Non-Abelian Zero Modes in Fractional Quantum Hall-Superconductor Heterostructure

TL;DR

This work demonstrates a concrete route to realize non-Abelian zero modes in Abelian topological phases by exploiting anyonic symmetries as extrinsic defects at the boundary of a FQH–superconductor heterostructure. Using a $K$-matrix Chern–Simons framework with $K= ext{diag}(4k,1,-1)$ and a folding boundary construction, it identifies charge-conjugation, fermion parity-flip, and their composite as universal anyonic symmetries for odd $k$, yielding parafermionic zero modes with quantum dimensions $ ext{QD}_{cc}= ext{√}{6}$, Majorana $ ext{QD}_{fpf}= ext{√}{2}$, and $ ext{QD}_{com}= ext{√}{3}$ for $k=3$. The authors construct explicit zero-mode operators at domain walls and derive corresponding effective Hamiltonians, showing robust non-Abelian statistics and predicting Josephson-current periodicities of $12 ext{π}$, $4 ext{π}$, and $6 ext{π}$ that serve as experimental signatures. The framework generalizes to arbitrary odd $k$, linking defect quantum dimensions to Josephson-phase periodicities and offering a scalable path to detect non-Abelian excitations in Abelian topological orders.

Abstract

We discuss the emergence of non-Abelian zero modes from twist defects in Abelian topological phases. We consider a setup built from a fractional quantum Hall (or a fractional Chern insulator)-superconductor heterostructure, which effectively induces a phase transition, leading to a topological phase endowed with new anyonic symmetries, and accordingly supporting distinct types of zero modes at fixed filling. These defects are modeled at the interface between two copies of the same heterostructure arranged side by side, which produces counterpropagating modes that can be gapped by interactions that realize the anyonic symmetries. We characterize the parafermions associated with each anyonic symmetry and discuss how their presence affect the periodicity of Josephson tunneling current.

Figures (4)

• Figure 1: Quantum Hall-Superconductor heterostructure, which is equivalent to a topological phase described by the $K$-matrix $K=\text{diag}(4k,1,-1)$.
• Figure 2: The gray cut lines connect a pair of defects, represented by the red triangles. A quasi-particle ${\bf l}$ crossing the cut line gets transmuted into $W{\bf l}$.
• Figure 3: The interface between the topological phases is displayed in gray. In the right, it is shown the folding process.
• Figure 4: Configuration with $2N=4$ alternating (black and gray) regions supporting zero modes, localized at the positions marked with red triangles. The gray part of the boundary corresponds to the region where anyons differing by the action of the symmetry $W$ (transmutation) are condensed, whereas the black region the condensed anyons are identical (backscattering). The lines represent the process of Fig. \ref{['DefectsGSD']}.