Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian quantum Hall states

TL;DR

This work proposes and analyzes fractional Majorana-like modes at the interfaces of superconducting and ferromagnetic regions along edges of fractional quantum Hall or fractional quantum spin Hall states. Using a bosonized edge theory and carefully constructed interface operators, the authors show a topologically protected ground-state degeneracy scaling as (2m)^{N−1} and derive explicit braiding unitary transformations. The resulting non-abelian statistics form a direct product of Ising anyons with an additional m-dimensional sector, enabling richer representations than Majorana-based systems but not universal quantum computation via braiding alone. They also discuss experimental signatures, such as a 4mπ fractional Josephson effect, and outline routes to realizing and detecting these fractionalized boundary anyons, as well as directions for future exploration of edge-based non-abelian phases.

Abstract

We study the non-abelian statistics characterizing systems where counter-propagating gapless modes on the edges of fractional quantum Hall states are gapped by proximity-coupling to superconductors and ferromagnets. The most transparent example is that of a fractional quantum spin Hall state, in which electrons of one spin direction occupy a fractional quantum Hall state of $ν= 1/m$, while electrons of the opposite spin occupy a similar state with $ν= -1/m$. However, we also propose other examples of such systems, which are easier to realize experimentally. We find that each interface between a region on the edge coupled to a superconductor and a region coupled to a ferromagnet corresponds to a non-abelian anyon of quantum dimension $\sqrt{2m}$. We calculate the unitary transformations that are associated with braiding of these anyons, and show that they are able to realize a richer set of non-abelian representations of the braid group than the set realized by non-abelian anyons based on Majorana fermions. We carry out this calculation both explicitly and by applying general considerations. Finally, we show that topological manipulations with these anyons cannot realize universal quantum computation.

Figures (7)

• Figure 1: Schematic setup. (a) A fractional topological insulator (FTI) realization. A FTI droplet with an odd filling factor $1/m$ is proximity coupled to ferromagnets (FM) and to superconductors (SC), which gap out its edge modes. The interfaces between the SC and FM segments on the edge of the FTI are marked by red stars. (b) A fractional quantum Hall (FQH) realization. A FQH droplet with filling factor $1/m$ is separated by a thin barrier into two pieces: an inner disk, and an outer annulus. On either side of the barrier, there are counter-propagating edge states, which are proximity coupled to superconductors and ferromagnets.
• Figure 2: Braiding process. (a) An FTI disk with six SC/FM segments. In stages I, II and III of the braiding process, quasi-particle tunneling (represented by blue solid lines) is turned on between the SC/FM interfaces. (b) Representation of the braiding procedure, involving interfaces 1, 2, 3 and 4. In the beginning of each stage, the two interfaces connected by a solid line are coupled; during that stage, the bond represented by a dashed line is adiabatically turned on, and simultaneously the solid bond is turned off. By the end of stage III, the system returns to the original configuration.
• Figure 3: Diagrammatic representation of the Yang-Baxter equations (Eq. \ref{['eq:YB1']}). Three interfaces 1,2,3 are braided in two distinct sequences. The Yang-Baxter equations state that the results of these two sequences of braiding operations are the same.
• Figure 4: Topological Spin. The process illustrated in (a) defines the TS, where the cup $\bigcup$ corresponds to creation of a particle -anti particle pair (two $X$'s fusing to $q=0$ or two charges with $q_1+q_2=0$), and the cap $\bigcap$ corresponds to projection on zero total charge. (b) TS of a composite object. When both lines are labelled by $X$ and fuse to charge $q$ , the phase acquired is $\theta_q$. The $4$-fold crossing, which is magnified in (c), is assumed to result in an exchange of two $q$'s. When the lines are labelled by $q_1$ and $q_2$, the phase acquired is $\theta_{q_1+q_2 \mod 2m}$. Importantly, the TS of a composite is equal to the phase accumulated in the process appearing in (d). This equality results from using the Yang-Baxter equation and the definition of the TS. (e) The evolution of the different segments in the process defining $\theta_X$. The green arrow corresponds to braiding, while the "M" corresponds to projection on zero charge. (f) The process defining $\theta_q$. The braids are ordered by color, green, blue and finally purple. The projections are on zero spin for the magnet segment intersecting the two SC segments, and zero total charge for these two segments.
• Figure 5: Braiding paths in Hamiltonian space. (a) Path $P_1$, for which we compute the braiding adiabatic evolution operator explicitly. (b) A different path $P_2$, whose Hamiltonians at the intermediate stages are assumed to be adiabatically connectable to those of $P_1$. $P_3$ is a path equivalent to $P_2$, in which each intermediate Hamiltonian of $P_2$ evolves to the corresponding Hamiltonian of $P_1$ and then back.