximitized edge is situated in the SC$_{N\neq 0}$ phase. (b,c) Four-terminal setups for probing the standard Hall resistance $R_H$ (with $R_H\equiv (V_3-V_4)/I$ for (b) and $R_H \equiv (V_1-V_2)/I$ for (c)). Irrespective of the nature of the proximitized edge, $R_H = 3h/(2e^2)$. The quantization of $R_d$ is thus a new and distinguished type of quantized transport uniquely arising from the interplay between superconductivity and fractional quantum Hall edge physics.