Exotic non-Abelian anyons from conventional fractional quantum Hall states

TL;DR

The paper proposes a realistic platform to realize exotic parafermions by interfacing conventional fractional quantum Hall edge states at filling ν=1/m with s-wave superconductors, producing domain walls that trap parafermion zero-modes. Through a clock-model mapping, it shows these zero-modes form a 2m-fold degenerate ground-state manifold and exhibit non-Abelian braiding with a richer gate set than Majorana systems, including a CP entangling operation. It predicts a 4πm periodic Josephson response as a hallmark of parafermions and outlines practical domain-wall braiding using gate-controlled geometries, such as a sack, to perform exchanges. The work also demonstrates how this framework yields a feasible Majorana platform in weakly spin-orbit-coupled materials and discusses experimental routes for detection and universal quantum computation potential.

Abstract

Non-Abelian anyons--particles whose exchange noncommutatively transforms a system's quantum state--are widely sought for the exotic fundamental physics they harbor as well as for quantum computing applications. There now exist numerous blueprints for stabilizing the simplest type of non-Abelian anyon, defects binding Majorana modes, by judiciously interfacing widely available materials. Following this line of attack, we introduce a device fabricated from conventional fractional quantum Hall states and s-wave superconductors that supports exotic non-Abelian anyons that bind `parafermions', which can be viewed as fractionalized Majorana fermions. We show that these modes can be experimentally identified (and distinguished from Majoranas) using Josephson measurements. We also provide a practical recipe for braiding parafermions and show that they give rise to non-Abelian statistics. Interestingly, braiding in our setup produces a richer set of topologically protected qubit operations when compared to the Majorana case. As a byproduct, we establish a new, experimentally realistic Majorana platform in weakly spin-orbit-coupled materials such as GaAs.

Figures (4)

• Figure 1: Schematic illustration of the parafermion chain Hamiltonian in Eq. (\ref{['Halpha']}) when (a) $J = 0$ and (b) $h = 0$. In the latter case the ends of the chain support 'unpaired' parafermion zero-modes that give rise to an $N$-fold ground-state degeneracy.
• Figure 2: (a) Experimental architecture realizing parafermion zero-modes. (b) Spatial profile for the pairing amplitude $\Delta(x)$ and tunneling strength $\mathcal{M}(x)$ induced by the superconductors and insulator in (a). (c) Schematic dependence of $\varphi(x)$ on the phase difference $\delta\phi_{sc}$ between the superconductors in (a), in the $m = 3$ case. As $\delta\phi_{sc}$ winds the larger mismatch between $\varphi(x)$ on the left and right increases the energy until $\delta\phi_{sc} = 6\pi$. Additional $2\pi$ cycles then 'untwist' $\varphi(x)$ until the ground state is again accessed at $\delta\phi_{sc} = 12\pi$. Remarkably, this implies that the Josepshon current exhibits $12\pi$ periodicity in $\delta\phi_{sc}$.
• Figure 3: (a) Setup allowing adiabatic domain wall transport via gating. (b) 'Sack' geometry that permits braiding of domain walls between tunneling-gapped (purple) and pairing-gapped (green) regions. Clockwise exchange of domain walls binding $\alpha_1$ and $\alpha_2$ proceeds as outlined in (c)-(f).
• Figure 4: Energy versus superconducting phase difference $\delta\phi_{sc}$ across the Josephson junction in the $m = 3$ case. The six curves shown correspond to the distinct values of $\hat{n}_\varphi^{(2)}$ characterizing the pinning of $\varphi$ under the right superconductor, assuming that $\varphi = 0$ beneath the left superconductor. Provided $\hat{n}_\varphi^{(2)}$ is conserved the energy and hence the current are both $12\pi$ periodic in $\delta\phi_{sc}$.