Introduction

The search for non-Abelian anyons has been a major focus of both theoretical and experimental efforts in the past decade, driven largely by their potential utility for topological quantum computation1. Historically, the first physical system predicted to harbour such exotic quasiparticles was a fractional quantum Hall state at 5/2 filling2. Chiral p+ip superconductors comprise a closely related platform for non-Abelian excitations3,4. Although the existence of intrinsic p+ip superconducting materials remains an open question, Fu and Kane5 provided a great insight by showing that one can mimic such physics by interfacing an ordinary s-wave superconductor with a topological insulator. Subsequent work distilled this architecture down to more conventional semiconductor-based setups6,7. In the above systems, non-Abelian anyons arise from Majorana zero modes bound to vortices3,4. Kitaev demonstrated that Majorana zero modes can also appear in a one-dimensional (1D) topological superconductor, and there have been several proposals applying the above ideas to 1D realizations8,9,10. Recent experiments have provided the first tantalizing signatures of Majorana modes in such devices11. Remarkably, even in networks of 1D wires Majorana zero modes generate non-Abelian statistics12,13,14.

The aforementioned systems feature non-Abelian anyons that are not computationally universal15. That is, braiding operations alone cannot approximate all unitary quantum gates. Several proposals to circumvent this problem exist16,17 but are either unrealistic or require non-topological operations, hence weakening the main advantage of using anyons—immunity from decoherence. Thus, whether realistic condensed matter systems can host other types of non-Abelian anyons remains an important question. Although there is some hope that certain quantum Hall plateaus may realize anyons with computationally universal braid statistics1,18, supporting experimental evidence has yet to appear.

Inspired by rapid advances in the pursuit of Majorana zero modes in quantum wires, here we ask whether one can engineer a quasi-1D platform featuring more exotic non-Abelian anyons. We stress that important work by Fidkowski and Kitaev19 appears to categorically rule out this possibility. Indeed, these authors proved that the only non-trivial zero modes supported by 1D electron systems—even with strong interactions—are Majorana fermions. We will show that this naively insurmountable restriction can be overcome by forming the 1D system out of the edge states of topologically non-trivial materials. Because the surrounding vacuum is then non-trivial, the properties of such systems are not constrained by Fidkowski and Kitaev’s classification.

Our study explores devices fabricated from superconductors and conventional quantum Hall states (for example, at 1/3 filling) that appear in many materials, including gallium arsenide (GaAs) quantum wells and graphene. We show that the quantum Hall edge states in such structures support parafermionic zero modes that reflect a further fractionalization of the electronic degrees of freedom beyond that found in Majorana platforms. These modes can be experimentally probed via Josephson and tunneling measurements. Using an efficient new formalism developed here, we demonstrate that the parafermions underpin richer non-Abelian statistics than Majoranas, rendering them better candidates for quantum computation. Specifically, braiding parafermionic zero modes allows entanglement of multiple quantum information registers (qudits). These results highlight the vast potential for designing novel phases of matter using conventional materials, and open numerous experimentally relevant directions in the search for exotic non-Abelian anyons.

Results

Parafermions from a clock model

To set the stage for our proposal, it is instructive to review a toy model discussed recently by Fendley20 that supports parafermionic zero modes akin to those that we will later realize in a physical electronic system. First, we recall the well-known connection between the transverse field Ising model and a spinless p-wave superconductor: the Hamiltonians for these physically distinct systems map to one another under a non-local Jordan–Wigner transformation that trades the bosonic spin variables for fermions. Although the Ising model exhibits only conventional paramagnetic and ferromagnetic phases, the corresponding superconducting system is far more exotic. Indeed, although the paramagnetic phase of the former maps to a trivial superconducting state in the latter, the ferromagnetic phase corresponds to a topological state featuring Majorana zero modes that generate non-Abelian statistics8,12,13,14.

Fendley’s insight (guided by the identification of parafermionic fields in the two-dimensional (2D) clock model21) is that one can access still more exotic zero modes by implementing a non-local transformation on the generalized N-state quantum clock model20,

Here, J≥0 couples neighboring clock states ferromagnetically, h≥0 is the transverse field, j labels sites of an L-site chain, and σj and τj are unitary operators defined on an N-state Hilbert space that satisfy . The only non-trivial commutation relation among these operators reads σjτj=τjσje2πi/N. When N=2, equation (1) reduces to the familiar transverse field Ising model, though the phases realized in this special case appear also for general N. For example, with J=0, h>0, there exists a unique paramagnetic ground state with τj=+1, whereas in the J>0, h=0 regime, an N-fold degenerate ferromagnetic ground state with σj=e2πiq/N emerges (q=1,…,N).

Consider now the non-local transformation20

The properties of σj and τj dictate that these new operators satisfy , and

The Methods section provides some additional useful relations. For N=2, the operators αj are self-Hermitian, anticommute and square to the identity—hence, they are Majorana fermions. At larger N, they define parafermions20,21,22.

In these variables, the Hamiltonian becomes

where h.c. is the hermitian conjugate. The phases of the clock model appear in this representation as follows. In the paramagnetic limit with J=0, parafermions dimerize as shown in Fig. 1a. Each dimer can be simultaneously diagonalized, leading to a unique, fully gapped ground state (see Methods). More interestingly, the ferromagnetic case h=0 produces the shifted dimerization shown in Fig. 1b. Again there is a bulk gap, but the ends of the chain now support unpaired zero modes α1 and α2L that encode the N-fold degeneracy of the clock model’s ferromagnetic phase admits N distinct eigenvalues that do not affect the energy). At N=2, the zero mode operators α1,2L form the unpaired Majoranas identified by Kitaev8; for N>2, they correspond to the parafermionic zero modes20 whose physical realization is the focus of this manuscript.

Figure 1: Parafermion chain.
figure 1

Schematic illustration of the parafermion chain Hamiltonian in equation (4) when (a) J=0 and (b) h=0. In the latter case, the ends of the chain support unpaired parafermionic zero modes that give rise to an N-fold ground-state degeneracy.

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Practical realization in 2D electron systems

The Majorana zero modes supported by equation (4) when N=2 are relatively easy to engineer, as the operators αj then satisfy familiar fermionic anticommutation relations (see equation (3)). This property allows one to rewrite the N=2 Hamiltonian in terms of ordinary fermion operators cj=(α2j−1+iα2j)/2, yielding a model for a spinless p-wave superconductor8 that can be realized in a variety of experimental architectures23,24. However, the non-standard commutation relations obeyed by αj with N>2 makes devising experimental realizations of parafermionic zero modes substantially more difficult. Our approach is inspired by the observation that commutation relations akin to those in equation (3) do occur among physical operators in a familiar system—a fractional quantum Hall edge. For Laughlin states at filling factor ν=1/m (m is an odd integer), the quasiparticle operators eiφ(x) that create right-moving charge e/m excitations at position x along the edge obey25

Such a system therefore provides a natural building block for a device supporting localized parafermion modes.

A single quantum Hall state is insufficient for this purpose, because its edge cannot be gapped out (and, hence, one cannot localize modes of any type at the edge). Thus, we consider the geometry of Fig. 2a, where two adjacent 2D electron gases (2DEGs), each at filling ν=1/m, produce a pair of counterpropagating edge states at their interface. These modes can acquire a gap via two different mechanisms: first, by electrons backscattering across the interface, and second, by electrons from each edge assembling into Cooper pairs. Such processes can be induced in numerous physical setups. We focus on the conceptually simplest example below but propose several viable alternatives in the Discussion.

Figure 2: Overview of proposed device.
figure 2

(a) Experimental architecture realizing parafermion zero modes. (b) Spatial profile for the electron pairing and backscattering amplitudes Δ(x) and induced by the superconductors and insulator in (a). (c) Schematic dependence of ϕ(x) on the phase difference δφsc between the superconductors in a in the m=3 case. As δφsc winds, the larger mismatch between ϕ(x) on the left and right increases the energy until δφsc=6π. Additional 2π cycles then untwist ϕ(x) until the ground state is again accessed at δφsc=12π. Remarkably, this implies that the Josephson current exhibits 12π periodicity in δφsc.

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To facilitate Cooper pair formation, we will assume that the 2DEGs exhibit opposite-sign g-factors, so that the red and blue edge states in Fig. 2a carry antiparallel spins. In practice, one can control the sign of the g-factor by various means26,27. The counterpropagating edge modes are then similar to those of 2D non-interacting topological insulators28,29,30 when m=1 and fractional topological insulators31 when m>1. This allows a pairing gap to open at the interface via the proximity effect with ordinary s-wave superconductors (green in Fig. 2a). The spin–orbit coupling necessary for a backscattering-induced gap can arise either directly from the 2DEGs or from the insulator (purple in Fig. 2a) that electrons traverse when crossing the interface.

Domain walls separating regions gapped by these different means are particularly interesting here. We explore their properties first by considering the domain structure of Fig. 2a, assuming for simplicity that electrons backscatter across the interface only via the central insulator and form Cooper pairs elsewhere. In terms of fields φR/L satisfying and , low-energy right- and left-moving e/m quasiparticles are created by operators that exhibit commutation relations of the form in equation (5). These properties ensure that the electron operators given by obey Fermi statistics. Below, it will prove useful to write φR/L=ϕ±θ; here, ρ=∂xθ/π is the electron density operator while

We model the interface with a Hamiltonian H=H0+H1, where25

describes gapless counterpropagating edge modes with speed v and H1 encodes couplings induced by the superconductors and spin–orbit-coupled insulator in Fig. 2a. For simplicity, we neglect Coulomb interactions between the edge states throughout; also, until specified otherwise, we set the superconducting phases to zero. In terms of electron operators, we have , where the profiles for the pairing and backscattering amplitudes Δ(x) and appear in Fig. 2b. One can alternatively express H1 using ϕ and θ variables as

Similar models have been studied previously in the m=1 case, where it is well-established that each domain wall binds a single Majorana fermion32,33. We now demonstrate that for m>1, the Hamiltonian supports precisely the localized parafermion zero modes that we seek.

Consider the limit where the induced pairing and backscattering terms are sufficiently strong that beneath each superconductor ϕ locks to one of the 2m minima of the first term in equation (8), whereas under the insulator, θ pins to the minima of the second. Using the coordinates specified in Fig. 2b, one can then write , and , where and are integer-valued operators. Importantly, equation (6) yields

whereas the other integer-valued operators commute. At low energies, one can focus on the intervals between xj and in Fig. 2b where Δ(x) and simultaneously vanish, allowing both ϕ and θ to fluctuate. These regions are governed by an effective Hamiltonian

subject to boundary conditions imposed by the adjacent gapped regions. As outlined in the Methods, the operators

commute with Heff, and thus represent zero modes bound to the domain walls in Fig. 2. Note that αj alters the charge by e/m; the first term simply involves , whereas the other terms similarly add charge e/m. Apart from these modes Heff also supports excitations with a finite-size gap of order . Upon projecting out these gapped excitations, the integral in equation (11) collapses to an unimportant constant (see Methods). One then obtains the elegant expression

that describe the action of the zero modes within the ground-state manifold.

From equations (9) and (12), it follows that

Thus, the zero modes bound to our domain walls indeed provide a physical realization of the parafermions supported by the model shown in Fig. 1 (with N=2m). In the present context, these modes produce a 2m-fold ground-state degeneracy corresponding to the distinct eigenvalues of One can gain intuition for this result by noting that because , tunnels between adjacent minima of the term induced by the spin–orbit-coupled insulator. The 2m distinct eigenstates of comprising the ground-state manifold can therefore equivalently be viewed as linear combinations of the 2m eigenstates of characterizing the central region.

In the limits studied so far, the degeneracy produced by α1 and α2 is exact. These modes will, however, inevitably hybridize in the more realistic situation where ϕ and θ are not perfectly pinned by the pairing and backscattering terms. Tunneling of e/m quasiparticles between the parafermions at lowest order produces a Hamiltonian

Because of the gap in the region between the parafermions, the coefficient A is exponentially small in and the spacing between domain walls. Thus, the ground states remain degenerate within exponential accuracy—enabling non-local storage of a 2m-state topological qudit.

As occurs with Majorana fermions8,5, the parafermions in our system mediate spectacular signatures in Josephson measurements. To illustrate this, let us revisit the setup of Fig. 2a in the limit where ϕ and θ are pinned by the superconductors and insulator. Suppose that after initializing the system into one of the resulting 2m ground states, the backscattering amplitude adiabatically decreases to zero, so that the parafermions strongly hybridize between the superconductors. We assume that the central region of Fig. 2a remains gapped by finite-size effects even when . Our goal is now to study the Josephson current flowing across the junction when the superconducting phase on the left side remains zero, whereas the phase on the right, δφsc, varies.

The Hamiltonian describing this configuration is , where H0 is again given in equation (7), whereas

Suppose first that when δφsc=0, we begin in a ground state with and . Upon smoothly increasing δφsc to 2π, our initial state evolves such that ϕ>=δφsc/2m to minimize the second term in equation (15). Crucially, this cycle raises the system’s energy because of the mismatch between ϕ< and ϕ>; the resulting twist of ϕ(x) between the superconductors costs energy due to the (∂xϕ)2 term in H0. The system returns to its original state only after δϕsc winds by 4πm, after which one obtains a value φ>=2π that is physically equivalent to our initial value of 0. Figure 2c shows the simplest non-trivial case with m=3. The Josephson current I0(δφsc)d〈H〉/dδφsc follows from the energy and, hence, also admits 4πm periodicity. (See Supplementary Methods for a more quantitative treatment.)

Interestingly, the current–phase relation depends sensitively on the initial state. Consider the more general situation where, before fusing the parafermions across the junction, we prepare the system into a ground-state characterized by , for some integer δn. One can generalize the above analysis to show that the current becomes Iδn(δφsc)=I0(δφsc+2πδn), which differentiates all physically distinct values of δn. Thus, the 4πm-periodic Josephson effect both provides a definitive signature of the parafermions in our setup and enables readout of the quantum information they store5.

Parafermion braiding

The results thus far extend straightforwardly to setups exhibiting arbitrarily many domain walls separating pairing- and backscattering-gapped regions. domain walls localize parafermion zero modes (numbered from left to right) that obey equation (13) and produce degenerate ground states. Next, we show that these modes generate non-Abelian statistics and allow one to perform a richer set of topologically protected operations on the ground-state manifold than are available in the Majorana case. As a first step, we introduce a practical recipe for transporting domain walls and a geometry that permits their meaningful exchange.

To mobilize the domain walls, we turn to the setup of Fig. 3a. Here the 2DEGs couple to a superconductor and spin–orbit-coupled insulator throughout the interface, whereas gates below control the potential along the edges. The Hamiltonian describing the interface is then H=H0+H1+Hμ, with H0,1 given by equations (7) and (8) but now with uniform and Δ. The induced potential μ(x) generates the third term, Hμ=−∫dxμ(x)∂xθ/π (recall that the electron density is ρ=∂xθ/π). We assume , so that when μ(x)=0, a gap arises from inter-edge tunneling. Suppose that starting from this regime we adjust the gates to raise μ uniformly. Shifting eliminates the resulting potential term Hμ but, crucially, also sends . The spatial oscillations render the backscattering term ineffective, so that the pairing energy Δ can then dominate. More physically, gating changes the momentum carried by low-energy quasiparticles, which affects inter-edge tunneling () but not Cooper pairing (ψRψL). The gates in Fig. 3a thereby allow one to dynamically control the nature of the gap in each region, and hence manipulate domain walls in real time.

Figure 3: Transport and braiding.
figure 3

(a) Setup allowing adiabatic domain wall transport via gating. (b) Sack-like geometry that permits braiding of domain walls between tunneling-gapped (purple) and pairing-gapped (green) regions. Parafermionic zero modes are denoted by αj. Clockwise exchange of domain walls binding α1 and α2 proceeds as outlined (c–f), where and describe the pinned ϕ and θ fields in a given region.

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Let us now deform the interface into the geometry sketched in Fig. 3b, where gates are tuned to localize parafermions α1,…,4. This setup permits exchange of any pair of domain walls shown. We will analyze the adiabatic clockwise braid of the domain walls binding α1 and α2 as outlined in Fig. 3c–f; other exchanges can be understood similarly. In the figures, and denote integer operators that describe the pinned ϕ and θ fields in a given region. Equation (6) implies that for j≥k, while for j<k, the operators commute. Because we are concerned here only with the ground-state sector, it suffices to express , , and similarly for α′1,2.

In the first step, (c)→(d), the right domain wall moves into the sack. We model this process with a Hamiltonian

The t′ term above represents charge e/m tunneling between the domain walls binding and in Fig. 3d. Similarly, t reflects e/m tunneling between the inner (red) edge states at the constriction in Fig. 3b, with β a system-dependent phase. Tunneling can also occur between the outer (blue) edge states. However, because the barrier between the 2DEGS does not support fractionalized excitations, only electrons can tunnel between these edges. Electron tunneling has no effect on our conclusions and shall henceforth be neglected. For details see Supplementary Methods. The configuration in Fig. 3c can be described by Hc→d with t=0 and t′≠0; and then hybridize while α1,2 comprise our initial parafermionic zero modes. We then pull and apart (thus reducing t′) while simultaneously increasing the coupling across the constriction (t>0), arriving at Fig. 3d. Throughout this process, α1 remains a zero mode. However, the zero mode initially given by α2 evolves nontrivially. In principle, one can track this evolution by explicitly solving Hc→d for general t and t′—a cumbersome task even for Majoranas12. Here we introduce a new approach that utilizes conserved quantities to obtain the result far more efficiently.

First, we observe that commutes with Hc→d for any choice of parameters and hence is conserved. This should be expected on physical grounds: were and to change throughout this process, zero modes well outside of the sack (for example, α1) would be affected as well. At the beginning of this step t′ pins . Consequently, χ projects onto the initial form of the zero mode of interest, χ→α2. Upon completing this step, t instead imposes the energy-minimizing condition

for some β-dependent integer k. Because χ is conserved, projecting onto the ground-state manifold yields the properly evolved zero mode: . Notice that this zero mode does not localize to the transported domain wall, except in the Majorana case where .

Next, we transport the domain wall binding α1 to arrive at Fig. 3e. This process can be similarly modeled by

where the parameters change from t≠0, t′′=0 at the start of this step to t=0, t′′≠0 at the end. In this case, the operator identified above remains a zero mode throughout, whereas now the zero mode initially given by α1 evolves. Proceeding as above, we define , which generically commutes with Hd→e and initially projects (using equation (17)) to α1. When the domain wall moves into the sack t′′ pins . The zero mode formerly described by α1 thus evolves to at the end of this step.

During the final step of the exchange, where the domain walls are transported to the configuration in Fig. 3f, both zero modes evolve trivially. Let U12 be the operator implementing the exchange. Comparing Fig. 3c to 3f, one finds that the parafermions transform as

Note that the analogue of parity in the Majorana case—is preserved here. We would now like to understand how the braid transforms the 2m degenerate ground states. Assuming α1,2 are the only zero modes, we denote these states by |q〉 where . Equation (19) implies that (up to an overall phase)

With additional domain walls, non-Abelian operations become available. For example, if U23 implements a clockwise exchange of the domain walls binding α2 and α3 in Fig. 3c, then , whereas . Remarkably, braiding parafermionic zero modes brings one closer to computational universality than braiding Majoranas. Indeed, we show in the Supplementary Methods that exchanging two pairs of parafermions produces a controlled phase gate CP=(U23U12U34U23)2 that can entangle the state of the pair α1,α2 with that of the pair α3,α4. Up to an overall phase, this operation yields

where q and q′ label the eigenvalues of and , respectively. Such an entangling gate is unavailable through braiding of Majorana modes, where m=1.

Discussion

In this paper, we introduced an experimental setup employing conventional quantum Hall edge states to localize exotic non-Abelian anyons. In the integer quantum Hall case (m=1), our proposal paves the way towards realizing Majorana wires in weakly spin–orbit-coupled systems such as GaAs. The fractional case (m>1) constitutes a much more important advance, as here our device provides a route to engineering networks of parafermion zero modes that can be moved along 1D channels. Although we focused on a setup with the virtue of conceptual clarity, numerous simplifications of the architecture introduced here are possible. The insulators in Figs 2 and 3 can be dispensed with entirely, as spin–orbit coupling can instead arise from the 2DEGs themselves. One also need not employ opposite-sign g-factors to generate Cooper pairing34; a proximity effect can still appear if spin is not conserved in either the 2DEGs or the parent s-wave superconductors35 or if one employs superconductors with a triplet component36. Graphene provides a particularly promising 2DEG, given the experimental evidence of superconducting proximity effect, even in the quantum Hall regime37. Yet another variation could employ bent quantum Hall systems pioneered by Grayson et al.38,39, which could provide an effective geometry for obtaining well-coupled Laughlin edge states.

We have shown that parafermion zero modes can be identified—and distinguished from Majoranas—via Josephson measurements. Tunneling experiments may provide a less definitive, though perhaps easier, probe. Indeed, if αj represents a parafermion zero mode then is a Majorana operator that can couple to electrons from a lead; thus, parafermions give rise to the same quantized zero bias anomaly as Majoranas40. Experimental verification may also be possible using resonant tunneling of e/m charges in a setting where a parafermion mode localizes inside a quantum Hall point contact (with tunneling charge detectable by, for example, noise measurements41).

In the future, it will be interesting to address whether one can create even more exotic non-Abelian anyons (for example, Fibonacci) using other fractional quantum Hall states beyond the Laughlin series considered here. Another worthwhile extension of our work would be to explore three-dimensional fractional topological insulators with proximity-induced superconductivity to generalize Fu and Kane’s proposal5 and search for realizations of novel models such as that of You and Wen42. And finally, an important question for future applications is whether one can utilize parafermions to more readily achieve universal quantum computation. To that end, we note that braiding of Majoranas must be supplemented by two additional gates to achieve universality1. The first is a single-qubit phase gate, which together with braid transformations completes the set of single-qubit unitary operations. The second is a multi-qubit entangling gate that may be obtained, for example, by measuring the topological charge of four Majoranas via interference. Although the first type of gate remains absent in the parafermion case, the CP braid described above eliminates the need for interferometry by generating entanglement through braiding alone. This gate, combined with arbitrary single-qudit operations, is universal for quantum computation. We leave to future work an investigation of how such single-qudit operations might be realistically performed.

Methods

Properties of parafermionic operators

Here we will enumerate several useful properties of parafermion operators that follow from the definitions provided in the main text. Because and , these operators are unitary and exhibit eigenvalues of the form e2πiq/N for integral q. The second equation together with the commutation relations in equation (3) imply that

Moving past , therefore, produces the opposite phase factor compared with moving past . Consequently, we obtain the following commutation relations,

which further imply that

so long as neither k nor l lie between i and j.

Equations (23) and (24) demonstrate that as claimed in our discussion of the 1D clock model, one can indeed simultaneously diagonalize each of the dimers sketched in Fig. 1, as well as the combination of zero-mode operators in Fig. 1b. To deduce the allowed eigenvalues, we note that one can show from the properties above that

which constrains the eigenvalues of to the form , where q is an integer. In the quantum clock model context, the eigenvalues of the relevant dimer operators can alternatively be found using the relations

that arise from the non-local transformation specified in equation (2). Equations (26) yield the same eigenvalue spectrum for the operators on the left-hand side as noted above, because τj and σj both exhibit non-degenerate eigenvalues e2πiq/N for q=1,…,N.

Finally, we consider the case where α1 and α2L represent zero modes and deduce the action of these operators on the ground-state manifold. Let |q〉 be a ground state satisfying . Using the parafermion commutation relations one can show that

for either j=1 or 2L. This equation implies that and , where the proportionality constants have unit magnitude. One can always fix the relative phases of the ground states such that

With this convention, α2L then acts as follows:

Solution for localized parafermion zero modes

Here we provide a detailed derivation of the localized parafermion zero-mode operators quoted in the main text (equation (11)). Consider again the static domain structure of Fig. 2a. As in the main text, we will assume that the Cooper pairing and inter-edge tunnelling terms induced at the interface are sufficiently strong that ϕ(x) is pinned beneath the superconductors, whereas θ(x) is pinned beneath the spin–orbit-coupled insulator. In the black regions of width in Fig. 2a, however, both fields can fluctuate because the pairing and tunnelling terms simultaneously vanish there. Our objective is to now understand the low-energy properties of these regions, which will eventually lead us to the parafermionic zero-mode operators of interest.

For clarity, we examine each domain wall separately, beginning with the one on the left. At low energies, this domain wall is governed by an effective Hamiltonian

Minimizing the energy of the adjacent gapped regions requires that these fields also satisfy boundary conditions and for integer-valued operators and (recall equation (8)). Note that and commute because according to equation (6).

Equation (30) can be diagonalized by expanding the ϕ and θ fields as

where and ak correspond to conventional bosonic operators satisfying . This decomposition simultaneously preserves the commutation relations among ϕ(x) and θ(x′) and encodes the boundary conditions specified above. Inserting equation (31) into the Hamiltonian yields

Thus, we see that the ak bosons exhibit a finite-size gap inversely proportional to . This does not, however, necessarily imply that the domain wall admits only gapped excitations. We will now demonstrate that, because the operators and both commute with the Hamiltonian, the domain wall supports zero modes that can be constructed from local operators present in our edge theory.

To do so, it is convenient to work with chiral fields ϕR/L instead of ϕ and θ. From equation (31), we have

which yields the useful relations

One can employ equations (32) and (34) to show, with the aid of various commutator identities and some algebra, that

As the right-hand side is a total derivative, the above equation integrates to

Finally, equations (35) and (37) allow one to demonstrate that the operator

commutes with the Hamiltonian and therefore represents a zero mode of the system.

Several points are worth emphasizing here. First, α1 is constructed purely from local e/m quasiparticle operators , and thus represents a physical zero mode bound to the domain wall. To see this explicitly, observe that and always appear in α1 via and . Second, the expression for the zero mode quoted above is not unique; one can always multiply α1 by allowed operators (such as H or that also commute with the Hamiltonian. This freedom—which is a generic feature of zero-mode operators—is, however, inconsequential for our purposes. We are concerned here only with the physics of the ground-state manifold and in operators that cycle the system among the various ground states. For this purpose, the form of α1 above suffices. Third, note from equation (34) that the integrand in equation (38) involves only bosonic operators (and not or ). This makes the projection of α1 into the ground-state manifold with very simple; upon discarding an overall constant one obtains the result

quoted in the main text.

The right domain wall in Fig. 2a can be analyzed very similarly. Here the effective low-energy Hamiltonian is given by

where now the fields satisfy boundary conditions and . Because the left and right domain walls of Fig. 2a are bridged by a single spin–orbit-coupled insulator, and θ(x2) must be pinned to identical values. Thus, is the same integer-valued operator that we introduced above. However, ϕ(x1) and are pinned by different superconductors, which necessitates the introduction of distinct integer-valued operators . Importantly, in this geometry and no longer commute because equation (6) yields .

Given our new boundary conditions, the appropriate decomposition for ϕ and θ reads

with and as before. With this expansion, our low-energy Hamiltonian once again takes the form in equation (32). One can then follow the steps outlined above (taking care to enforce the non-trivial commutation relations between and ) to show that the right domain wall binds a zero mode described by an operator

Projecting onto the ground-state manifold yields, up to a constant,

Additional information

How to cite this article: Clarke, D.J. et al. Exotic non-Abelian anyons from conventional fractional quantum Hall states. Nat. Commun. 4:1348 doi: 10.1038/ncomms2340 (2013).