Universal Topological Quantum Computation from a Superconductor-Abelian Quantum Hall Heterostructure

PHYSICAL REVIEW X 4, 011036 (2014)

Roger S. K. Mong,1 David J. Clarke,1 Jason Alicea,1 Netanel H. Lindner,1,2 Paul Fendley,3 Chetan Nayak,4,5 Yuval Oreg,6 Ady Stern,6 Erez Berg,6 Kirill Shtengel,7,8 and Matthew P. A. Fisher5 1Department of Physics, California Institute of Technology, Pasadena, California 91125, USA 2Department of Physics, Technion, 32000 Haifa, Israel 3Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA 4Microsoft Research, Station Q, University of California, Santa Barbara, California 93106, USA 5Department of Physics, University of California, Santa Barbara, California 93106, USA 6Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel 7Department of Physics and Astronomy, University of California, Riverside, California 92521, USA 8Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA (Received 2 August 2013; published 12 March 2014) Non-Abelian anyons promise to reveal spectacular features of quantum mechanics that could ultimately provide the foundation for a decoherence-free quantum computer. A key breakthrough in the pursuit of these exotic particles originated from Read and Green’s observation that the Moore-Read quantum Hall state and a (relatively simple) two-dimensional p þ ip superconductor both support so-called Ising non-Abelian anyons. Here, we establish a similar correspondence between the Z3 Read-Rezayi quantum Hall state and a novel two-dimensional superconductor in which charge-2e Cooper pairs are built from fractionalized quasiparticles. In particular, both phases harbor Fibonacci anyons that—unlike Ising anyons—allow for universal topological quantum computation solely through braiding. Using a variant of Teo and Kane’s construction of non-Abelian phases from weakly coupled chains, we provide a blueprint for such a superconductor using Abelian quantum Hall states interlaced with an array of superconducting islands. Fibonacci anyons appear as neutral deconfined particles that lead to a twofold ground-state degeneracy on a torus. In contrast to a p þ ip superconductor, vortices do not yield additional particle types, yet depending on nonuniversal energetics can serve as a trap for Fibonacci anyons. These results imply that one can, in principle, combine well-understood and widely available phases of matter to realize non-Abelian anyons with universal braid statistics. Numerous future directions are discussed, including speculations on alternative realizations with fewer experimental requirements.

DOI: 10.1103/PhysRevX.4.011036 Subject Areas: Condensed Matter physics, Quantum Information, Strongly Correlated Materials

I. INTRODUCTION nature in principle permits, which by itself strongly motivates their pursuit. Non-Abelian anyons are further The emergence of anyons that exhibit richer exchange statistics than the constituent electrons and ions in a coveted, however, because they provide a route to fault- – material is among the most remarkable illustrations of tolerant topological quantum computation [14 18]. Here, ’ “more is different.” Such particles fall into two broad qubits are embedded in the system s ground states and, by categories: Abelian and non-Abelian. Interchanging virtue of non-Abelian statistics, manipulated through anyon Abelian anyons alters the system’s wave function by a exchanges. The nonlocality with which the information is phase eiθ that is intermediate between that acquired for stored and processed elegantly produces immunity against bosons and fermions [1,2]. Richer still are non-Abelian decoherence stemming from local environmental perturba- anyons, whose exchange rotates the system’s quantum state tions. One thereby sidesteps the principal bottleneck facing among a degenerate set of locally indistinguishable ground most quantum-computing approaches, but does so at the states produced by the anyons [3–13]. The latter variety expense of introducing a rather different challenge: iden- realizes the most exotic form of exchange statistics that tifying suitable platforms for non-Abelian excitations. The quantum Hall effect catalyzed numerous break- throughs in the search for anyons in physical systems Published by the American Physical Society under the terms of [18,19]. Quantum Hall states supporting fractionally the Creative Commons Attribution 3.0 License. Further distri- charged Abelian anyons are, by now, widely believed to bution of this work must maintain attribution to the author(s) and surface in a myriad of settings, including GaAs [20], the published article’s title, journal citation, and DOI. graphene [21,22], oxide interfaces [23,24], and CdTe

2160-3308=14=4(1)=011036(41) 011036-1 Published by the American Physical Society ROGER S. K. MONG et al. PHYS. REV. X 4, 011036 (2014)

[25], among others. Moreover, Moore and Read suggested physical electrons form Cooper pairs. In particular, both in 1991 that the quantum Hall regime could support systems exhibit a chiral Majorana edge mode at their non-Abelian anyons and constructed a candidate state—a boundary and support Ising non-Abelian anyons in the quantum Hall fluid in which composite fermions undergo bulk. Several important distinctions between these phases p þ ip pairing [26]. This phase supports chiral edge states do, nevertheless, persist: (i) Their edge structures are not consisting of a neutral Majorana sector coupled to a identical—a p þ ip superconductor lacks the chiral bosonic charge mode [27], along with Ising non-Abelian bosonic charge mode found in the Moore-Read state. anyons [28] carrying charge e=4 in the bulk [29–36].A (ii) Different classes of topological phenomena arise in variety of experiments support the onset of the Moore-Read each case. On one hand, a p þ ip superconductor realizes a state (or its particle-hole conjugate [37,38]) at filling factor topological superconducting phase with short-range entan- ν ¼ 5=2 in GaAs quantum wells [39–47]. It is important to glement; the Moore-Read state, on the other, exhibits true remark, however, that braiding Ising anyons does not topological order, long-range entanglement, and hence produce a gate set sufficient for universal topological nontrivial ground-state degeneracy on a torus. This impor- quantum computation. Thus, more exotic non-Abelian tant point closely relates to the next two distinctions. (iii) In phases that do not suffer from this shortcoming are highly contrast to the paired state of composite fermions, an desirable. electronic p þ ip superconductor is characterized by a Quantum Hall systems can, in principle, host nonlocal order parameter. Defects in that order parameter—i.e., Abelian anyons with universal braid statistics (i.e., that neutral h=2e vortices—bind Majorana zero modes allow one to approximate an arbitrary unitary gate with and, accordingly, constitute the Ising anyons akin to braiding alone). In this context, the Z3 Read-Rezayi state charge-e=4 quasiparticles in the Moore-Read state [48], which generalizes the pairing inherent in the Moore- [54,56]. (iv) Because of the energy cost associated with Read phase to clustering of triplets of electrons [49], local order-parameter variations, superconducting vortices constitutes the “holy grail.” Chiral edge states with a very are, strictly speaking, confined (unlike e=4 quasiparticles). interesting structure appear here: A charged boson sector Confinement does not imply inaccessibility of non-Abelian that transports electrical current (as in all quantum Hall anyons in this setting, since the “user” can always supply states) in this case coexists with a neutral sector that carries the energy necessary to separate vortices by arbitrary only energy and is described by the chiral part of Z3 distances. Non-Abelian braiding statistics is, however, parafermion conformal field theory. As a by-product realized only projectively [57,58] as a result—i.e., up to of this neutral sector, the bulk admits vaunted an overall phase that, for most purposes, is fortunately “Fibonacci” anyons—denoted as ε—that obey the fusion inessential. The existence of an order parameter may rule ε × ε ∼ 1 þ ε. This fusion rule implies that the low- actually prove advantageous, as experimental techniques energy Hilbert space for n Fibonacci particles with trivial for coupling to order parameters can provide practical total topological charge has a dimension given by the means of manipulating non-Abelian anyons in the (n − 1)th Fibonacci number. Consequently, the asymptotic laboratory. ’ dimension per particle, usually calledpffiffiffi the quantum dimen- Shortly after Read and Green s work, Kitaev showed sion, is the golden ratio φ ≡ ð1 þ 5Þ=2. Perhaps the most that a 1D spinless p-wave superconductor forms a closely remarkable feature of Fibonacci anyons is that they related topological superconducting phase [59] (which one allow for universal topological quantum computation in can view as a 2D p þ ip superconductor squashed along which a single gate—a counterclockwise exchange of two one dimension). Here, domain walls in the superconductor Fibonacci anyons—is sufficient to approximate any unitary bind Majorana zero modes and realize confined Ising transformation to within desired accuracy (up to an anyons whose exotic statistics can be meaningfully har- inconsequential overall phase). Such particles remain vested in wire networks [60–63]. Although such nontrivial elusive, although the Z3 Read-Rezayi state and its par- one-dimensional (1D) and two-dimensional (2D) super- ticle-hole conjugate [50] do provide plausible candidate conductors are unlikely to emerge from a material’s ground states for fillings ν ¼ 13=5 and 12=5. Intriguingly, a intrinsic dynamics, numerous blueprints now exist for plateau at the latter fraction has indeed been measured in engineering these phases in heterostructures fashioned GaAs, although little is presently known about the under- from ingredients such as topological insulators, semicon- lying phase; at ν ¼ 13=5, a well-formed plateau has so far ductors, and s-wave superconductors [64–70]. (See eluded observation [51–53]. Refs. [71,72] for recent reviews.) These proposals highlight Read and Green [54] laid the groundwork for the pursuit the vast potential that “ordinary” systems possess for of non-Abelian anyons outside of the quantum Hall effect designing novel phases of matter and have already inspired by demonstrating a profound correspondence between the a flurry of experiments. Studies of semiconducting wires Moore-Read state and a spinless 2D p þ ip superconductor interfaced with s-wave superconductors have proven par- [55]. Many properties that stem from composite-fermion ticularly fruitful, delivering numerous possible Majorana pairing indeed survive in the vastly different case where signatures [73–78].

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These preliminary successes motivate the question of trap Majorana modes that, owing to broken time-reversal whether one can—even in principle—design blueprints for symmetry, rather naturally couple to form a 2D spinless non-Abelian anyons with richer braid statistics compared to p þ ip superconducting phase within the fluid. In other the Ising case. Several recent works demonstrated that this words, imposing Cooper pairing provides a constructive is indeed possible using, somewhat counterintuitively, means of generating the non-Abelian physics of the Moore- Abelian quantum Hall states as a canvas for more exotic Read state starting from the comparatively trivial integer non-Abelian anyons [58,79–84]. (See also Refs. [85,86].) quantum Hall effect. This result is fully consistent with Most schemes involve forming a fractionalized “wire” out earlier studies of Refs. [92,93] that explored similar physics of counterpropagating Abelian quantum Hall edge states. from a complementary perspective. This wire can acquire a gap via competing mechanisms, One can intuitively anticipate richer behavior for a e.g., proximity-induced superconductivity or electronic superconducting array embedded in an Abelian fractional backscattering. Domain walls separating physically distinct quantum Hall state. In particular, since here charge-2e gapped regions bind Zn generalizations of Majorana Cooper pairs derive from conglomerates of multiple frac- zero modes [87,88] and consequently realize non- tionally charged quasiparticles, such a setup appears natural Abelian anyons of a more interesting variety than those for building in the clustering properties of Read-Rezayi in a 1D p-wave superconductor. Unfortunately, however, states. This more interesting case is addressed in the they too admit nonuniversal braid statistics, although remainder of the paper. We focus specifically on the achieving universal quantum computation requires fewer experimentally observed spin-unpolarized ν ¼ 2=3 state unprotected operations [79,89]. [94]—also known as the (112) state—for which super- In this paper, we advance this program one step further conducting islands bind Z3 generalizations of Majorana and pursue a similar strategy toward non-Abelian anyons modes. This phase is ideal for building in the physics of the with universal braid statistics. More precisely, our goal is to Z3 Read-Rezayi state, since coupling to an s-wave super- construct a new 2D superconductor that bears the same conductor can generate Cooper pairs built from three 2e=3 relation to the Z3 Read-Rezayi state as a spinless p þ ip charge- excitations [95]. [Note that various other superconductor bears to the Moore-Read state. With this quantum Hall phases, e.g., the bosonic (221) state, yield analogy in mind, it seems reasonable to demand that such a the same physics.] Hybridization of these modes is sub- phase satisfy the following basic properties. First, the stantially more difficult to analyze since the problem boundary should host a chiral Z3 parafermion edge mode cannot, in contrast to the integer case, be mapped to free but lack the Read-Rezayi state’s bosonic charge sector. fermions. Burrello et al. recently addressed a related setup Second, the bulk should exhibit essentially the same non- consisting of generalized Majorana modes coupled on a 2D Abelian content as the Read-Rezayi phase—particularly lattice, capturing Abelian phases including a generalization Fibonacci anyons. of the toric code [96]. We follow a different approach We show that one can nucleate a phase with precisely inspired by Teo and Kane’s method of obtaining non- these properties, not in free space but rather in the interior Abelian quantum Hall phases from stacks of weakly of a fractionalized medium. Our approach resembles that of coupled Luttinger liquids [97]. Although their specific Refs. [90,91], which demonstrated that hybridizing a finite coset construction is not applicable to our setup, a variant density of non-Abelian anyons produces new descendant of their scheme allows us to leverage theoretical technology — phases in the bulk of a parent non-Abelian liquid. In the for 1D systems i.e., bosonization and conformal field — most experimentally relevant cases of the Moore-Read state theory to controllably access the 2D phase diagram. and a 2D spinless p þ ip superconductor, these descend- With the goal of bootstrapping off of 1D physics, ants were found to be Abelian. We describe what amounts, Secs. III and IV develop the theory for a single chain of ν 2=3 in a sense, to an inverse of this result. The specific superconducting islands in a ¼ state. There we show, construction we follow relies on embedding an array of by relating the setup to a three-state quantum clock model, superconducting islands in an Abelian quantum Hall that this chain can be tuned to a critical point described by a system to proximity-induce Cooper pairing in the fluid. nonchiral Z3 parafermion conformal field theory. Section V When the islands remain well separated, each one binds then attacks the 2D limit coming from stacks of critical localized zero modes that collectively encode a macro- chains. (A related approach in which the islands are scopic ground-state degeneracy spanned by different “smeared out” is discussed in Sec. VII.) Most importantly, charge states on the superconductors. Hybridizing these we construct an interchain coupling that generates a gap in zero modes can then lift this degeneracy in favor of novel the bulk but leaves behind a gapless chiral Z3 parafermion non-Abelian 2D superconducting phases—including the sector at the boundary, thereby driving the system into a Read-Rezayi analogue that we seek. superconducting cousin of the Z3 Read-Rezayi state that As an illustrative warm-up, Sec. II explores the simplest we dub the “Fibonacci phase.” trial application corresponding to an integer quantum Hall The type of topological phenomena present here raises system at filling ν ¼ 1. Here, the superconducting islands an intriguing question. Should one view this state as

011036-3 ROGER S. K. MONG et al. PHYS. REV. X 4, 011036 (2014) analogous to a spinless p þ ip superconductor (which Moore-Read realizes short-ranged entanglement) or rather an intrinsic Majorana Majorana non-Abelian quantum Hall system (which exhibits true + charged boson topological order)? Interestingly, although superconductiv- Superconducting islands ity plays a key role microscopically for our construction, we argue that the Fibonacci phase is actually topologically ordered with somewhat “incidental” order-parameter physics. We indeed show that Fibonacci anyons appear as Z deconfined quantum particles, just like in the 3 Read- phase Rezayi state, leading to a twofold ground-state degeneracy (a) on a torus that is the hallmark of true topological order. Moreover, superconducting vortices do not actually lead to Spin-unpolarized Read-Rezayi new quasiparticle types, in sharp contrast to a p þ ip superconductor where vortices provide the source of Ising parafermion parafermion + charged boson anyons. In this sense, the fact that the Fibonacci phase Superconducting exhibits an order parameter is unimportant for its universal islands topological physics. Vortices can, however, serve as one mechanism for trapping Fibonacci anyons—depending on nonuniversal energetics—and thus might provide a route to manipulating the anyons in practice. Section VI provides a “Fibonacci phase” topological quantum field theory interpretation of the (b) Fibonacci phase that sheds light on the topological order present and establishes a connection between our con- FIG. 1. Abelian quantum Hall states interlaced with an array of struction and that of Refs. [90,91]. superconducting islands (left column) realize analogues of exotic Figure 1 summarizes our main results for the ν ¼ 1 and non-Abelian quantum Hall states (right column). The interface ν ¼ 2=3 architectures as well as their relation to “intrinsic” between the superconducting regions and surrounding Abelian non-Abelian quantum Hall states. (For a more complete quantum Hall fluids supports chiral modes similar to those on the technical summary, see the beginning of Sec. VIII.) On a right, but without the bosonic charge sector. (We suppress the edge states at the outer boundaries of the Abelian quantum Hall conceptual level, it is quite remarkable that a phase with states for simplicity.) Solid circles denote deconfined non- Fibonacci anyons can emerge in simple Abelian quantum Abelian excitations, while open circles connected by dashed Hall states upon breaking charge conservation by judi- lines represent confined h=2e superconducting vortices. Quasi- ciously coupling to ordinary superconductors. Of course, particle charges are also listed for the non-Abelian quantum Hall experimentally realizing the setup considered here will be states. In (a), σ particles represent Ising anyons, which in the very challenging— certainly more so than stabilizing Ising p þ ip phase on the left correspond to confined vortex excita- anyons. It is worth, however, providing an example that tions. In (b), ε is a Fibonacci anyon that exhibits universal braid puts this challenge into proper perspective. As shown in statistics. The superconducting Fibonacci phase is topologically ordered and supports deconfined ε particles—similar to the Read- Ref. [98], a 128-bit number can be factored in a fully fault- ’ ≈103 Rezayi state. Vortices in this nontrivial superconductor do not tolerant manner using Shor s algorithm with lead to new quasiparticle types, but can, in principle, trap Fibonacci anyons. In contrast, performing the same com- Fibonacci anyons and/or electrons. putation with Ising anyons would entail much greater overhead since the algorithm requires π=8 phase gates that would need to be performed nontopologically and then interesting to explore. The abundance of systems known distilled, e.g., according to Bravyi’s protocol [99]. For a to host Abelian fractional quantum Hall phases and the π=8 phase gate with 99% fidelity, the scheme analyzed in large potential payoff together provide strong motivation Ref. [98] requires ≈109 Ising anyons to factor a 128-bit for further pursuit of this avenue toward universal topo- number [100]. Thus, overcoming the nontrivial fabrication logical quantum computation. challenges involved could prove enormously beneficial for p ip quantum-information applications. In this regard, inspired II. TRIAL APPLICATION: þ by recent progress in Majorana-based systems, we are SUPERCONDUCTIVITY FROM THE INTEGER optimistic that it should similarly be possible to distill the QUANTUM HALL EFFECT architecture we propose to alleviate many of the practical The first proposal for germinating Ising anyons in an difficulties toward realizing Fibonacci anyons. Section VIII integer quantum Hall system was introduced by Qi, proposes several possible simplifications—including alter- Hughes, and Zhang [92]; these authors showed that in nate setups that do not require superconductivity—along the vicinity of a plateau transition, proximity-induced with numerous other future directions that would be Cooper pairing effectively generates spinless p þ ip

011036-4 UNIVERSAL TOPOLOGICAL QUANTUM COMPUTATION ... PHYS. REV. X 4, 011036 (2014) superconductivity in the fluid. In this section, we will establish a similar link between these very different phases from a viewpoint that illustrates, in a simplified setting, the basic philosophy espoused later in our pursuit of a Read- Rezayi-like superconductor that supports Fibonacci anyons. Specifically, here we investigate weakly coupled critical 1D superconducting regions embedded in a ν ¼ 1 quantum Hall system, following the spirit of Ref. [97] (see also Ref. [101]). This quasi-1D approach gives one a convenient window from which to access various states present in the phase diagram—including a spinless 2D p þ ip superconductor analogous to the Moore-Read state [54]. There are, of course, experimentally simpler ways of designing superconductors supporting Ising anyons, but we hope that this discussion is nonetheless instructive and interesting. Two complementary approaches will be pur- (a) sued as preliminaries for our later treatment of the fractional quantum Hall case.

A. Uniform-trench construction Consider first the setup in Fig. 2(a), wherein a ν ¼ 1 quantum Hall system contains a series of trenches (labeled by y ¼ 1; …;N) filled with some long-range-ordered superconducting material. As the figure indicates, the boundary of each trench supports spatially separated right- or left-moving integer quantum Hall edge states described by operators fR=LðyÞ. We assume that adjacent counterpropagating edge modes hybridize and are therefore generically unstable, due either to ordinary electron back- (b) scattering or Cooper pairing mediated by the supercon- p ip ductors [102]. Let the Hamiltonian governing these edge FIG. 2. (a) Setup used to nucleate a þ superconducting state with Ising anyons inside of a ν ¼ 1 quantum Hall fluid. The modes be H ¼ H þ δH þ H⊥. Here, KE arrows indicate integer quantum Hall edge states. Uniform N XN Z superconductors fill each of the trenches shown. The edge H −ivf† y ∂ f y ivf† y ∂ f y (1) states opposite a given trench can hybridize either through KE ¼ ½ Rð Þ x Rð Þþ Lð Þ x Lð Þ t Δ y 1 x electron backscattering or Cooper pairing mediated by the ¼ intervening superconductor; both processes favor gapping the edge modes, but in competing ways. Adjacent trenches are captures the kinetic energy for right and left movers, with x assumed to couple weakly via electron tunneling t⊥. With t⊥ ¼ a coordinate along the trenches (which we usually leave 0 and t ¼ Δ, each trench resides at a critical point at which the implicit in operators throughout this section). The second adjacent quantum Hall edge states evolve into counterpropagat- δH term includes electron-tunneling and Cooper-pairing ing Majorana modes. Turning on t⊥ then mixes these modes in perturbations acting separately within each trench: such a way that “unpaired” chiral Majorana edge states survive at p ip Z the boundary, thus triggering a þ phase. (b) Phase diagram XN for the weakly coupled trenches near criticality. States that † δH ¼ ½−tfRðyÞfLðyÞþΔfRðyÞfLðyÞþH:c:; (2) smoothly connect to the limit of decoupled chains are labeled x y¼1 “trivial”; see the text for a more detailed description of their properties. where t>0 and Δ > 0 denote the tunneling and pairing strengths. Finally, H⊥ incorporates electron tunneling Hereafter, we assume jt⊥j ≪ t, Δ, corresponding to the between neighboring trenches with amplitude t⊥, limit of weakly coupled trenches. It is then legitimate Z H δH XN−1 to first treat KE þ , which is equivalent to the † N H⊥ ¼ −t⊥ ½fLðyÞfRðy þ 1ÞþH:c:: (3) Hamiltonian for independent copies of quantum spin x y¼1 Hall edge states with backscattering generated by a magnetic field and proximity-induced pairing [65].Asin Figure 2(a) illustrates all of the above processes. the quantum spin Hall problem, the t and Δ perturbations

011036-5 ROGER S. K. MONG et al. PHYS. REV. X 4, 011036 (2014) favor physically distinct gapped phases that cannot be of Majorana modes at the left and right ends of the trenches smoothly connected without crossing a phase transition. that form a dispersing band due to small t⊥. We also refer For Δ >t, each trench realizes a 1D topological super- to the resulting 2D superconductor as trivial since it conductor with Majorana zero modes bound to its end smoothly connects to the decoupled-chain limit. (This points, while for Δ 0 Cooper pairing Δ produces a chain calculation reveals that the Majorana operator for domain

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Here ξðx − xjÞ denotes the decay length for the Majorana mode and is given either by v=t or v=Δ, depending on the sign of x − xj. The 2D array of zero modes present in this limit underlies a macroscopic ground-state degeneracy, since one can combine each pair of Majoranas into an ordinary fermion that can be vacated or filled at no energy cost. Next, imagine shrinking the width of the tunneling- and pairing-gapped regions, as well as the spacing between trenches, such that domain walls couple appreciably. Our objective here is to investigate how the resulting hybridization among nearby Majorana modes resolves the massive degeneracy present in our starting configuration. Focusing again on the weakly-coupled-chain limit, we first incorporate hybridization within each trench. The (a) simplest intrachain perturbation consistent with the symmetries of the problem tunnels right- and left-moving electrons between neighboring domain walls and reads [108]

1 XN X H λ −if† x ;y f x ;y intra ¼ 4 j½ Rð j Þ Rð jþ1 Þ y¼1 j if† x ;y f x ;y : : : (8) þ Lð j Þ Lð jþ1 ÞþH c

[This Hamiltonian encodes a discrete version of the kinetic energy in Eq. (1).] The x coordinate in the argument of fR=L, usually left implicit, has been explicitly displayed since it is now crucial. We define the real couplings appearing above as (b) λj ≡ λΔ for j even and λj ≡ λt for j odd. Physically, λΔ and λt respectively arise from coupling adjacent pairing- and FIG. 3. (a) Variation on the setup of Fig. 2(a) that also supports tunneling-gapped regions [see Fig. 3(a)] and thus clearly a p þ ip superconducting state with Ising anyons. Here, a ν ¼ 1 need not be identical. We assume, however, that λΔ, λt ≥ 0. quantum Hall system hosts spatially modulated trenches whose H edge states are gapped in an alternating fashion by backscattering According to Eq. (7), projection of intra into the low- t and Cooper pairing Δ. When the trenches decouple and the energy manifold spanned by the Majorana operators is gapped regions are “large,” each domain wall binds a Majorana achieved (up to an unimportant overall constant that we will zero mode. Electron hopping across the domains hybridizes the neglect) by replacing chain of Majorana modes in each trench through couplings λΔ λ j and t shown above. These couplings favor competing gapped fRðxj;yÞ → γjðyÞ;fLðxj;yÞ → ið−1Þ γjðyÞ: (9) phases, and when λΔ ¼ λt, each chain realizes a critical point with — counterpropagating gapless Majorana modes in the bulk similar This projection yields the following effective Hamiltonian to the uniform-trench setup of Fig. 2(a). Turning on weak 0 0 for the decoupled trenches: coupling t⊥ðj − j Þ between domain walls j and j in adjacent trenches then generically drives the system into a p þ ip phase XN X (or a p − ip state with opposite chirality). (b) Phase diagram for H → −i ½λtγ2j−1ðyÞγ2jðyÞþλΔγ2jðyÞγ2j 1ðyÞ; the 2D array of coupled Majorana modes near criticality. Here, λ⊥ intra þ 0 y¼1 j and λ⊥ represent interchain couplings between gapless Majorana 0 fermions at the critical point, which follow from t⊥ðj − j Þ (10) according to Eq. (15). which is equivalent to N independent Kitaev chains [59]. As written above, λΔ and λt favor distinct dimerization wall j at position xj in trench y takes the form (up to patterns for the Majorana operators that cannot be smoothly normalization) connected without closing the bulk gap. Alternatively, one can view the problem in more physical terms by Z implementing a basis change to ordinary fermions γ y ∝ e−jx−xjj=ξðx−xjÞ f y − i −1 jf y : : : jð Þ ½ Rð Þ ð Þ Lð ÞþH c cjðyÞ¼½γ2j−1ðyÞþiγ2jðyÞ=2. Equation (10) then x describes decoupled 1D p-wave-paired wires. If λΔ domi- (7) nates the superconducting wires reside in a gapped

011036-7 ROGER S. K. MONG et al. PHYS. REV. X 4, 011036 (2014) Z topological phase with protected Majorana end states, XN−1 H ∼ i λ γ y γ y 1 λ0 γ y γ y 1 : whereas if λt dominates a trivially gapped state emerges. ⊥ ½ ⊥ Lð Þ Rð þ Þþ ⊥ Rð Þ Lð þ Þ x The transition separating these 1D phases arises when y¼1 λΔ ¼ λt. Viewed in terms of superconductors this limit (14) corresponds to the situation where the chemical potential for the cj fermions is fine-tuned to the bottom of the band, so The coupling constants here are defined as thatgaplessbulkexcitationsremainatzeromomentumdespite X X the p-wave pairing. As in the preceding subsection we will 0 j λ⊥ ∝ t⊥ðjÞ; λ⊥ ∝ ð−1Þ t⊥ðjÞ (15) concentrate on this transition point, since here even weak j j intertrench coupling (to which we turn shortly) can qualita- tively affect the physics. When λΔ ¼ λt one can solve either and, importantly, differ in magnitude unless fine-tuned. Eq. (10) directly, or the equivalent superconducting problem The full low-energy theory describing our weakly H H by diagonalizing the Hamiltonian in momentum space. This coupled, spatially modulated trenches is eff ¼ intraþ exercise shows that at criticality right- and left-moving H⊥, with the terms on the right side given in Eqs. (12) and γ y 0 Majorana fields R=Lð Þ form the relevant low-energy degrees (14). When λ⊥ ¼ 0, this effective Hamiltonian is essentially of freedom—precisely as in the uniform-trench construction identical to Eq. (6) [109]. The phase diagram thus mimics examined earlier. Moreover, these continuum fields relate to that of the uniform-trench case and can again be inferred the lattice Majorana operators via from considering extreme cases. When the mass term j m~ ∝ λΔ − λt dominates over all other couplings, we obtain γjðyÞ ∼ γRðyÞþð−1Þ γLðyÞ: (11) superconducting states that smoothly connect to the Using Eq. (11) to rewrite Eq. (10) and taking the continuum decoupled-chain limit; the cases λΔ < λt and λΔ > λt limit yields respectively correspond to the trivial and “trivial*” phases λ Z discussed in the previous subsection. If instead ⊥ domi- XN nates, then the interchain coupling gaps out all Majorana H ∼ −iv~γ y ∂ γ y iv~γ y ∂ γ y intra ½ Rð Þ x Rð Þþ Lð Þ x Lð Þ fields in the bulk but leaves a gapless right mover at the top y 1 x ¼ edge and a gapless left mover at the bottom edge. This þim~ γRðyÞγLðyÞ; (12) regime realizes the spinless p þ ip superconducting phase that supports Ising anyons. Finally, by examining Eq. (14), v~ where the velocity follows from the tunnelings in Eq. (10) we see that when λ0 provides the leading term, we simply m~ ∝ λ − λ ⊥ and Δ t reflects small deviations away from criti- obtain a spinless p − ip superconductor with gapless edge cality. Note that Eq. (12) exhibits an identical structure to the states moving in the opposite direction. All of these phases intrachain terms in Eq. (6), which were derived for spatially exhibit a bulk gap; the transitions between them occur uniform trenches. The appearance of common physics near 0 when jm~ j¼jλ⊥ − λ⊥j, at which this gap closes. Figure 3(b) criticality in the two setups is quite natural; indeed, in a coarse- illustrates the corresponding phase diagram. It is worth grained picture appropriate for the critical point the spatial stressing that when the trenches are each tuned to criticality modulations in the trenches are effectively blurred away. (so that m~ ¼ 0), interchain coupling generically drives the One can now readily restore weak coupling between 0 system to either the p þ ip or p − ip phase since λ⊥ − λ⊥ neighboring trenches. Consider the following intertrench vanishes only with fine-tuning. Hamiltonian: To summarize, we have shown in this section that XN−1 X depositing superconducting islands (either uniformly or H − t j−j0 f† x ;y f x ;y 1 : : ; ⊥ ¼ ⊥ð Þ Lð j Þ Rð j0 þ ÞþH c nonuniformly) within integer quantum Hall trenches allows y¼1 j;j0 one to access nontrivial 2D superconducting states support- (13) ing Ising anyons. This outcome emerges quite naturally from weak interchain perturbations when the individual which encodes generic electron hoppings from the bottom trenches are tuned to criticality, which can be traced to the of domain wall j in one trench to the top of domain wall j0 fact that time-reversal symmetry is absent and the carriers in the trench just below. We have assumed that the in the quantum Hall fluid derive from a single fermionic 0 tunneling strengths t⊥ðj − j Þ above are real and depend species. So far, the weakly-coupled chain approach was only on the spacing j − j0 between domain walls. These convenient but by no means necessary since this section hoppings should be reasonably short ranged as well; see dealt only with free fermions. One can readily verify, for Fig. 3(a) for examples of significant processes. Since we instance, that the Ising-anyon phases we captured survive are interested in weak interchain coupling near criticality, it well away from this regime and persist even in an isotropic is useful to filter out high-energy physics, employing system. The remainder of this paper treats analogous setups Eqs. (9) and (11) to project H⊥ onto the low-energy where the ν ¼ 1 state is replaced by a strongly correlated manifold: fractional quantum Hall fluid. Throughout, numerous

011036-8 UNIVERSAL TOPOLOGICAL QUANTUM COMPUTATION ... PHYS. REV. X 4, 011036 (2014) Y μ τ ; ν σ†σ parallels will arise with the simpler treatment described j ¼ k j ¼ j jþ1 (18) here. We should point out that in the fractional case, the k≤j weakly-coupled chain approach provides the only analyti- σ → μ cally tractable window currently at our disposal, although that satisfy the same relations as in Eq. (16) with j j τ → ν we similarly expect isotropic relatives of the physics we and j j, the Hamiltonian takes on an identical form, capture to exist there as well. X X H −h μ†μ : : − J ν† ν ; dual ¼ ð j jþ1 þ H c Þ ð j þ jÞ (19) j j III. OVERVIEW OF Z3 PARAFERMION CRITICALITY with h and J interchanged. Equation (17) additionally One useful way of viewing Sec. IIB is that we dissected a exhibits a number of other symmetries that play an ν 1 ¼ quantum Hall system to construct a nonlocal repre- important role in our analysis. Spatial symmetries include sentation of the transverse-field Ising model—i.e., a simple lattice translations Tx and parity P (which sends Majorana chain. In preparation for treating the more σj → σ−j and τj → τ−j). The model also preserves a Z3 theoretically challenging ν 2=3 fractionalized setup, here i2π=3 ¼ transformation (σj → e σj) and a corresponding dual we review an analogous Z3-invariant chain corresponding dual i2π=3 operation Z3 (μj → e μj). Finally, there exists a time- to the three-state quantum clock model. This clock model reversal symmetry T that squares to unity (σj → σj, realizes a critical point described by a Z3 parafermion † τj → τj ) and a charge-conjugation symmetry C that flips conformal field theory (CFT), which provides the building the sign of the Z3 charge carried by the clock-model blocks for the Read-Rezayi wave function and plays a † † operators (σj → σj , τj → τj ). central role in describing the edge modes of this state. Like the closely related transverse-field Ising model, the Studying the chain will enhance our understanding of the clock Hamiltonian supports two symmetry-distinct phases. symmetries, phase structure, and perturbations of this CFT. When J dominates, a ferromagnetic phase emerges with Furthermore, much of the groundwork necessary for our hσji ≠ 0, thus spontaneously breaking Z3; increasing h subsequent quantum Hall analysis will be developed here. drives a transition to a paramagnetic state that in dual The Z3 quantum clock model is comprised of a chain dual language yields hμji ≠ 0 and a broken Z . Hence, one of three-component “spins.” Here, we assume an infinite 3 can view σj as an order parameter and μj as a “disorder number of sites (to avoid subtleties with boundary con- parameter.” Duality implies that the phase transition occurs ditions) and define operators σj and τj that act nontrivially at the self-dual point J ¼ h, and indeed the exact solution on the three-dimensional Hilbert space capturing the spin shows that this point is critical [110]. The scaling limit of at site j. These operators satisfy a generalization of the the self-dual clock Hamiltonian is described by a Z3 Pauli-matrix algebra parafermion (or, equivalently, three-state Potts) CFT [111], σ3 τ3 ; σ† σ2; τ† τ2; σ τ ei2π=3τ σ ; (16) whose content we discuss further below. j ¼ j ¼ 1 j ¼ j j ¼ j j j ¼ j j We will describe in the next section a new physical route to this CFT. In particular, our approach uses ν ¼ 2=3 while all other commutators aside from the last equation Z σ ; τ σ ; σ τ ; τ 0 quantum Hall states to construct a chain of 3 generalized above are trivial: ½ j j0≠j¼½ j j0 ¼½ j j0 ¼ .It Majorana operators that arise from the clock model follows that σj and τj can point in three inequivalent “ ” 2π=3 via a Fradkin-Kadanoff transformation [112]. This directions separated by an angle of , similar to a clock transformation—which is analogous to the more familiar hand that takes on only three symmetric orientations. Jordan-Wigner mapping in the transverse-field Ising chain Noncommutation of these operators implies that τj “winds” — σ also lends useful intuition for the physical meaning of j and vice versa. In other words, each operator can be parafermion fields, as we will see. The Fradkin-Kadanoff represented by a matrix with eigenvalues 1, ei2π=3, and −i2π=3 transformation in the clock model allows for two closely e , but one cannot simultaneously diagonalize σj and related forms of these Z3 generalized Majorana operators: τj. The simplest quantum clock Hamiltonian bears a similar either structure to the transverse-field Ising model and reads X X i2π=3 αR;2j−1 ¼ σjμj−1; αR;2j ¼ e σjμj (20a) H −J σ†σ : : − h τ† τ ; ¼ ð j jþ1 þ H c Þ ð j þ jÞ (17) j j or J; h ≥ 0 † −i2π=3 † where we assume couplings . This 1D Hamiltonian αL;2j−1 ¼ σjμj−1; αL;2j ¼ e σjμj ; (20b) can be found by taking an anisotropic limit of the 2D classical three-state Potts model, and so the two share which differ only in the string of operators encoded in the essentially identical physical properties. disorder parameter μj. Note that when applying a Jordan- The quantum clock model in Eq. (17) exhibits the useful Wigner transformation to the Ising chain, there is no such property of nonlocal duality symmetry. Indeed, upon freedom since there the string is Hermitian. The above introducing dual operators operators satisfy

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3 † 2 αA;j ¼ 1; αA;j ¼ αA;j (21) to consider purely chiral primary fields. These fields exhibit nonlocal correlations; local operators are found by combin- for A ¼ R=L, similar to the clock operators from which ing left and right movers in a consistent way. they derive. Because of the strings, however, they exhibit When the conformal symmetry algebra is extended nonlocal commutation relations by a spin-three current into the so-called “W3 algebra” [111,113], the Z3 parafermion CFT possesses six right- ið2π=3Þsgnðj0−jÞ α α 0 e α 0 α ; moving primary fields. These consist of the identity field R;j R;j ¼ R;j R;j † IR σR σ −ið2π=3Þsgnðj0−jÞ , the chiral parts of the spin field and R, parafermion αL;jαL;j0 ¼ e αL;j0 αL;j: (22) † fields ψ R and ψ R, and the chiral part ϵR of the “thermal” operator. The left-moving sector contains an identical set of Equations (21) and (22) constitute the defining proper- fields, labeled by replacing R with L. The CFT analysis ties for the Z3 generalized Majorana operators that will — L R yields the exact scaling dimensions of these operators the appear frequently in this paper. By using the labels and , chiral spin fields each have dimension 1=15, the para- we have anticipated the identification of these operators fermions each have dimension 2=3, while ϵR=L has dimen- with left- and right-moving fields in the CFT. On the lattice, 2=5 α α sion . however, Rj and Lj are not independent, as one can Perturbing the critical Hamiltonian by the thermal readily verify that operator—which changes the ratio of J=h away from — α† α ei2π=3α† α ; criticality provides a field-theory description of the clock R;2jþ1 R;2j ¼ L;2jþ1 L;2j Hamiltonian’s gapped ferromagnetic and paramagnetic † † αR;2j−1αR;2j ¼ αL;2jαL;2j−1: (23) phases. Note that in the Ising case, the thermal operator is composed of chiral Majorana fields, which also form the Despite this redundancy, it is nevertheless very useful to analogue of the parafermions ψ R=L. The fact that here the consider both representations since αRj and αLj transform parafermions and thermal operator constitute independent into one another under parity P and time reversal T . fields allows for additional relevant perturbations, which in In terms of αRj, the clock Hamiltonian of Eq. (17) reads part underlies the interesting behavior we describe in this X paper. More precisely, perturbing the critical Hamiltonian H −J ei2π=3α† α : : instead by ψ Lψ R þ H:c: violates Z3 symmetry but still ¼ ð R;2jþ1 R;2j þ H c Þ j results in two degenerate ground states that are not X symmetry related [114,115]; see Sec. VA for further − h ei2π=3α† α : : : (24) ð R;2j R;2j−1 þ H c Þ discussion. The analogous property in our quantum Hall j setup is intimately related to the appearance of Fibonacci anyons. An equivalent form in terms of αL;j follows from exploiting Eqs. (23). The ferromagnetic and paramagnetic phases of All of the symmetries introduced earlier in the lattice model are manifested in the CFT. Particularly noteworthy the original clock model correspond here to distinct dimer dual are the Z3 and Z3 symmetries, whose existence is patterns for αR;j (or αL;j) favored by the J and h terms above. On a finite chain, the symmetry-related degeneracy actually more apparent in the CFT due to the independence Z of the left- and right-moving fields. The former trans- of the ferromagnetic phase is encoded through 3 zero ψ → ei2π=3ψ σ → ei2π=3σ modes bound to the ends of the system [87], similar to the formation sends A A and A A, where A ¼ L or R. (As usual, the conjugate fields acquire a phase Majorana end states in a Kitaev chain [59]. The dimeriza- −i2π=3 dual e instead.) The dual transformation Z3 similarly tion appropriate for the paramagnetic phase, by contrast, i2π=3 i2π=3 takes ψ R → e ψ R and σR → e σR but alters left supports no such edge zero modes, consistent with the −i2π=3 −i2π=3 Z movers via ψ L → e ψ L and σL → e σL. Under onset of a unique ground state. In this representation 3 ϵ ϵ parafermion criticality arising at J ¼ h corresponds to the either symmetry, the fields L and R remain invariant, limit where these competing dimerizations balance, leaving which is required in order for the Hamiltonian to preserve Z Zdual J h the system gapless. For the remainder of this section we both 3 and 3 for all couplings and . provide an overview of this well-understood critical point. The relation between the lattice operators and primary fields at the critical point provides valuable insight into the The Z3 parafermion CFT has central charge c ¼ 4=5 and is rational. One of the very useful properties of a rational physical content of the CFT. Reference [116] establishes CFT is that a finite set of operators—dubbed primary such a correspondence by appropriately matching the spin fields—characterizes the entire Hilbert space. That is, all and symmetry properties carried by a given microscopic states in the Hilbert space can be found by acting with the operator and the continuum fields. This prescription yields primary fields and the (possibly extended) conformal the following familiar expansions for the lattice order and symmetry generators on the ground state. With appropriate disorder parameters: boundary conditions, the theory admits independent left- σ ∼ σ† σ† ; μ ∼ σ† σ ; and right-moving conformal symmetries, and so it is useful j R L þ j R L þ (25)

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† where the ellipses denote terms with subleading scaling with those for σR=L given by conjugation. A sum on the dimension. One can similarly express the thermal operator right-hand side indicates that two particular fields can fuse as to more than one type of field, signaling degeneracies. Finally, the chiral part of the thermal operator exhibits a σ†σ : : ∼ 1 − ϵ ϵ : “ ” j jþ1 þ H c R L þ (26) Fibonacci fusion rule ϵ ϵ ∼ I ϵ: Most crucial to us here is the expansion of the Z3 × þ (30) generalized Majorana operators [116], which will form the fundamental low-energy degrees of freedom in our Equation (30) is especially important: It underlies why the “ ” quantum Hall construction: decorated fractional quantum Hall setup to which we turn next yields Fibonacci anyons with universal non-Abelian j ϵ I αR;j ∼ aψ R þð−1Þ bσRϵL þ; (27a) statistics. (To be precise, we reserve and for CFT operators; the related Fibonacci anyon and trivial particle that appear in the forthcoming sections will be respectively j αL;j ∼ aψ L þð−1Þ bσLϵR þ; (27b) denoted as ε and 1.)

a b with and denoting real constants. [The phases in the IV. Z3 PARAFERMION CRITICALITY VIA definition of αR=L in Eqs. (20) and (20b) are paramount in ν ¼ 2=3 QUANTUM HALL STATES this lattice operator-CFT field correspondence.] The above equations endow clear meaning to the parafermion fields— Our goal now is to illustrate how one can engineer the they represent long-wavelength fluctuations in the gener- nonlocal representation of the clock model in Eq. (24), and Z alized Majorana operators at the critical point. Importantly, with it a critical point described by the 3 parafermion ν 2=3 however, these lattice operators also admit an oscillating CFT, using edge states of a spin-unpolarized ¼ component involving products of σ and ϵ fields, which in system in the so-called (112) state. As a primer, fact yield a slightly smaller scaling dimension than the Sec. IVA begins with an overview of the edge theory for parafermion fields. In Sec. V, we will use the link between this quantum Hall phase (see Ref. [117] for an early Z ultraviolet and infrared degrees of freedom encapsulated in analysis). Section IV B then constructs 3 generalized Eqs. (27a) and (27b) to controllably explore the phase Majorana zero modes from counterpropagating sets of ν 2=3 diagram for coupled critical chains. ¼ edge states, while Sec. IVC hybridizes these Z The physical meaning of the chiral primary fields is modes along a 1D chain to generate 3 parafermion further illuminated by their fusion algebra, which describes criticality. Results obtained here form the backbone of how the fields behave under operator products. This our coupled-chain analysis carried out in Sec. V. Note that property is constrained strongly but not entirely by com- much of the ensuing discussion applies also to the bosonic mutativity, associativity, and consistency with the Z3 (221) state with minor modifications; this bosonic setup symmetries. Any fusion with the identity of course is will be briefly addressed later in Secs. VD and VI. trivial. As a more enlightening example, two parafermion † fields obey the fusion rule ψ R × ψ R ∼ ψ R (and similarly for A. Edge theory ψ L). That is, taking the operator product of two parafer- Edge excitations at the boundary between a spin-unpo- mion fields contains something in the sector of the larized ν ¼ 2=3 droplet and the vacuum can be described ⃗ conjugate parafermion (i.e., the conjugate parafermion with a two-component field ϕðxÞ¼½ϕ↑ðxÞ; ϕ↓ðxÞ, where itself or some descendant field obtained by acting with x is a coordinate along the edge and the subscripts indicate the symmetry generators on the parafermion). This fusion physical electron spin. In our conventions, ϕαðxÞ is is natural to expect, given the properties in Eq. (21) compact on the interval ½0; 2πÞ; hence, physical operators ⃗⃗ exhibited by the lattice analogues αR=Lj. The complete involve either derivatives of ϕ⃗or take the form eil·ϕ for set of fusion rules involving ψ R or ψ L reads some integer vector l⃗. Commutation relations between these fields follow from an integer-valued K matrix that ψ I ∼ ψ; ψ ψ ∼ ψ †; ψ ψ † ∼ I; × × × encodes the charge and statistics for allowed quasiparticles ψ × σ† ∼ ϵ; ψ × σ ∼ σ†; ψ × ϵ ∼ σ; (28) in the theory [118]. For the case of interest here we have

ϕ x ; ϕ x0 iπ K−1 x − x0 iσy ; here and below, the fields in such expressions implicitly all ½ αð Þ βð Þ ¼ ½ð Þαβ sgnð Þþ αβ (31) † belong to either the L or R sectors. Fusion rules for ψ R=L simply follow by conjugation or by fusing again with ψ R=L. with The remaining rules for fusion with σR=L are 12 K ¼ : (32) σ × σ ∼ σ† þ ψ †; σ × ϵ ∼ σ þ ψ; σ × σ† ∼ I þ ϵ; (29) 21

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The term involving the Pauli matrix σy corresponds to a Spin-unpolarized Klein factor as discussed below. Since det K<0 the ν ¼ 2=3 edge supports counterpropagating modes; these can be viewed, roughly, as ν ¼ 1 and ν ¼ 1=3 modes running in opposite directions. In terms of the “charge vector” q⃗¼ð1; 1Þ, the total ⃗ electron density for the edge is q⃗· ∂xϕ=ð2πÞ. Since we are dealing with an unpolarized state, it is also useful to consider the density for electrons with a definite spin α ¼ ↑, ↓, which is given by FIG. 4. Spin-unpolarized ν ¼ 2=3 setup with a long, narrow ∂xϕα trench producing counterpropagating sets of edge states de- ρα ¼ : (33) ⃗ ⃗ 2π scribed by fields ϕ1 on the top and ϕ2 on the bottom. One way of gapping these modes is through electron backscattering Equations (31) and (33) allow one to identify across the interface—which essentially “sews up” the trench. A second gapping mechanism can arise if an s-wave superconduc- iKαβϕβ tor mediates spin-singlet Cooper pairing of electrons from the top ψ α ¼ e (34) and bottom sides of the trench, as illustrated above. These as spin-α electron operators. Indeed, these operators add processes lead to physically distinct gapped states that cannot be smoothly connected, resulting in the formation of Z3 gener- one unit of electric charge and satisfy appropriate anti- alized Majorana zero modes at domain walls separating the two. commutation relations. (Note that anticommutation between ψ ↑ and ψ ↓ requires the Klein factor introduced above.) One can further, with the aid of Eq. (33), define a same fields connected at the right end of the trench.) It Hamiltonian incorporating explicit density-density inter- follows that the electron operators for the top and bottom actions via sides of the trench are respectively Z 1 X ψ eiKαβϕ1β ; ψ eiKαβϕ2β : H ¼ ð∂xϕαÞVαβð∂xϕβÞþ; (35) 1α ¼ 2α ¼ (38) x 4π α;β¼↑;↓ Similarly to Eq. (35), one can express the Hamiltonian where Vαβ is a positive-definite matrix describing screened for the edge interface as Coulomb interactions and the ellipsis denotes all other Z 1 X allowed quasiparticle processes. H ∂ ϕ V ∂ ϕ δH: (39) ¼ 4π ð x aαÞ aα;bβð x bβÞþ These preliminary definitions allow us to readily treat the x α;β¼↑;↓ following more interesting setup. Suppose that one carves a;b¼1;2 out a long, narrow trench from the system as sketched in Of crucial importance here are the additional terms present Fig. 4, thus generating two identical (but oppositely δH oriented) sets of ν ¼ 2=3 edge states in close proximity in . Since the interface carries identical sets of counter- to each other. To describe this “doubled” edge structure, we propagating modes, it is always possible for perturbations ⃗ to gap out the edges entirely. We will invoke two physically employ fields ϕ1 ¼ðϕ1↑; ϕ1↓Þ for the top side of the trench ⃗ distinct gapping mechanisms, similar to our earlier ν ¼ 1 and ϕ2 ¼ðϕ2↑; ϕ2↓Þ for the bottom. The corresponding electron densities for spin α are defined as setup: (i) spin-conserving electron tunneling across the interface and (ii) spin-singlet Cooper pairing of electrons ∂ ϕ ∂ ϕ on opposite sides of the trench, mediated by an s-wave ρ x 1α ; ρ − x 2α ; 1α ¼ 2π 2α ¼ 2π (36) superconductor. These processes are schematically illustrated in Fig. 4 and lead to the following perturbations: while the commutation relations read Z † † δH ¼ −tðψ 1↑ψ 2↑ þ ψ 1↓ψ 2↓ þ H:c:Þ 0 −1 0 y x ½ϕ1αðxÞ; ϕ1βðx Þ ¼ iπ½ðK Þαβ sgnðx − x Þþiσαβ; Δ ψ ψ − ψ ψ : : ; 0 −1 0 y þ ð 1↑ 2↓ 1↓ 2↑ þ H c Þ (40) ½ϕ2αðxÞ; ϕ2βðx Þ ¼ iπ½−ðK Þαβ sgnðx − x Þþiσαβ; 0 −1 y ½ϕ1αðxÞ; ϕ2βðx Þ ¼ iπ½−ðK Þαβ þ iσαβ: (37) where t and Δ are the tunneling and pairing amplitudes. It is important to emphasize that in this setup tunneling and (The relative minus sign for the density on the bottom side pairing of fractional charges across the trench is not of the trench, along with the commutation relations above, possible—such processes are unphysical since the inter- ⃗ ⃗ can be understood by viewing ϕ1 and ϕ2 as essentially the vening region separating the top and bottom sides by

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ˆ construction supports only electronic excitations. Later, θσ ¼ πM; however, we will encounter edges separated by a ν ¼ 2=3 2π π π θ mˆ Mˆ − ; (44) quantum Hall fluid, and in such a geometry interedge ρ ¼ 3 þ 3 6 ðtunneling gapÞ fractional charge tunneling can arise. Before discussing the fate of the system in the presence to minimize the energy, thus fully gapping the charge and δH of the couplings in , it is useful to introduce a basis spin sectors. Note that both fields are simultaneously change to charge- and spin-sector fields: pinnable since θσ and θρ commute with each other. If the pairing term dominates, however, a gap arises from 1 θ ϕ ϕ − ϕ − ϕ ; pinning ρ ¼ 2 ð 1↑ þ 1↓ 2↑ 2↓Þ ˆ 1 θσ ¼ πM; ϕ ϕ ϕ ϕ ϕ ; ρ ¼ 2 ð 1↑ þ 1↓ þ 2↑ þ 2↓Þ 2π π π ϕ nˆ Mˆ ; (45) 1 ρ ¼ 3 þ 3 þ 6 ðpairing gapÞ θ ϕ − ϕ − ϕ ϕ ; σ ¼ 2 ð 1↑ 1↓ 2↑ þ 2↓Þ 1 where nˆ is another integer operator. Both fields are ϕ ϕ − ϕ ϕ − ϕ : (41) σ ¼ 2 ð 1↑ 1↓ þ 2↑ 2↓Þ again simultaneously pinnable, but note that Eqs. (44) and (45) cannot be simultaneously fulfilled in the same 0 region of space since θρ x ; ϕρ x ≠ 0. Consequently, the ρ ∂ θ =π S ∂ θ =π ½ ð Þ ð Þ Here, þ ¼ x ρ and þ ¼ x σ respectively denote tunneling and pairing terms compete with one another ρ the total edge electron density and spin density, while − ¼ [120]. The physics is directly analogous to the competing ∂ ϕ =π S ∂ ϕ =π x ρ and − ¼ x σ are respectively the difference in ferromagnetic and superconducting instabilities in a the electron density and spin density between the quantum spin Hall edge; there, domain walls separating top and bottom sides of the trench. Equations (32) imply regions gapped by these different means bind Majorana that the only nontrivial commutation relations among these zero modes [65]. Because of the fractionalized nature fields are of the ν ¼ 2=3 host system, in the present context domain walls generate more exotic zero modes—as in 2πi – — 0 0 Refs. [58,79 81,83,85,86,121,122] that will eventually ½θρðxÞ; ϕρðx Þ ¼ − Θðx − xÞ; 3 serve as our building blocks for a Z3 parafermion CFT. 0 0 ½θσðxÞ; ϕσðx Þ ¼ 2πiΘðx − xÞ; 0 B. Z3 zero modes ½ϕρðxÞ; ϕσðx Þ ¼ −2πi; (42) As an incremental step toward this goal, we would like to now capture these zero modes by studying an infinite array where Θ is the Heaviside step function. (Contrary to the of long domains alternately gapped by tunneling and first two lines, the third is nontrivial only because of Klein pairing, as displayed in Fig. 5 [123]; note the similarity factors.) to the integer quantum Hall setup analyzed in Sec. IIB.(For In this basis, δH becomes simply illuminating complementary perspectives on this problem, Z see the references cited at the end of the previous para- graph.) In each tunneling- and pairing-gapped segment, the δH ¼ ½4t cos θσ sinð3θρÞ − 4Δ cos θσ sinð3ϕρÞ: (43) x fields are pinned according to Eqs. (44) and (45), respectively. Since θσ is pinned everywhere, in the ground-state ˆ The scaling dimensions of the operators above depend on sector the integer operator M takes on a common value Mˆ the matrix Vaα;bβ in Eq. (39) specifying the edge density- throughout the trench. (Nonuniformity in requires θ density interactions. In the simplest case Vaα;bβ ¼ vδabδαβ, energetically costly twists in σ.) Conversely, the pinning both the tunneling and pairing terms have scaling dimen- of θρ and ϕρ is described by independent operators mˆ j and sion two and hence are marginal (to leading order). nˆ j in different domains—see Fig. 5 for our labeling Following Ref. [119] we have verified that upon tuning conventions. The commutation relations between the inte- Vaα;bβ away from this limit, t and Δ can be made ger operators follow from Eqs. (42), which yield simultaneously relevant. Hereafter, we assume that both 3 ij>j0 terms can drive an instability, either because they are 2π ½nˆ j; mˆ j0 ¼ (46) explicitly relevant or possess “order-one” bare coupling 0 j ≤ j0; constants. Suppose first that interedge tunneling dominates. In while all other commutators vanish. terms of integer-valued operators Mˆ and mˆ , this coupling The zero-mode operators of interest can be obtained ⃗ ⃗ ⃗ ⃗ pins from quasiparticle operators eiðl1·ϕ1þl2·ϕ2Þ acting inside of a

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ˆ ið2π=3Þðnˆ jþM−1Þ ið2π=3Þmˆ j−1 Spin-unpolarized αR;2j−1 ¼ e e ðtop edgeÞ; ˆ ið2π=3Þðnˆ jþMÞ ið2π=3Þmˆ j αR;2j ¼ e e ðtop edgeÞ;

ið2π=3Þðnˆ jþ1Þ −ið2π=3Þmˆ j−1 αL;2j−1 ¼ e e ðbottom edgeÞ;

ið2π=3Þnˆ j −ið2π=3Þmˆ j αL;2j ¼ e e ðbottom edgeÞ: Domain wall (49)

Spin-unpolarized Above, we denote whether a given zero-mode operator adds charge 2e=3 (mod 2e) to the top or bottom edge. The importance of the spatial separation between αRj and αLj evident here is hard to overstate and will prove exceedingly ν 2=3 FIG. 5. Schematic of a spin-unpolarized ¼ system valuable in the following section. Equation (46) implies that hosting a trench in which the edge modes are alternately gapped Z by electron backscattering t and Cooper pairing Δ. The integer the 3 zero-mode operatorsin our quantumHall setup satisfy precisely the properties in Eqs. (21–23) introduced in the operators mˆ i and nˆ i in each domain characterize the pinning of the charge-sector fields as specified in Eqs. (44) and (45). quantum clock-model context. Once again, αRj and αLj are Physically, mˆ i − mˆ i−1 quantifies the total charge (top plus not independent, but as we will see describing physical þ bottom) Qi on the intervening superconducting-gapped region, processes for coupled trenches in a simple way requires nˆ − nˆ while iþ1 i quantifies the charge difference (top minus retaining both representations because of their spatial − bottom) Qi on the intervening tunneling-gapped segment. The separation. Z remaining low-energy physics is captured by 3 generalizations The Z3 zero modes encode a ground-state degeneracy α of Majorana operators R=L;j bound to each domain wall labeled that admits a simple physical interpretation. First, we note Q as above. These operators cycle the values of i on the domains mˆ 2e=3 2e that gauge-invariant quantities involve differences in the j by adding charge (mod ) to the top and bottom trench nˆ edges, as illustrated in the figure. Charge-2e=3 tunneling between or j operators on differentR domains. Consider then the iπ x0 ρ x″ dx″ neighboring domain walls hybridizes these modes and can be þð Þ i θ x0 −θ x quantity Aðx − x0Þ¼e x ¼ e ½ ρð Þ ρð Þ, where described by a 1D Hamiltonian [Eq. (57)] intimately related to the ρ ∂ θ =π x x0 again þ ¼ x ρ denotes the total density. If and three-state quantum clock model. The critical point of this nˆ Hamiltonian, as in the clock-model context, is described by straddle a pairing-gapped domain in which j is pinned, Z3 parafermion conformal field theory. then Eq. (44) yields a ground-state projection

2π 2π † 0 i ðmˆ j−mˆ j−1Þ −i domain wall, simply by projecting into the ground-state PAðx − x ÞP ¼ e 3 ¼ e 3 αR;2j−1αR;2j: (50) manifold. To project nontrivially, the dependence on the iϕ field ϕσ must drop out since e σ creates a kink in θσ that Hence, costs energy. This condition is satisfied provided 2 þ Qj ≡ ðmˆ j − mˆ j−1Þ (51) l1↑ − l1↓ þ l2↑ − l2↓ ¼ 0. (47) 3

2e Projectionoftheremainingfieldsisachievedbyreplacingθσ, specifies the total charge (mod ) on the pairing-gapped θρ, and ϕρ by their pinned values on the adjacent domains. segment. A comparison with the more familiar case of The complete set of projected quasiparticle operators Majorana zero modes along a quantum spin Hall edge is ⃗ ⃗ ⃗ ⃗ obeying Eq. (47) can be generated by eiðl1·ϕ1þl2·ϕ2Þ with useful here. In that context, the Majoranas encode a twofold l1↑ ¼ l1↓ ¼ 1, l2↑ ¼ l2↓ ¼ 0 and l1↑ ¼ l1↓ ¼ 0, degeneracy between even- and odd-parity ground states of ⃗ l2↑ ¼ l2↓ ¼ 1. Crucially, these values of l1;2 correspond to a superconducting-gapped region of the edge. Here the charge-2e=3 quasiparticle operators acting on the top and physics is richer—a superconducting segment of the ν ¼ bottom edges of the trench, respectively. Suppose that P is 2=3 interface supports ground states with charge 0, 2=3,or 4=3 2e ρ ∂ ϕ =π the ground-state projector while xj denotes a coordinate (mod ). From the density difference − ¼ x ρ inside of domain wall j. We then explicitly get between the top and bottom edgesR of the trench, one can x0 ″ ″ iπ ρ−ðx Þdx i ϕ x0 −ϕ x B x − x0 e x e ½ ρð Þ ρð Þ i ϕ x ϕ x j similarly define ð Þ¼ ¼ . Pe ½ 1↑ð jÞþ 1↓ð jÞP ≡ −1 α ; 0 ð Þ Rj With x and x now straddling an mˆ j-pinned tunneling-

i½ϕ2↑ðxjÞþϕ2↓ðxjÞ j iπ=3 Pe P ≡ ð−1Þ e αLj; (48) gapped region, one obtains

i2π nˆ −nˆ −i2π † PB x − x0 P e 3 ð jþ1 jÞ e 3 α α : whereontherightsidewehaveinsertedphasefactorsforlater ð Þ ¼ ¼ R;2j R;2jþ1 (52) convenience and defined Z3 generalized Majorana zero- mode operators We thus see that

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− 2 Finally, we note a curious feature implicit in the zero Q ≡ ðnˆ j 1 − nˆ jÞ (53) j 3 þ modes and ground states: Although a ν ¼ 2=3 edge supports charge-e=3 excitations, they are evidently frozen represents the charge difference (again mod 2e) across out in the low-energy subspace in which we are working. the trench in a tunneling-gapped region, which can The doubling of the minimal charge arises because the also take on three distinct values. If desired,P one can spin sector is uniformly gapped throughout the trench. mˆ 3 Qþ e=3 use theseP definitions to express j ¼ 2 i≤j i and Charge- excitations must therefore come in opposite- 3 − spin pairs to circumvent the spin gap. As a corollary, one nˆ j ¼ 2 i 0. The first term reflects a fractional Josephson coupling between adjacent superconducting k 2j − 1 2j for ¼ or , while segments [59,79–81], mediated by charge-2e=3 tunneling

− − across the intervening tunneling-gapped region. This cou- iπðQj 2=3Þ iπQj e αR=L;k ¼ αR=L;ke (55) pling favors pinning mˆ j to uniform values in all super- þ conducting regions, resulting in Qj ¼ 0 throughout. for k ¼ 2j or 2j þ 1. (At other values of k the zero modes Similarly, the second (competing) term represents a “dual do not affect Qj .) Notice that αR;k and αL;k increment the fractional Josephson” [125–128] coupling favoring uni- − nˆ Q− 0 charge difference Qj in opposing directions because they form j in tunneling-gapped regions and hence j ¼ .In add quasiparticles to opposite sides of the trench. terms of generalized Majorana operators defined in One can now intuitively understand why two nontrivial Eq. (43), the effective Hamiltonian becomes R=L representations exist for the Z3 zero modes whereas X γ H −J ei2π=3α† α : : the Majorana operators j discussed in Sec. IIB are eff ¼ t ð R;2jþ1 R;2j þ H c Þ j uniquely defined, up to a sign. For concreteness, let us X work in a basis where the ground states are labeled by the − J ei2π=3α† α : : ; (57) þ Δ ð R;2j R;2j−1 þ H c Þ set of charges fQj g on the superconducting regions. The j key point is that in the fractional quantum Hall case there are two physically distinct processes that transform the which exhibits precisely the same form as the Fradkin- system from one such ground state to another. Namely, the Kadanoff representation of the quantum clock model total charge on a given superconducting segment can be in Eq. (24). incremented by adding fractional charge either to the upper The connection to the quantum clock model can be or lower trench edges. This distinction is meaningful since further solidified by considering how the various sym- fractional charge injected at one edge cannot pass to the metries present in the former are manifested in our ν ¼ 2=3 other because only electrons can tunnel across the trench. setup. Appendix A discusses this important issue and These two processes are implemented precisely by αRj and shows that all of these symmetries in fact have a transparent αLj, as illustrated in Fig. 5. By contrast, in the integer physical origin (including the time-reversal operation T quantum Hall case no such distinction exists. The Majorana that squares to unity). To streamline the analysis, we have operators add one unit of electric charge (mod 2e) that can defined the generalized Majorana operators in Eqs. (49) readily meander across the trench, so that their representa- such that under each symmetry they transform identically tion is essentially unique. to those defined in the clock model.

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dual The Z3 and Z3 transformations, which send V. FIBONACCI PHASE: A SUPERCONDUCTING ANALOGUE OF THE Z3 READ-REZAYI STATE i2π=3 αR=Lj → e αR=Lj ðZ3Þ; (58a) Consider now the geometry of Fig. 6(a) in which a spin- unpolarized ν ¼ 2=3 quantum Hall system hosts an array of N i2π=3 dual trenches of the type studied in Sec. IV. Edge excitations αR=Lj → e αR=Lj ðZ Þ; (58b) 3 on the top and bottom of each trench can similarly be described with fields ϕ1α x; y and ϕ2α x; y , where α warrant special attention. Clearly the Hamiltonian in ð Þ ð Þ Eq. (57) preserves both operations. In our quantum Hall problem these symmetries relate to physical electric charges. More precisely, they reflect global conservation of the “triality” operators P P þ − − iπQþ iπ Qj iπQ iπ Qj e tot ≡ e j ;etot ≡ e j ; (59) which generalize the notion of parity and take on three distinct values. The trialities respectively constitute con- dual served Z3 and Z3 quantities that specify (modulo 2e) the sum and difference of the total electric charge on each side of the trench. According to Eqs. (58a) and (58b), αRj and αLj dual carry the same Z3 charge but opposite Z3 charge; this property is sensible, given that these operators increment the charge on opposite trench edges [see also Eqs. (54) and (55)]. The correspondence with the clock model allows us to directly import results from Sec. III to the present setup. (a) Most importantly, we immediately conclude that the limit JΔ ¼ Jt realizes a self-dual critical point described by a Z3 parafermion CFT. Furthermore, at the critical point the primary fields relate to the lattice operators through Gapped phase? Eqs. (27a) and (27b), repeated here for clarity: Critical phase j αRj ∼ aψ R þð−1Þ bσRϵL þ ðtop edgeÞ; (60a) Fibonacci phase α ∼ aψ −1 jbσ ϵ : (60b) Lj L þð Þ L R þ ðbottom edgeÞ Gapped phase? An important piece of physics that is special to our ν ¼ 2=3 setup is worth emphasizing here. First, we note that ϵA, with A ¼ R or L, represents an electrically neutral field that (b) modifies neither the total charge nor the charge difference FIG. 6. (a) Multichain generalization of Fig. 5 in which a across the trench. This fact can be understood either from y 1; :::::; N ϵ ϵ ∼ 1 ϵ— ϵ sequence of trenches labeled by ¼ is embedded in a the fusion rule × þ which implies that A carries spin-unpolarized ν ¼ 2=3 quantum Hall system. Once again, the — the same (trivial) charge as the identity or by recalling edge modes opposite each trench are alternately gapped by ϵ Z from Sec. III that R=L remains invariant under both 3 and electron backscattering and Cooper pairing, with mˆ iðyÞ and dual Z3 . It follows that ψ R=L and σR=L must carry all of the nˆ iðyÞ characterizing the pinned charge-sector fields in a given physical charge of the lattice operators αR=Lj. That is, like domain [see Eqs. (44) and (45)]. We assume that the Z3 their lattice counterparts, ψ R and σR add charge 2e=3 to the generalized Majorana operators bound to each domain wall top edge of the trench, while ψ L and σL add charge 2e=3 to hybridize strongly within a trench and weakly between neighbor- 2e=3 the bottom trench edge. In this sense the ψ and σ fields ing trenches. Underlying this hybridization is tunneling of charges, which can only take place through the fractional inherit the spatial separation exhibited by αR=Lj. The next quantum Hall fluid; examples of allowed and disallowed proc- section explores stacks of critical chains, and there this esses are illustrated above. (b) Phase diagram for this system of property will severely restrict the perturbations that couple weakly coupled chains starting from the limit where each chain is fields from neighboring chains, ultimately enabling us to tuned to a critical point described by Z3 parafermion conformal access a superconducting analogue of the Read-Rezayi field theory. The couplings λa=b represent interchain perturbations state in a rather natural way. defined in Eq. (67).

011036-16 UNIVERSAL TOPOLOGICAL QUANTUM COMPUTATION ... PHYS. REV. X 4, 011036 (2014) denotes spin, x is a coordinate along the edges, and y XN X ¼ i2π † H − J e 3 α y α y : : 1; …;N labels the trenches. In the charge- and spin-sector intra ¼ t ½ R;2jþ1ð Þ R;2jð ÞþH c basis defined in Eqs. (41), the nontrivial commutation y¼1 j X relations now read i2π † þ JΔ ½e 3 αR;2jðyÞαR;2j−1ðyÞþH:c: : (64) j − 2πi Θðx0 − xÞ y ¼ y0 0 0 3 ½θρðx; yÞ; ϕρðx ;yÞ ¼ J J “ ” − 2πi Θðy0 − yÞ y ≠ y0; Here, Δ and t denote superconducting and dual frac- 3 tional Josephson couplings, respectively, mediated by 2πiΘ x0 − x y y0 0 0 ð Þ ¼ charge-2e=3 tunneling across the domains. ½θσðx; yÞ; ϕσðx ;yÞ ¼ 2πiΘðy0 − yÞ y ≠ y0; Interchain couplings are encoded in H⊥ and similarly 0 0 arise from the tunneling of fractional charges between ½ϕρðx; yÞ; ϕσðx ;yÞ ¼ −2πi: (61) adjacent trenches. Consider, for example, the perturbations

For y ¼ y0, one simply recovers Eqs. (42). The additional XN−1 X ⃗⃗ ⃗⃗ 0 −il·ϕ2ðxj;yÞ il·ϕ1ðxj0 ;yþ1Þ commutators for y ≠ y ensure proper anticommutation ½ζjj0 e e þ H:c: (65) relations between electron operators acting at different y¼1 j;j0 trenches but play no important role in our analysis. ⃗ We assume that the sets of counterpropagating edge with l ¼ð1; 1Þ and xk corresponding to a coordinate in modes opposite each trench are alternately gapped by the domain wall k in a given chain. These terms transfer charge Cooper-pairing and electron-backscattering mechanisms 2e=3 between the top edge of domain wall j0 on trench discussed in Sec. IV. At low energies, the pinning of the y þ 1 and the bottom edge of domain wall j on trench y. charge- and spin-sector fields in each gapped region is Such processes are indeed physical since the intervening again described by Eqs. (44) and (45). Using the labeling quantum Hall fluid supports fractionalized excitations. As scheme in Fig. 6(a), we denote the integer operators emphasized earlier 2e=3 tunneling across a trench is, by characterizing θσ, θρ, and ϕρ in a given domain by contrast, not permitted since the charge would necessarily ˆ ˆ MðyÞ, mˆ jðyÞ, and nˆ jðyÞ, respectively. [Note that MðyÞ pass through trivial regions that support only electrons. For depends only on y since the spin sector is gapped uniformly instance, hopping of charge-2e=3 quasiparticles from the in each trench.] It follows from Eqs. (61) that Mˆ ðyÞ bottom edge of trench y þ 1 to the top edge of trench y is commutes with all integer operators while disallowed for this reason. Figure 6(a) schematically illustrates such physical and unphysical processes. 8 3 iy>y0; Symmetry partially constrains the tunneling coefficient < 2π ζjj0 in Eq. (65). Specifically, enforcing charge conjugation 0 3 0 nˆ j y ; mˆ j0 y iy y0 j>j; (62) C Zdual ½ ð Þ ð Þ ¼ : 2π ¼ and (up to a 3 transformation) allows one to take ζ ei2π=3ζ 0 y

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possible sign structure in tj−j0 , which encodes phases and furthermore closely resembles that arising in Teo and acquired by quasiparticles upon tunneling from domain Kane’s construction of Read-Rezayi quantum Hall states wall j in one chain to j0 in another. Note also the from coupled Luttinger liquids [97]. In the present context, † conspicuous absence of terms that couple αR;jðyÞ with provided λa gaps the bulk (which requires λa > 0 as αL;j0 ðy þ 1Þ—which importantly are unphysical. As discussed below), the system enters a superconducting stressed in Sec. IVB, αRj and αLj respectively add frac- analogue of the Z3 Read-Rezayi phase that possesses edge tionalized quasiparticles to the top and bottom edges of a and bulk quasiparticle content similar to its non-Abelian given trench. Consequently, such terms would implement quantum Hall cousin. For brevity, we hereafter refer to this disallowed processes similar to that illustrated in Fig. 6(a). state as the “Fibonacci phase”—the reason for this nomen- Suppose that Jt ¼ JΔ so that in the decoupled-chain clature will become clear later in this section. limit each trench resides at a critical point described by a Z3 One can deduce rough boundaries separating the phases parafermion CFT. Again, this limit is advantageous since driven by λa and λb from scaling. To leading order, these arbitrarily weak intertrench couplings can dramatically couplings flow under renormalization according to impact the properties of the coupled-chain system. At ∂ λ 2 − Δ λ ; low energies it is then legitimate to expand the lattice l a=b ¼ð a=bÞ a=b (69) operators αR=L;jðyÞ in terms of critical fields using where l is a logarithmic rescaling factor and Δa 4=3, Eqs. (60a) and (60b). Inserting this expansion into the ¼ Δb ¼ 14=15 represent the scaling dimensions of the interchain Hamiltonian yields respective terms. The physics will be dominated by XN−1 Z whichever of these relevant couplings first flows to strong † coupling (i.e., values of order some cutoff Λ). Equating the H⊥ ∼ − ½λaψ LðyÞψ Rðy þ 1Þ x λ y¼1 renormalization-group scales at which a=b reach strong † coupling yields the following phase boundary: þ λbσLðyÞϵRðyÞσRðy þ 1ÞϵLðy þ 1ÞþH:c:; (67) 8=5 jλbj ∝ jλaj (70) with real couplings λ X X with a=b the bare couplings at the transition. Figure 6(b) 2 j 2 λa ¼ a ð−1Þ tj; λb ¼ b tj: (68) sketches the resulting phase diagram, which we expound j j upon below. Naturally, we are especially interested in the Fibonacci Insight into the phases driven by these interchain pertur- phase favored by λa > 0 and flesh out its properties in the bations—both of which are relevant at the decoupled-chain remainder of this section. We do so in several stages. First, fixed point—can be gleaned by examining certain extreme Sec. VA analyzes the properties of a single “ladder” limits. consisting of left movers from one trench and right movers Consider first the case with λa ¼ 0, λb ≠ 0. Since λb from its neighbor. As we will see, this toy problem is hybridizes both the right- and left-moving sectors of a given already extremely rich and contains seeds of the physics for chain with those of its neighbor, we conjecture that this the 2D Fibonacci phase. Section VB then bootstraps off of coupling drives a flow to a fully gapped 2D phase with no the results of Sec. VA to obtain the Fibonacci phase’s low-energy modes “left behind.” It is unclear, however, ground-state degeneracy and quasiparticle content. The whether this putative gapped state smoothly connects to properties of superconducting vortices in this state are that generated by moving each individual trench off of addressed in Sec. VC, and finally Sec. VD discusses the criticalityP R by turning on the thermal perturbation edge structure between the Fibonacci phase and the vacuum (as opposed to the interface with the ν 2=3 fluid). HT ¼ y x λTϵRðyÞϵLðyÞ, where λT ∼ Jt − JΔ. This in- ¼ triguing question warrants further investigation but will not be pursued in this paper. A. Energy spectrum of a single ladder Instead, we concentrate on the opposite limit λa ≠ 0, Until specified otherwise, we study the critical trenches λb ¼ 0, where a more immediately interesting scenario perturbed by Eq. (67) assuming λb ¼ 0. This special case arises. Here, the parafermion fields hybridize in a nontrivial allows us to obtain various numerical and exact analytical way—left movers from chain one couple only to right results that will be used to uncover universal topological movers in chain two, left movers from chain two couple properties of the Fibonacci phase that persist much more only to right movers in chain three, and so on. “Unpaired” generally. Tractability here originates from the fact that right- and left-moving Z3 parafermion CFT sectors thus with λb ¼P0 one can rewrite the coupled-chain Hamiltonian H Hy;yþ1 remain at the first and last chains, respectively. The as ¼ y ladder, where the ladder Hamiltonian involves structure of this perturbation parallels the coupling that only left-moving fields from trench y and right movers produced spinless p þ ip superconductivity from critical from trench y þ 1. (Nonzero λb clearly spoils this decom- Hy;yþ1 chains in the integer quantum Hall case studied in Sec. II position.) More explicitly, ladder can be written as

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y;y 1 H þ ¼ HL ðyÞþHR ðy þ 1Þ ladder CFTZ CFT † − ½λaψ LðyÞψ Rðy þ 1ÞþH:c:; (71) x H with CFT terms describing the dynamics for the unper- turbed left and right movers from trenches y and y þ 1, respectively. Although the ladder Hamiltonians at different values of y act on completely different sectors, the problem does not quite decouple: There remains an important (a) constraint between their Hilbert spaces which will become crucial in Sec. VB. For the rest of this subsection, we Hy;yþ1 explore the structure of ladder for a single ladder. The information gleaned here will then allow us to address the full 2D problem. Although λa as defined earlier is real, it will be useful to now allow for complex values—not all of which yield distinct spectra. Because correlators in the critical theory (b) with λa ¼ 0 are nonzero only when each of the total Z3 charges is trivial, perturbing around the critical point shows that the partition function can only depend on the combi- 3 3 2 nations ðλaÞ , ðλaÞ , and jλaj . Thus, Hamiltonians related i2π=3 by the mapping λa → e λa are equivalent. The physics does, however, differ dramatically for λa positive and negative [114,115].Forλa < 0, the model flows to another critical point, which turns out to fall in the universality class of the tricritical Ising model. In CFT language, this is an (c) example of a flow between minimal models via the Φ1;3 FIG. 7. (a) Phase diagram of the “ladder” Hamiltonian in operator [130]; here the flow is from central charge Eq. (71) for complex λa.Atλa 0, the ladder resides at a Z3 c 4=5 c 7=10 ¼ ¼ to ¼ theories. The solid lines in Fig. 7(a) parafermion critical point. Along the three solid lines, the ladder correspond to λa values for which the ladder remains remains gapless but flows instead to the tricritical Ising point. gapless. These results imply that the full coupled-chain Everywhere else the system is gapped and exhibits two sym- model with λa < 0 and λb ¼ 0 realizes a critical phase, as metry-unrelated ground states together with the Fibonacci kink denoted in Fig. 6(a). spectrum described in the main text. The dotted lines indicate For λa non-negative (and not with phase π=3), the integrability. (b) Effective double-well Ginzburg-Landau poten- spectrum of a single ladder is gapped. We focus on this case tial of the ladder Hamiltonian, which provides an intuitive picture for the ground-state degeneracy and Fibonacci kink spectrum. from now on—especially the limit of λa real and positive 2π=3 The equal-depth wells represent the two ground-state sectors. (modulo a phase of ), where the field theory is Excitations in these sectors are nondegenerate and correspond to integrable [114]. These special values are indicated by massive modes about the asymmetric well minima. Kinks and dotted lines in Fig. 7(a). Integrability provides a valuable antikinks interpolate between ground states, and turn out to have tool for understanding the physics, as it allows one to obtain the same energy as the oscillator excitations in one of the ground exact results for the ladder spectrum. Namely, the spectrum states. This is the hallmark of the Fibonacci kink spectrum. can be described in terms of quasiparticles with known (c) Energy versus momentum obtained via the truncated con- ¯ scattering matrices and degeneracies. References [131,114] formal space approach for each superselection sector. (The ½ε1 determined these properties via the indirect method of spectrum is identical to that of ½1ε¯ with k → −k.) Notice the two ground states, the nearly identical single-particle bands in ½1ε¯ finding the simplest solution of the integrability constraints εε¯ adhering to known properties of a Hamiltonian equivalent and ½ , as well as the multiparticle continuum in all sectors. to Eq. (71). This analysis is fairly technical, using tools from the representation theory of quantum groups [132]. While this language is probably unfamiliar to most case of interest here, the implications of integrability are condensed-matter physicists, the results are not: They striking but quite simple to understand. are the rules for fusing anyons. The connection between To illustrate the results, it is useful to first characterize the quasiparticle spectrum and scattering matrix of a the Hilbert space for a critical clock chain reviewed in 1 þ 1D integrable quantum field theory and the fusing Sec. III and then identify the (related but not identical) and braiding of anyons in a 2 þ 1D topological phase is Hilbert space for a single ladder. Consider for the moment explained in depth in Ref. [133]. For the Z3 parafermion the familiar three-state quantum clock model. As discussed

011036-19 ROGER S. K. MONG et al. PHYS. REV. X 4, 011036 (2014) in Sec. III, the entire spectrum at the critical point can be action of the perturbation on these states. By truncating the organized into sectors labeled by the chiral primary fields. Hilbert space, one obtains a finite-dimensional matrix that With periodic boundary conditions, the allowed left- and can be diagonalized numerically. Results of this analysis right-moving Hilbert spaces correspond to conjugate pairs appear in Fig. 7(c), which displays the energy E versus HL ⊗ HR F I ψ k F F † , where signifies one of the six fields , , momentum for three of the physical superselection † † ¯ ψ , ϵ, σ, and σ . PerturbingR the critical clock model with a sectors (the spectrum of the fourth ½ε1 follows from that † perturbation δH ∝ ðψ ψ R þ H:c:Þ analogous to the λa of ½1ε¯Þ. These plots clearly reveal a degeneracy between x L ¯ term in our ladder Hamiltonian mixes these sectors, but not the ground states in the ½11 and ½εε¯ sectors, and a gap to all completely. Two decoupled sectors remain, which follows excited states. Since there is no symmetry of the fusion from the fusion algebra described in Sec. III: The key algebra between the identity and ε sectors, however, gapped property here is that fusing with ψ or ψ † does not mix the excitations about the two ground states are not degenerate. first three of the six fields above with the last three. Thus, This property too is readily apparent from our TCSA when the critical clock Hamiltonian is perturbed by δH, the numerics in Fig. 7(c). Hilbert space can still be divided into the following To understand the situation more intuitively, it is useful “superselection” sectors: to imagine a Ginzburg-Landau-type effective potential following Refs. [136,137], where the same spectrum as 11¯ HL ⊗ HR⊕HL ⊗ HR ⊕HL ⊗ HR; ½ ¼ I I ψ ψ† ψ† ψ the ladder Hamiltonian arises (but starting from a different εε¯ HL ⊗ HR⊕HL ⊗ HR ⊕HL ⊗ HR: (72) model). Two non-symmetry-related vacua together with the ½ ¼ ϵ ϵ σ σ† σ† σ low-lying excitations can be described by a double-well potential, where the two wells have the same depth but Next, we return to the ladder Hamiltonian given in exhibit different curvature as in Fig. 7(b). In the figure, Φ is Eq. (71). In this case the superselection sectors above still † roughly the field ðσ þ σ Þ, with σ defined in Sec. III [138]. appear, but now the left- and right-moving Hilbert spaces From this effective potential, one can understand the four correspond to different trenches. For this reason the sectors in the ladder spectrum as follows. Two of the sectors constraints between the left and right movers are relaxed, ¯ ½11 and ½εε¯ correspond to the degenerate minima and resulting in sectors not present in the periodic clock chain. massive fluctuations thereabout. The different curvature of Specifically, there are two additional superselection sectors the wells leads to nondegenerate massive modes—similar given by to our TCSA numerical data where ½εε¯ exhibits the smaller 1ε¯ HL ⊗ HR⊕HL ⊗ HR ⊕HL ⊗ HR; gap. In fact, there “one-particle” states occur, whereas the ½ ¼ I ϵ ψ σ† ψ† σ ¯ gap in the ½11 sector is about twice as large and appears to ε1¯ HL ⊗ HR⊕HL ⊗ HR ⊕HL ⊗ HR; (73) ½ ¼ ϵ I σ ψ † σ† ψ consist of a multiparticle continuum. The remaining two sectors correspond to “kinks” interpolating between the where again L and R refer to different trenches. Note that ground states. A kink is a field configuration where the field L R we forbid combinations such as HI ⊗ Hψ that would takes on one minimum to the left of some point in space and require net fractional charge in the ν ¼ 2=3 strip between a different minimum on the right; the excitation energy is the trenches; for a more detailed discussion, see Sec. VC. then localized at the region where the field changes. There The upshot of this perturbed CFTanalysis is that the Hilbert are two possible configurations, related by parity, and we space for a single ladder can be split into the four distinct will label these here as kinks and antikinks. It is natural to ¯ sectors defined in Eqs. (72) and (73). expect that these parity conjugates occur in the ½1ε¯ and ½ε1 Exploiting the integrability of the Hamiltonian in sectors. This expectation is indeed consistent with our Eq. (71) at λa > 0 both provides an intuitive way of numerical work displayed in Fig. 7(c). understanding the spectrum and reveals remarkable degen- Aside from the two ground states, there exists another eracies among the sectors that are far from apparent a priori. remarkable degeneracy between two very different quasi- One important feature is that the integrable model admits particle excitations: The gap in the ½εε¯ sector is the same as two degenerate ground states not related by any local the minimum kink or antikink energy [131,114]. One can symmetry. (Actually, this property survives for rather see this either directly from the numerics in Fig. 7(c) or general λa—see below.) We confirm the presence of two from an analysis exploiting integrability. The latter shows ground states by analyzing the spectrum numerically in two that the kink, antikink, and “oscillator” excitation in the ½εε¯ complementary ways. The first method employs the den- sector exhibit identical dispersion as well. The entire sity-matrix renormalization group on an integrable lattice spectrum is then built up from these fundamental excita- ¯ model; this analysis will be detailed elsewhere [116]. The tions. For instance, the lowest excited states in the ½11 second method utilizes the truncated conformal space sector form a two-particle continuum originating from approach (TCSA), which directly simulates the field theory kink-antikink pairs (as opposed to another species of [134,135]. Here, the eigenstates and operator-product rules single-particle excitations), consistent with the numerically of the CFTare used to characterize the Hilbert space and the determined spectrum.

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Even though there are three flavors of excitations, the Eq. (71) and was studied for a single y in the last number of states in the spectrum with N quasiparticles subsection. actually grows more slowly than 3N.(By“quasiparticle,” Given that for a single ladder Eq. (71) already exhibits a we mean a localized excitation that takes the form of either twofold ground-state degeneracy, one might naively expect 2N N a kink, an antikink, or an oscillator mode.) The reason is a -fold degeneracy for the full -trench system. This that the spatial order in which different excitation flavors conclusion is incorrect, however, as such naive counting occur is constrained. Viewing the problem in terms of the ignores Hilbert-space constraints between the left and right movers within a given trench. In particular, combinations double-well potential described above, the following rules R L † 0 † H ðyÞ ⊗ H 0 ðyÞ with F ∈ fI;ψ; ψ g and F ∈ fϵ; σ; σ g are evident. Going (say) left to right, a kink can be followed F F (or vice versa) are forbidden for any physical boundary by an antikink or an oscillator excitation, an oscillator can conditions on trench y [139]. Here, we have explicitly be followed by an antikink or another oscillator, and an denoted that HR=L correspond to the same chain y to avoid antikink can only be followed by a kink. Because of these possible confusion with the previous subsection (where the restrictions, the number of states growsffiffiffi asymptotically with right- and left-moving Hilbert spaces correspond to differ- N p N as φ , where again φ ≡ ð1 þ 5Þ=2 is the golden ratio. ent trenches). Thus, the allowed CFT superselection sectors We therefore dub the features described here as the in each chain must have either F, F 0 ∈ fI;ψ; ψ †g,orF, “Fibonacci kink” spectrum. F 0 ∈ fϵ; σ; σ†g; in other words, Integrability turns out to provide a sufficient but not necessary condition for these striking degeneracies. We CFT sectorRðyÞ ∼ CFT sectorLðyÞ mod ψ: (74) have verified numerically using the TCSA method that the R L two symmetry-unrelated ground states and the Fibonacci Note that this set includes sectors such as HI ðyÞ ⊗ Hψ ðyÞ, kink spectrum persist even for λa lying away from the which are physical since fractional charges can hop dashed lines in Fig. 7(a) that mark the integrable points between trenches. iπ=5 [116]. For instance, with λa ¼ e the spectra are nearly Now recall from Sec. VA that the ground states for a ¯ indistinguishable from those in Fig. 7(c). Hence, for almost single ladder occur in the sectors ½11 and ½εε¯ as defined in R L all λa (the exception occurring where the the system is Eq. (72), where again H and H correspond to chains y critical), the ladder Hamiltonian realizes a gapped phase and y þ 1. In order for the 2D coupled-trench system to with the properties noted above. It is useful to comment that reside in a ground state, the superselection sectors between one can, in principle, spoil this structure: Terms such as adjacent chains must therefore match; i.e., σLðyÞσRðy þ 1ÞþH:c: break the degeneracies—but are nonlocal in our setup and thus do not reflect physical CFT sectorLðyÞ ∼ CFT sectorRðy þ 1Þ†: (75) perturbations. We should emphasize here that the preceding discussion Combining with Eq. (74), this constraint locks the Hilbert applies only to a single ladder Hamiltonian defined in spaces of every chain together, yielding two ground states Eq. (71). By itself, this 1D model does not support as claimed. We label the ground states as j1i and jεi, which Fibonacci anyons as stable excitations in any meaningful denote the corresponding sectors in the chains. sense. Nevertheless, the tantalizing similarities are by no Our aim next is to unambiguously identify the anyon means accidental. In fact, the remarkable Fibonacci kink content of our coupled-chain phase. The Fibonacci kink spectrum should be viewed as a precursor to both the spectrum identified in the ladder problem in Sec. VA topological order and Fibonacci anyons that do appear in already strongly hints that a Fibonacci anyon is present, the full 2D coupled-trench system. This will be elucidated although we will derive this explicitly in what follows. To in the next subsection, which uses the results obtained here do so, it will be instructive to review a few facts regarding to deduce the ground-state degeneracy and particle content topological states on a cylinder (instead of a torus). On an of the Fibonacci phase. infinite cylinder, the ground-state degeneracy equals the number of anyon types. For every anyon α, there is an B. Ground-state degeneracy and quasiparticle content associated ground state jαi, the set of which forms an We now show that in the 2D Fibonacci phase, the orthogonal basis for the ground-state Hilbert space. coupled-chain system exhibits a twofold ground-state Physically, these states are defined with a fixed anyon degeneracy on a torus. Consider N parallel trenches labeled charge at infinity, or equivalently, as eigenstates of Wilson- by y, coupled to their neighbors via λa > 0. (We continue to loop or anyon-flux operators around the circumference of assume λb ¼ 0.) To form the torus geometry, each chain the cylinder. (They are also referred to as “minimally is itself periodic, and the first and last chains at y ¼ 1, N entangled states” [140].) Anyon excitations are trapped at coupleP as well. The system is therefore described by the domain wall between ground states that are consistent H N Hy;yþ1 y ¼ y¼1 ladder with periodic boundary conditions along with the fusion rules. More precisely, using as a the x and y directions; the ladder Hamiltonian is defined in coordinate in the infinite direction of the cylinder, let the

011036-21 ROGER S. K. MONG et al. PHYS. REV. X 4, 011036 (2014) wave function for y>0 be jαþi and for y<0 be jα−i. Spin-unpolarized SE At least one anyon must be trapped on the circle y ¼ 0, with total topological charge β satisfying the fusion − relation α × β ∼ αþ þ. 3 Applying this discussion to our setup, we now consider an infinite number of trenches, each forming a ring around the 2 cylinder. This geometry gives us an infinite number of chains labeled by y ∈ Z, coupled via Eq. (71). By the same logic as 1 for the torus geometry, there are again two ground states j1i and jεi that arise from a different superselection sector on Lx each chain. Keep in mind that, for the time being, 1 and ε are 4 8 12 merely labels derived from the coupled-chain construction; Trenches Partition we have not yet made the association with anyons. (a) (b) Recall in our argument for the two ground states that FIG. 8. (a) Bipartition of the superstructure that cuts between Eq. (74) is an unyielding requirement that follows from the two chains on a cylinder. (b) Entanglement entropy SE of the j1i boundary condition, while Eq. (75) follows from ener- (red dots) and jεi (blue dots) ground states of the 2D Fibonacci getics. Hence, when studying excited states, we can relax phase as a function of the cylinder circumference Lx, computed the second condition on specific ladders where localized numerically via the truncated conformal space approach. Fitting excitations exist. Let us examine the three flavors of SE for state j1i to the form sLx − γ at large Lx, we extract the fundamental ladder excitations—kink, antikink, and intercept −γ ≈ −0.65; see the solid line in the figure. This yields a oscillator—identified in Sec. VA. Suppose first that there total quantum dimension D ≈ 1.9 for the Fibonacci phase. Taking S ε − S 1 d is a single kink between trenches y ¼ 0,1—i.e., that the the difference E½j i E½j i ¼ log ε, we deduce the quantum d ≈ 1 62 ≈ φ ε corresponding ladder resides in the ½1ε¯ sector defined in dimension ε . , which confirms that corresponds to Eq. (73). The chains then lie in the 1 sector for y ≤ 0 and the Fibonacci anyon. the ε sector for y ≥ 1. For an antikink, the sectors are ε and L S ∼ sL − 1 for y ≤ 0 and y ≥ 1, respectively. Finally, for an oscillator linearly with the cylinder circumference x: E x γ L excitation every chain must be in the ε sector. (That þ(up to terms that decay exponentially with x). The s excitation type exists only in the ½εε¯ ladder sector.) slope is identical for all ground states of the same Since the three excitations possess the same mass and Hamiltonian but depends on nonuniversal microscopic γ dispersion, it is natural to identify all of these as the same details. By contrast, the intercept defines the topological — nontrivial anyon (which we label as • for the time being). entanglement entropy [141,142] a universal topological The discussion above then implies that a • anyon can occur invariant of the ground state used in the computation. This γ D=d at a domain wall between jεi and j1i on the cylinder, or invariant can be further decomposed as ¼ logð ΨÞ, d simply between two jεi regions—but not between two j1i where Ψ is the quantum dimension of the quasiparticle states. Accordingly, the allowed fusion channels follow as corresponding to the state jΨi, and D is the “total quantum ” – 1 × • ∼ ε and ε × • ∼ 1 þ ε, whereas 1 × • → 1 is forbid- dimension of the phase [141 143]. a In the geometry illustrated in Fig. 8(a), the only con- den. We can rewrite these rules as a tensor N•;b with integer a tribution to entanglement comes from the left movers of entries, where N•;b ¼ 1 if b × • → a is admissible and zero 1 ε chain y ¼ yc and right movers of chain y ¼ yc þ 1, as all otherwise. In the basis of and ground states, the fusion λ 0 matrix for the excitation is other degrees of freedom decouple at b ¼ . Hence, the entanglement entropy arising from a bipartition of the 01a a cylinder is equivalent to that arising from a bipartition of a N•;b ¼ (76) 11b single ladder into left and right movers. (This setup bears much resemblance to the Affleck-Kennedy-Lieb-Tasaki with the dominant eigenvalue, or quantumpffiffiffi dimension, equal spin-one chain [144]. There, each spin fractionalizes into d φ ≡ 1 5 =2 1 1 to the golden ratio: ε ¼ ð þ Þ . Hence, in addi- a pair of spin-2’s, and in the ground state the “right” spin-2 1 1 tion to being associated with CFT sectors, we can identify for a given site forms a singlet with the “left” spin-2 at the as the trivial anyon and ε ¼ • as the Fibonacci anyon. next site over. An entanglement cut between two adjacent We further corroborate this result through numerical sites thus breaks apart exactly one spin singlet into its left 1 evaluation of the “topological entanglement entropy.” and right spin-2’s.) Suppose that we partition the cylinder between chains yc We use our TCSA simulations of Eq. (71) to evaluate SE and yc þ 1 as illustrated schematically in Fig. 8(a). The for the two ground states j1i and jεi; the data appear in S − ρ ρ S L 1 entanglement entropy is given by E ¼ Try>yc ½ log , Fig. 8(b). By fitting E versus x for ground state j i ρ Ψ Ψ d 1 where ¼ Try≤yc j ih j is the reduced density matrix that (which corresponds to 1 ¼ ), we extract the total quan- comes from a partial trace of the wave function jΨi. For a tum dimension D ¼ 1.9 0.1. One can, in principle, ground state of any gapped system, this quantity scales perform a similar fit for the other ground state jεi to

011036-22 UNIVERSAL TOPOLOGICAL QUANTUM COMPUTATION ... PHYS. REV. X 4, 011036 (2014) extract dε=D. However, a far more precise value for dε λb ≠ 0, either analytically or numerically, is highly non- follows from the difference δSE ≡ SE½jεi − SE½j1i of trivial since we then lose integrability and can no longer entanglement entropies for the two ground states; the distill the problem into individual ladders with a Hilbert- linear term in Lx cancels here, leaving δSE ¼ space constraint. Progress could instead be made by logðdε=d1Þ. In this way, we obtain quantum dimension employing density-matrix renormalization-group simula- dε ¼ 1.619 0.002. These values are in excellent agree- tions to map out the phase diagram more completely, which ment with those of a Fibonacci-anyon modelqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with just would certainly be interesting to pursue in follow-up work. 2 2 one nontrivial particle, for which D ¼ d þ dε ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 C. Superconducting vortices 1 þ φ ≈ 1.902 and dε ¼ φ ≈ 1.618. The ground-state degeneracy on the torus, fusion rules, Since the Fibonacci phase arises in a superconducting and topological entanglement entropy computed above are system, it is also important to investigate the properties of sufficient in this case to uniquely identify the 2D topo- h=2e vortices—despite the fact that, unlike Fibonacci logical phase that the system enters. Indeed, there are only anyons, they are confined. Before turning to this problem, two topological phases of fermions with twofold ground- it will be useful to briefly recall the corresponding physics state degeneracy on the torus [145]. The nontrivial particle in a spinless 2D p þ ip superconductor [54,150–152].One can be either a semion or a Fibonacci anyon. We can way of understanding the nontrivial structure of vortices distinguish between these possibilities with either the there is by considering the chiral Majorana edge states of a fusion rules or topological entanglement entropy; both p þ ip superconductor on a cylinder. Finite-size effects indicate that our coupled-trench system supports the quantize their energy spectrum in a manner that depends Fibonacci anyon—which justifies the name “Fibonacci on boundary conditions exhibited by the edge Majorana phase” christened here. fermions. With antiperiodic boundary conditions the spec- Given the particle types and fusion rules, the universal trum is gapped, while in the periodic case an isolated zero topological properties of this phase can be determined by mode appears at each cylinder edge. Threading integer solving the pentagon and hexagon identities; they may be multiples of h=2e flux through the cylinder axis toggles summarized as follows. (For a concise review, see Ref. between these boundary conditions, thereby creating and [146].) The Fibonacci phase admits only the two particle removing zero modes. This reflects the familiar result that types deduced above: the trivial particle 1 and a Fibonacci h=2e p ip ε θ 1 θ e4πi=5 vortices in a planar þ superconductor bind anyon . They have topological spins 1 ¼ , ε ¼ Majorana zero modes and consequently form Ising anyons. and satisfy the fusion rule ε × ε ∼ 1 ε [147]. As a result þ We will deduce the properties of vortices in the of this fusion rule, the dimension of the low-energy Hilbert Fibonacci phase by similarly deforming our ν 2=3 space of (n þ 1) ε particles with total topological charge 1 ¼ quantum Hall setup into a cylinder as sketched in Fig. 9. is the nth Fibonacciffiffiffi number Fn, which grows asymptoti- n p cally as φ = 5; thus, its quantum dimension is dε ¼ φ,as In principle, the physics can be analyzed by deriving the we saw previously. (This is the same quantity that enters the influence of flux on boundary conditions for the Z3 formulas for the entanglement entropy used above.) When parafermionic edge modes supported by this state, although two Fibonacci anyons are exchanged, the resulting phase such an approach will not be followed here. Instead, we εε −4πi=5 εε 3πi=5 acquired is either R1 ¼ e or Rε ¼ e , depending develop a related adiabatic flux-insertion argument that on the fusion channel of the two particles denoted in the allows us to obtain the result with minimal formalism. We subscript. The result of an exchange can thereby be proceed by first assuming that the Fibonacci phase is deduced if we can bring an arbitrary state into a basis in bordered by “wide” ν ¼ 2=3 regions on the upper and ε which the two particles in question have a definite fusion lower parts of the cylinder, as Fig. 9 indicates. This channel. This basis change can be accomplished with the F assumption will allow us to separately address the effect symbols. The only nontrivial one is of flux on (i) the gapless Z3 parafermion modes at the φ−1 φ−1=2 interface between the Fibonacci phase and ν ¼ 2=3 regions εεε Fε ¼ (77) and (ii) the outermost ν ¼ 2=3 edge states that border the φ−1=2 −φ−1 vacuum. One can then couple these sectors to determine the 1; ε final vortex structure. Following this logic, we will show written in the basis f g for the central fusion channel. p ip h=2e From these relatively simple rules follows a remarkable that in contrast to the þ case, flux does not fact: These anyons support universal topological quantum introduce new topological anyons beyond the trivial and computation [148,149]. Fibonacci particles already discussed. A vortex may, While the aforementioned analysis was carried out however, provide a local potential that happens to trap a for λb ¼ 0, the gapped topological phase that we have deconfined Fibonacci anyon, although whether or not this constructed must be stable up to some finite λb. Rough transpires is a nonuniversal question of energetics. (Note phase boundaries for this state were estimated earlier; see that the same could be said for, say, an impurity, so one Fig. 6(a). However, directly exploring the physics with should not attach any deep meaning to this statement.)

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Q− −2=3 for each trench to tot ¼ mod 2. The only allowed sectors consistent with this charge arrangement are † † ψ R × ψ L and σR × σL. Finally, we turn on the interchain perturbation λa in Eq. (67) to enter the Fibonacci phase. Spin- The CFT sectors in the bulk that are gapped by this unpolarized coupling will then clearly mix. However, the gapless right movers from the top trench and left movers from the bottom remain unaffected by λa; the former necessarily realizes ψ σ ψ † σ† Trenches either R or R, while the latter realizes L or L. Focusing on the top half of the system, this argument shows that an h=2e superconducting vortex traps an Abelian

Fibonacci phase ψ or non-Abelian σ particle at the interface between the ν ¼ 2=3 fluid and the Fibonacci phase. Importantly, we must additionally account for the quantum Hall edge at the top of the cylinder, which also responds to the flux and influences the structure of a vortex in a crucial way, as we will see. Figure 9 shows that the flux induces charge þe=3 at the uppermost cylinder edge. Together, we see that an h=2e vortex gives rise to edge excitations hψ; 1=3i or hσ; 1=3i when the Fibonacci phase is bordered by a wide Abelian FIG. 9. Cylinder geometry used to deduce the properties of quantum Hall fluid. Here and below, hF;qi indicates that the h=2e superconducting vortices in the Fibonacci phase. We interface between the ν 2=3 liquid and Fibonacci phase ν 2=3 ¼ initially assume that pure ¼ quantum Hall states border traps particle type F, while the quantum Hall edge bordering the Fibonacci phase from above and below. This results in two the vacuum binds charge q mod 1. Recalling the 2e=3 charge types of well-defined boundaries: the Fibonacci phase-to-quan- ψ σ h=2e tum Hall interface, and the quantum Hall-to-vacuum edge. associated with and , we conclude that the vortex Adiabatically inserting h=2e flux through the cylinder (which carries total charge e mod 2e—which is not fractional. Next, is topologically equivalent to an h=2e vortex in the bulk of a we discuss the fate of the ψ and σ particles at the Fibonacci- planar Fibonacci phase) pumps charge e=3 across each quantum phase boundary when we include coupling to the outer Hall region, as shown above. Because the charge difference quantum Hall edge. across the trenches then changes, the upper Fibonacci phase-to- If one assumes that the Z3 parafermion edge states and quantum Hall interface binds either a ψ or σ excitation that carries outer ν ¼ 2=3 edge modes decouple, then the system can, 2e=3 2e charge mod . The upper quantum Hall-to-vacuum edge, in principle, reside in six possible edge sectors: hI;0i, however, binds charge e=3 so that, in total, the vortex carries only † † hψ; 1=3i, hψ ; 2=3i, hϵ; 0i, hσ; 1=3i, and hσ ; 2=3i. (This fermion parity. If one shrinks the pure quantum Hall regions so that the two boundaries hybridize, ψ and σ lose their meaning statement is independent of vorticity and simply tells one since other sectors mix in. The final conclusion is that an h=2e which states have physical charge configurations.) Suppose vortex traps either a trivial particle or a Fibonacci anyon now that the pure quantum Hall region at the top of Fig. 9 depending on nonuniversal details, but does not lead to new shrinks to allow fractional charge tunneling between the quasiparticle types. parafermion and ν ¼ 2=3 edge modes. Some of the edge sectors above then mix and hence are no longer distinguish- Let us first consider a cylinder with no flux, in the limit able. For instance, transferring e=3 charge from the vacuum where each trench is tuned to Z3 parafermion criticality and edge to the boundary of the Fibonacci phase can send interchain coupling is temporarily turned off. For concrete- hσ; 1=3i → hϵ; 0i. In fact, only two inequivalent edge † ness, we also assert that each ν ¼ 2=3 edge contains no net sectors remain—the triplet hI;0i, hψ; 1=3i, and hψ ; 2=3i electric charge mod 2e. The sum and difference of the total that is associated with the identity particle and the remain- Q ϵ; 0 σ; 1=3 σ†; 2=3 ε charge on the two sides of each trench tot [see Eq. (59)] ing set h i, h i, and h i associated with the must also then vanish mod 2e. This constraint restricts non-Abelian anyon. the possible CFT sectors present in the trenches to either Applying the above discussion to vortices, we infer that h=2e ψ σ IR × IL or ϵR × ϵL; all other physical sectors contain the flux does not generically bind a or in any wrong charge. Next, we adiabatically increase the flux meaningful way once the parafermion and outer ν ¼ 2=3 through the cylinder from 0 to h=2e [153]. Because of the edge modes hybridize. The vortex can trap a trivial or nontrivial Hall conductivity in the ν ¼ 2=3 fluids, charge Fibonacci anyon but exhibits no finer Z3 structure—which e=3 pumps from the bottom to the top edge of each is entirely consistent with the fact that it carries only quantum Hall region in response to the flux insertion, as fermion parity. Which of the two particle types occurs in Qþ Fig. 9 illustrates. The pumping leaves the total charge tot practice depends on nonuniversal microscopic details, on each trench intact but alters the total charge difference although both cases are guaranteed to be possible because

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ε is deconfined. (If a vortex binds a trivial particle, one can Since the Fibonacci phase has only two particle types 1 always bring in a Fibonacci anyon from elsewhere and and ε, the minimal possible edge theory describing the attach it to the vortex to obtain the ε case, or vice versa.) boundary with the vacuum has two primary fields that we In fact, a similar state of affairs occurs for any phase that denote as 1 and ϵ~. (The tilde is used to distinguish from supports a Fibonacci anyon, including the Z3 Read-Rezayi the field ϵ that lives at the boundary between the state. Because of the fusion rule ε × ε ∼ 1 þ ε, the Fibonacci phase and the parent quantum Hall fluid.) Fibonacci anyon ε must carry the same local quantum At first glance, however, our quantum Hall-superconduc- numbers (such as charge and vorticity) as the trivial tor heterostructure appears to exhibit a much more particle. Thus, any Abelian anyon A can fuse with the complicated edge structure than the quasiparticle content neutral Fibonacci anyon to form a non-Abelian particle suggests. The interface between the Fibonacci phase and with identical local quantum numbers: A × ε ∼ Aε [154]. the spin-unpolarized ν ¼ 2=3 state is described by a Z3 For example, in the case of the Z3 Read-Rezayi state at parafermion CFT, and the boundary between the ν ¼ 2=3 filling ν ¼ 13=5, there are two anyons with electric charge state and the vacuum is described by a CFT for two e=5: one Abelian and the other non-Abelian with quantum bosons with K matrix dimension φ. The latter quasiparticle may be obtained by 12 fusing the former with a neutral Fibonacci anyon. K ¼ ½fermionic ð112Þ state (78) 21 Equivalently, the former may be obtained from the latter by fusing two non-Abelian e=5 quasiparticles with a −e=5 The former CFT has six primary fields, while the latter quasihole. Which of these e=5 excitations has the lowest has three. One can obtain a direct interface between energy is a priori nonuniversal. Details of such energetics the Fibonacci phase and vacuum by simply shrinking the issues are interesting but left to future work. outer ν ¼ 2=3 fluid until it disappears altogether; the Finally, we remark that the Z3 structure at the edge resulting boundary is then naively characterized by a between the Fibonacci phase and the ν ¼ 2=3 state arises product of these two edge theories. However, in the solely from the fractional quantum Hall side. The corre- previous subsection we argued that of the 18 primary sponding fractionally charged quasiparticles indeed do not fields in the product CFT, only a subset of six are exist within the Fibonacci phase, as evidenced by the physical from charge constraints, and these combine to absence of ψ or σ particles in the bulk. Our coupled-chain just two primary fields. Here we explicitly construct a construction provides an intuitive way of understanding chiral CFT with exactly these two primary fields. this result: 2e=3 excitations are naturally confined in the Furthermore, we demonstrate that upon edge Fibonacci phase since the trenches provide a barrier that reconstruction, the Fibonacci-phase-to-vacuum interface prevents fractional charge from tunneling between adjacent is described by this CFT combined with unfractionalized quantum Hall regions. The Fibonacci anyon is neutral, by fermionic edge modes, in precise correspondence with contrast, and thus suffers no such obstruction. the bulk quasiparticle types supported by the Fibonacci phase. D. Excitations of the edge between the It is useful to first examine the simpler case of a ν ¼ 2=3 Fibonacci phase and the vacuum state built out of underlying charge-e bosons. To describe this setup one replaces the K matrix of Eq. (78) with Bulk properties strongly constrain the edge excitations of a topological phase. In particular, the edge bordering the 21 K ¼ ½bosonic ð221Þ state: (79) vacuum must support as many anyon types as the bulk. This 12 correspondence is simplest when the bulk is fully chiral. Edge excitations are then described by a CFT (possibly For brevity, we refer to this bosonic quantum Hall phase as deformed by marginal perturbations so that some of the the (221) state. Most of the preceding analysis, including velocities are unequal) that exhibits precisely the same the appearance of a descendant Fibonacci phase, is number of primary fields as the bulk has anyon types. unchanged by this modification. However, by working These fields possess fractional scaling dimensions, and all with a bosonic theory, we can appeal to modular invariance other fields have scaling dimensions that differ from these to connect the bulk quasiparticle structure to the edge chiral by integers. Therefore, one can view an arbitrary field as central charge cR − cL: creating an anyon (via a primary operator) together with 1 X some additional bosonic excitations. It is important to note θ d2 eð2πi=8ÞðcR−cLÞ; D a a ¼ (80) that the edge may have additional symmetry generators a beyond just the Virasoro generators derived from the energy-momentum tensor. These additional symmetry where a sums over the two anyon types and cR=L denote the generators have their scaling dimensions fixed to 1 central charges for right or leftp movers.ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Using results from 2 (Kac-Moody algebras) or some other integer (e.g., W Sec. VB—in particular, D ¼ 1 þ φ , d1 ¼ 1, dε ¼ φ, 4πi=5 algebras) [155]. and θ1 ¼ 1, θε ¼ e —the chiral central charge follows

011036-25 ROGER S. K. MONG et al. PHYS. REV. X 4, 011036 (2014) as cR − cL ≡ 14=5 mod 8. Thus, the minimal edge theory describing the boundary with the vacuum is purely chiral with cR ¼ 14=5 and cL ¼ 0. We now show that the bosonic Fibonacci-phase-to-vacuum edge is consistent with these scaling dimensions and central charges. The key physical observation was made in the previous subsection: Fractional charge and the resulting Z3 structure are features of the ν ¼ 2=3 state, not the Fibonacci phase. Equivalently, not all of the excitations of the combined Z3 parafermion CFT and the ð221Þ edge states are allowed in the Fibonacci phase because we cannot transfer fractional charge from one edge of the system to the other via the bulk. Fractional charge can pass only between the Fibonacci-phase-to-(221)-state and the (221)- state-to-vacuum interfaces; together, these two edges form the Fibonacci-to-vacuum edge. As such, the total charge of the Fibonacci-phase-to-vacuum edge must be an integer that dictates the set of physical operators that appear. In terms of the Z3 parafermion operators and the edge fields ϕ↑; ϕ↓ of the (221) state, the most relevant operators In the axes, the vector l⃗represents the argument of a given that transfer fractional charge within an edge are ⃗⃗ (221)-state operator written as eil·ϕ [e.g., l⃗¼ð2; 1Þ for e2iϕ↑þiϕ↓ ψeiϕ↑ ; ψeiϕ↓ ; ψe−iϕ↑−iϕ↓ ; ]. As an extension to the Virasoro algebra, this Kac-Moody algebra has c ¼ 14=5 and only two primary † −iϕ † −iϕ † iϕ iϕ ψ e ↑ ; ψ e ↓ ; ψ e ↑þ ↓ : (81a) fields, the identity 1 and ϵ~ ¼ σ†eiϕ↑þiϕ↓ [156]. All other fields of the CFT can be constructed by combining one of Note that these fields all have scaling dimension one. There the primaries with the generators in Eqs. (81); e.g., ϵ arises are six additional dimension-one operators that add integer from the operator product expansion between ϵ~ and −iϕ −iϕ charge to an edge: ψe ↑ ↓ . The identity field has scaling dimension h1 ¼ 0 and transforms trivially under the G2 action, while the nontrivial field ϵ~ has scaling dimension hϵ~ ¼ 2=5 and eiϕ↑þ2iϕ↓ ;e2iϕ↑þiϕ↓ ;eiϕ↑−iϕ↓ ; belongs in the seven-dimensional fundamental representa- −iϕ↑−2iϕ↓ −2iϕ↑−iϕ↓ −iϕ↑þiϕ↓ e ;e ;e : (81b) tion of G2. Here we can see that the bulk-edge correspondence is consistent with our identification of the bulk as the 1 Finally, the two charge-current operators Fibonacci phase; for example, the topological spins of and ε are related to the scaling dimensions of the fields 1 2πih rffiffiffi and ϵ~ via θ1;ε ¼ e 1;ϵ~ . 3 i ffiffiffi p We now return to the fermionic case, where the ν 2=3- i ∂ϕ↑; pffiffiffi ∂ϕ↑ þ i 2∂ϕ↓ (81c) ¼ 2 2 to-vacuum edge is characterized by the K matrix in Eq. (78). The allowed operators that transfer charge in also have scaling dimension one. The 14 operators in the fermionic Fibonacci-phase-to-vacuum edge are once Eqs. (81a–81c) satisfy the Kac-Moody algebra for the Lie again given by Eqs. (81). Unlike in the bosonic case, group G2 at level one: however, these operators are nonchiral because the fermionic ν ¼ 2=3 state supports counterpropagating edge b abc c modes at the interface with the vacuum. Nevertheless, δa¯ f J ðwÞ Ja z Jb w ; (82) they remain spin-one operators as in the bosonic setup. ð Þ ð Þ¼ z − w 2 þ z − w þ ð Þ Moreover, the fermionic Fibonacci-to-vacuum edge exhibits a phase that bears a simple relation to the bosonic edge, abc where f are the structure constants for the G2 Lie as we now demonstrate. algebra, normalized such that the Killing form This phase occurs when the edge reconstructs such that facdðfbcdÞ ¼ 8δab. The two charge currents form the an additional nonchiral pair of unfractionalized modes Cartan subalgebra for G2, while the operators in comes down in energy and hybridizes with the modes of Eqs. (81a) and (81b) correspond to the nonzero roots of the ν ¼ 2=3-to-vacuum edge. In the limit where these G2 as follows: modes are gapless, the K matrix becomes

011036-26 UNIVERSAL TOPOLOGICAL QUANTUM COMPUTATION ... PHYS. REV. X 4, 011036 (2014) 0 1 120 0 Both have Fibonacci anyons as well as fermionic B C B 210 0C excitations [158]. Ke ¼ B C: (84) @ 001 0A VI. TOPOLOGICAL QUANTUM FIELD 000−1 THEORY INTERPRETATION We will now provide an alternative topological quantum The ν ¼ 2=3-to-vacuum edge is then described by the effective field theory field theory (TQFT) interpretation of the Fibonacci phase introduced in the preceding sections. Although less con- Z 1 nected to microscopics, the perspective developed here cuts S Ke ∂ ϕ ∂ ϕ − V ∂ ϕ ∂ ϕ : ¼ 4π ½ IJ t I x J IJ x I x Jþ (85) more directly to the elegant topological properties enjoyed t;x by this state. Our discussion will draw significantly on the earlier works of Gils et al. [90] and especially Ludwig et al. Here the ellipsis represents quasiparticle tunneling proc- [91]. As already mentioned in the Introduction, our con- I J esses, indices and label the field components such that struction of the Fibonacci phase from superconducting ϕ ϕ 1 and 2 denote the original spin-up and spin-down islands embedded in a ν 2=3 quantum Hall state bears ϕ ϕ ¼ modes while 3 and 4 represent the new counterpropagat- some resemblance to these studies. Starting from parent V ing modes added to the edge, and IJ is a symmetric matrix non-Abelian systems, Refs. [90,91] investigated descend- that characterizes density-density interactions among all ant phases emerging in the interior of the fluid due to V I 1 J 3 four modes. If IJ is small for ¼ , 2 and ¼ , 4, then interaction among a macroscopic collection of non-Abelian ϕ the additional 3;4 fields generically acquire a gap because anyons. We follow a similar approach, in that the domain ϕ ϕ one of the tunneling perturbations cosð 3 4Þ will be walls in our spatially modulated trenches correspond to relevant [157]. However, when these off-diagonal entries in extrinsic non-Abelian defects [79–81,58,83] by virtue of V IJ are appreciable the edge can enter the new phase that the Z3 zero modes that they bind; moreover, we likewise we seek. hybridize these defects to access the (descendant) To describe this phase, it is convenient to invoke a basis Fibonacci phase within a (parent) ν ¼ 2=3 state. This K~ e WKeWT V~ WVWT change to ¼ and ¼ , where common underlying philosophy suggests a deep relation- 0 1 1010 ship with Refs. [90,91]. B C Of course the most glaring difference stems from the B 01−10C W ¼ B C (86) Abelian nature of our parent state. We will show below that @ −11−10A one can blur this (certainly important) distinction, however, 0001 by developing a nonstandard view of the spin-unpolarized ν ¼ 2=3 quantum Hall state—namely, as emerging from and some non-Abelian phase upon condensation of a boson that 0 1 confines the non-Abelian particles. Such an interpretation 21 0 0 might initially seem rather unnatural but provides an B C e B 12 0 0C illuminating perspective in situations where one can exter- K~ ¼ B C: (87) @ 00−10A nally supply the energy necessary to generate these confined non-Abelian excitations in a meaningful way. This is 00 0 −1 indeed precisely what we accomplish by forcing superconducting islands into the ν ¼ 2=3 fluid to nucleate the V~ 0 I 1 Suppose, for the moment, that IJ ¼ for ¼ , 2 and domain walls that trap Z3 zero modes. We will employ such J ¼ 3, 4. By comparing Eqs. (79) and (87) one sees that the a picture to sharpen the connection with earlier work and, in fermionic edge is then equivalent to the bosonic case the process, develop a TQFT view of the Fibonacci phase examined earlier, supplemented by two Dirac fermion generated within a ν ¼ 2=3 state. In the discussion to modes running in the opposite direction relative to the follow, we ignore the fermion present in the (112) state, chiral modes of the (221) state. This correspondence allows which leads to subtle consequences that we address at the us to immediately deduce that the fermionic Fibonacci-to- end of this section. [In fact, our conclusions will apply vacuum edge is described by the G2 Kac-Moody theory at more directly to the analogous bosonic (221) state.] level one together with two backward-propagating Dirac As a first step, we summarize the results from Ref. [91] fermions (or, equivalently, four backward-propagating that will be relevant for our discussion. Consider a parent ~ Majorana fermions). More generally, when VIJ is small non-Abelian phase described by an SUð2Þ4 TQFT. Table I but nonzero for I ¼ 1, 2 and J ¼ 3, 4, the G2 theory and the lists the properties of the gapped topological excitations of backward-propagating fermions hybridize through the this phase—including the SUð2Þ spin j, conformal spin h, ~ marginal couplings VIJ. Once again, we find a correspon- quantum dimension d, and nontrivial fusion rules for each dence between the bulk and the edge with the vacuum: field. Ludwig et al. found that antiferromagnetically

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TABLE I. Fields of SUð2Þ4, along with their corresponding TABLE II. Properties of SUð2Þ3 and SUð2Þ1 topological SU(2) label j, conformal spin h, quantum dimension d, and quantum field theories that describe the descendant phase on nontrivial fusion rules. The chiral central charge associated with the left side of Fig. 10. In the table, c is the chiral central charge, j c 2 SUð2Þ4 is ¼ . The parent state on the left side of Fig. 10 is is an SU(2) spin label, h denotes conformal spin, d represents the described by this TQFT. quantum dimension, and φ is the golden ratio. c 2 SUð2Þ4 ¼ SUð2Þ3 c ¼ 9=5 0 Field 1 XYXZ Field 1 ε0 εξ j 0 1=2 1 3=2 2 j 0 1=2 1 3=2 h 1=81=35=8 0 pffiffiffi pffiffiffi 1 h 0 3=20 2=53=4 d 1 3 2 3 1 d 1 φφ1

Fusion rules Fusion rules X X ∼ 1 YX0 X0 ∼ 1 Y × þ × þ ε × ε ∼ 1 þ εε0 × ε0 ∼ 1 þ ε 0 0 0 X × Y ∼ X þ X X × Y ∼ X þ X ε × ξ ∼ ε0 ε × ξ ∼ ε X Z ∼ X0 X0 Z ∼ X × × ε × ε0 ∼ ξ þ ε0 ξ × ξ ∼ 1 X × X0 ∼ Z þ YY× Z ∼ Y Y × Y ∼ 1 þ Y þ ZZ× Z ∼ 1 SUð2Þ1 c ¼ 1 Field 1 η coupling a 2D array of non-Abelian anyons in this parent j 0 1=2 state produces a gapped descendant phase described by an h 0 1=4 SUð2Þ3 ⊗ SUð2Þ1 TQFT, as sketched in the left half of d 11 Fig. 10. See Table II for the corresponding properties of SUð2Þ3 and SUð2Þ1. The interface between these parent Fusion rule and descendant phases supports a gapless SUð2Þ3×SUð2Þ1 edge η η ∼ 1 SUð2Þ4 × state, which exhibits central charge c ¼ 4=5 and ten fields corresponding exactly to those of the so-called Mð6; 5Þ minimal model. Note that this edge theory is distinct from (like the non-Abelian defects in our trenches), and the the Z3 parafermion CFT arising in our setup, which descendant SUð2Þ3 ⊗ SUð2Þ1 region supports a Fibonacci possesses only six fields. Nevertheless, there are already anyon (as in our Fibonacci phase). hints here of a relation with our present work: SU 2 ðpÞffiffiffi4 At this point, it is worth speculating on the field content supports non-Abelian anyons with quantum dimension 3 expected from a hypothetical TQFT describing our ν ¼ 2=3 state with domain walls binding Z3 zero modes. First, one should have Abelian fields Y1 and Y2 corresponding to charge-2e=3 and 4e=3 excitations (which can live either on the gapped regions of the trenches or in the bulk of the quantum Hall fluid). Conservation of charge mod 2e suggests the fusion rules Y1 × Y1 ∼ Y2, Y2 × Y2 ∼ Y1, and Y1 × Y2 ∼ 1, where 1 denotes the neutral identity channel. One also might expect non-Abelian fields X~ FIG. 10. Boson condensation picture leading to a TQFT corresponding to domain walls separating pairing- and interpretation of the Fibonacci phase. On the left, a parent tunneling-gapped regions of the trenches. Recalling that the 2e=3 4e=3 non-Abelian SUð2Þ4 phase hosts a descendant SUð2Þ3 ⊗ Cooper-paired regions can carry charge 0, ,or SUð2Þ1 state arising from interacting anyons within the fluid mod 2e, the merger of two adjacent superconducting [91]. Condensing a single boson throughout the system produces islands in a trench should be captured by the fusion rule the setup on the right in which an Abelian Z3 parent state gives X~ X~ ∼ 1 Y Y X~ × þ 1 þ 2. From this perspective, pffiffiffi quite rise to a descendant phase described by a pure Fibonacci TQFT. clearly possesses a quantum dimension of d ¼ 3 (con- The latter system very closely relates to our spin-unpolarized sistent with deductions based on ground-state counting), ν ¼ 2=3 state with superconducting islands that generate the since 1, Y1, and Y2 are Abelian fields with d ¼ 1. No other Fibonacci phase inside of the quantum Hall medium, in that the quasiparticle content (modulo the electron) is identical. An even fields are immediately evident. This picture cannot possibly more precise analogy occurs in the case where the Fibonacci be complete, however, as there is no TQFT with four fields phase resides in a bosonic (221) quantum Hall state; here, the obeying these fusion rules [159]. TQFTs from the right side of the figure exactly describe the The difficulty with identifying a TQFT using the preced- universal topological physics. ing logic stems from the fact that X~ differs fundamentally

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TABLE III. Field content and fusion rules for SUð2Þ4 upon TABLE IV. The fields of Fib, along with their corresponding condensing the bosonic Z field listed in Table I. As in the other conformal spin h, quantum dimension d, and nontrivial fusion tables, j is an SU(2) spin label, h denotes conformal spin, and d rule. This TQFT arises from SUð2Þ3 ⊗ SUð2Þ1 upon condensing represents the quantum dimension for each particle. The X~ field is the boson ðξ; ηÞ in Table II and describes the topologically confined by the condensation and hence exhibits an ill-defined ordered sector of the Fibonacci phase in our ν ¼ 2=3 setup. conformal spin; this field obeys the same fusion rules and projective non-Abelian statistics as the (also confined) domain- Fib c ¼ 14=5 wall defects in our ν ¼ 2=3 trenches. Additionally, Y1 and Y2 Field 1 ε represent Abelian fields that correspond to charge-2e=3 and 4e=3 h 2=5 excitations in our quantum Hall setup. If one ignores the confined 0 ~ d 1 φ excitation X, the remainder is a pure Abelian Z3 theory with only 1, Y1, and Y2 particles. Fusion rule Z SUð2Þ4 with boson condensed ε ε ∼ 1 ε ∼ × þ Field 1 X Y1 Y2 h 1=31=3 0 Ill-definedpffiffiffi d 1 3 11with the new vacuum and therefore must be confined by a physically measurable string. Fussion rules Let us now apply this discussion to the parent SUð2Þ4 ∼ ∼ TQFT described earlier, assuming the Z field condenses. Y1 × Y2 ∼ 1 X × X ∼ 1 þ Y1 þ Y2 ∼ ∼ (From Table I, we see that Z is the only nontrivial boson in Y Y ∼ Y X ∼ Y ∼ X 1 × 1 2 ∼ 1 ∼ the TQFT.) The resulting theory was already discussed Y Y ∼ Y X Y 2 × 2 1 × 2 ∼ X extensively by Bais and Slingerland and will be briefly summarized here. First of all, the fusion rules tell us that condensation of Z identifies X and X0; anticipating a from the other fields in that it does not represent a connection with our ν ¼ 2=3 extrinsic defects, we will ~ pointlike excitation. Rather, X occurs only at the end of a label the corresponding excitation by X~. Indeed, X~ is “string” formed by a superconducting region within our confined (because the conformal spins of X and X0 differ ~ pffiffiffi trenches; since these strings are physically measurable, X by a noninteger), possesses a quantum dimension of 3, is confined and exhibits only projective non-Abelian and exhibits the same projective non-Abelian braiding statistics. One could—at least in principle—envision statistics as our quantum Hall domain-wall defects quantum mechanically smearing out the location of the [58,79–81]. As for the Y field, it can fuse into the vacuum ~ superconductors to elevate X to the status of a deconfined in two different ways when Z condenses (since Z → 1) and pointlike quantum particle belonging to some genuine so must split into two Abelian fields with conformal spin- non-Abelian TQFT. Or by turning the problem on its 2=3 mod 1 [160]. We will denote these two fields as Y1 and ~ head, one can instead view confined excitations like X as Y2, as they exhibit the same characteristics as the charge- remnants of that non-Abelian TQFT after a phase tran- 2e=3 and 4e=3 excitations in our quantum Hall problem. sition. In the latter viewpoint, the mechanism leading to The properties of this “broken SUð2Þ4” theory [160], the transition—and the accompanying confinement—is including the confined X~ excitation, appear in Table III. boson condensation, which was described in detail by From the table, it is apparent that this condensed theory Bais and Slingerland in the context of topologically reproduces exactly the structure anticipated from our ordered phases [160]. ν ¼ 2=3 setup decorated with superconducting islands that To be precise, we will define a boson here as a field generate Z3 zero modes. Hence the fusion rules and possessing integer conformal spin and quantum dimension braiding statistics for our parent state can be viewed as d ¼ 1 [161]. Suppose that a boson B with these properties inherited (projectively) from SUð2Þ4. Note, however, that condenses. When this happens the condensed boson is broken SUð2Þ4 is not a pure TQFT; focusing only on identified with the vacuum 1, and any fields related to one deconfined excitations, we are left with a simple Z3 another by fusion with B are correspondingly identified Abelian theory with only 1, Y1, and Y2. with each other. For instance, if A × B ∼ C then fields A So far, we have shown that the parent SUð2Þ4 theory and C are equivalent in the condensed theory. The fate of discussed by Ludwig et al. recovers the particle content of such fields that are related by the boson B depends on their our parent ν ¼ 2=3 system upon condensing the Z field. relative conformal spin. If their conformal spins differ by an Next we explore the fate of their descendant SUð2Þ3 ⊗ integer, they braid trivially with the new vacuum and SUð2Þ1 phase upon boson condensation. Let us denote represent deconfined excitations. Otherwise it is no longer fields from SUð2Þ3 ⊗ SUð2Þ1 as ðA; BÞ, where A and B possible to define in a gauge-invariant manner the con- respectively belong to SUð2Þ3 and SUð2Þ1, and explore the formal spin for that type of excitation; it braids nontrivially consequences of ðξ; ηÞ condensing. (According to Table II,

011036-29 ROGER S. K. MONG et al. PHYS. REV. X 4, 011036 (2014) this field is indeed bosonic.) Aside from the identity, we phase—exhibits a chiral central charge of c ¼ 2 and is need only consider three fields after condensation—ðε; 1Þ, described by a Z3 TQFT with no modification. The Fib ðε; ηÞ, and ðξ; 1Þ—since all others are related to these by the TQFT denoted on the right side of Fig. 10 also does not condensed boson. The latter two are, however, confined as exactly describe our Fibonacci phase because our state one can deduce by examining their conformal spin before exhibits a local order parameter (and hence is not strictly and after fusing with ðξ; ηÞ. The lone deconfined field that described by any TQFT). This difference poses a far more remains is ðε; 1Þ, which is described by a pure Fibonacci minor issue than those noted above. Recall from Sec. VC theory. Table IV summarizes the main features of this that superconducting vortices do not generate additional TQFT, denoted here by “Fib.” This theory is analogous to nontrivial quasiparticles in the Fibonacci phase. that describing the descendant Fibonacci phase that we Consequently, the order-parameter physics “factors out” obtained by hybridizing arrays of Z3 zero modes in our and essentially decouples from the topological sector. More parent ν ¼ 2=3 system. formally, one can envision quantum disordering the super- While it is not yet apparent, the condensation transitions conductor by condensing vortices to eradicate the order that we discussed separately in the parent and descendant parameter altogether without affecting the quasiparticles phases are, in fact, intimately related. This connection supported by the Fibonacci phase that we have con- becomes evident upon examining (from a particular point structed [162]. of view) the structure of the Mð6; 5Þ minimal model The TQFT perspective on our results espoused in this describing the boundary between the pure SUð2Þ4 and section has a number of virtues. For one, it clearly SUð2Þ3 ⊗ SUð2Þ1 phases prior to the transitions. illustrates the simplicity underlying the end product of Appendix B shows that at that boundary the Z and our construction and also unifies several related works that ðξ; ηÞ bosons are identified, which is reasonable since their may, at first glance, appear somewhat distantly related. SU(2) spin labels, conformal spins, and quantum dimen- Another benefit is that the condensation picture used sions all match. Thus, one can move the Z boson smoothly along the way naturally captures the confined non- ν 2=3 from the parent to the descendant region, where it Abelian domain-wall defects supported by ¼ trenches with superconductivity. More generally, viewing “becomes” ðξ; ηÞ—or vice versa. It follows that the tran- Abelian phases as remnants of non-Abelian TQFTs as we sitions in the parent and descendant phases are not have done here may be useful in various other settings as a independent but rather can be viewed as arising from the way of similarly identifying nontrivial phases accessible condensation of a single common boson. from interacting extrinsic defects. Figure 10 summarizes the final physical picture that we obtain. The left-hand side represents the parent SUð2Þ 4 VII. FIBONACCI PHASE FROM with a descendant SUð2Þ3 ⊗ SUð2Þ1 setup analyzed by Ludwig et al. [91], which exhibits quite different physics UNIFORM TRENCHES from what we captured in this paper. Condensing a single In Sec. II, we identified two closely linked routes to boson throughout that system leads to the parent Z3 with spinless p þ ip superconductivity from an integer quantum descendant Fib configuration illustrated on the right side of Hall system. The first utilized trenches with spatially the figure. These parent and descendant states do, by uniform Cooper-pairing and electron-backscattering per- contrast, closely relate to our ν ¼ 2=3 quantum Hall setup turbations present simultaneously; the second considered with superconducting islands that drive the interior into the trenches alternately gapped by pairing and backscattering, Fibonacci phase, in the sense that both systems exhibit the yielding chains of hybridized Majorana modes. In either same deconfined bulk excitations in each region. There are, case the trenches could be tuned to an Ising critical point, at however, subtle differences between the system on the right which interchain coupling then naturally generates side of Fig. 10 and our specific quantum Hall architecture p þ ip superconductivity. To construct a superconducting that deserve mention. Z3 Read-Rezayi analogue (the Fibonacci phase), Secs. IV First, the Abelian Z3 TQFT technically does not quite and V adopted the second approach and analyzed chains of describe the spin-unpolarized ν ¼ 2=3 state: The theory Z3 generalized Majorana modes nucleated in a ν ¼ 2=3 must be augmented to accommodate the electron in this fractional quantum Hall fluid. This route enabled us to fermionic quantum Hall phase [159]. Moreover, the edge exploit the results of Ref. [116], which derived the relation structure for the Z3 TQFT admits a chiral central charge between lattice and CFT operators at the Z3 parafermion c ¼ 2, whereas the ν ¼ 2=3 state has c ¼ 0 (because there critical point for a single chain, to controllably study the 2D are counterpropagating modes). Both of these issues are coupled-chain system. Here we will argue that as in the relatively minor for the purposes of our discussion, how- integer quantum Hall case the same physics can also be ever, and in any case can easily be sidestepped by obtained from spatially uniform ν ¼ 2=3 trenches. This considering a bosonic parent system. In particular, as outcome is eminently reasonable since on the long length alluded to earlier, the bosonic (221) state—which provides scales relevant at criticality the detailed structure of the an equally valid backdrop for the descendant Fibonacci trenches should become unimportant.

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The analysis proceeds in two stages. First, we will use similarities hint at common critical behavior for the two results from Lecheminant, Gogolin, and Nersesyan (LGN) models. [163] to argue that a ν ¼ 2=3 trench with uniform pairing The simplest way to make this relation precise is to and backscattering perturbations also supports a Z3 paraf- include a perturbation that explicitly gaps the spin sector ermion critical point. The relation between bosonized fields (while leaving the charge sector intact). One such pertur- and CFT operators at criticality will then be deduced by bation arisesR from correlated spin-flip processes described † † coarse graining the corresponding relationship obtained in by δH ¼ xðΓψ 1↑ψ 2↓ψ 2↑ψ 1↓ þ H:c:Þ, where ψ 1α and ψ 2α Sec. IV in the spatially nonuniform case. At that stage our are spin-α electron operators acting on the top and bottom results from Sec. V carry over straightforwardly, allowing sides of the trench, respectively. In bosonized language this us to immediately deduce the existence of a Fibonacci term takes the form phase in the uniform-trench setup. We start by reviewing the critical properties [163] for a Z toy Hamiltonian of the form δH ¼ uσ cosð2θσÞ: (91) Z x v H ∂ ϕ 2 ∂ θ 2 u 3θ LGN ¼ ½ð x Þ þð x Þ þ 1 cosð Þ x 2π Suppose that the coupling uσ dominates over t and Δ and drives an instability in which θσ is pinned by the cosine þ u2 cosð3ϕÞ ; (88) potential above. At low energies the Hamiltonian H in Eq. (90) that describes the remaining charge degrees of where the fields satisfy [164] freedom then maps onto the LGN Hamiltonian in Eq. (88). Consequently, the self-dual critical point at which jtj¼jΔj 2πi is likewise described by Z3 parafermion CFT. θ x ; ϕ x0 − Θ x0 − x : ½ ð Þ ð Þ ¼ 3 ð Þ (89) For the following reasons, we believe that it is likely that the same critical physics arises without explicitly invoking u H u t Δ The 1;2 perturbations in LGN are both relevant at the the σ perturbation. Recall that both and favor pinning Gaussian fixed point and favor locking θ and ϕ to the three the spin-sector field θσ in precisely the same fashion but distinct minima of the respective cosines. Because of the gap the charge sector in incompatible ways [see Eqs. (44) nontrivial commutator above, however, these terms com- and (45)]. Suppose that we start from a phase in which pete and favor physically distinct gapped phases—very tunneling t gaps both sectors. Increasing Δ at fixed t must much like the tunneling and pairing terms in our quantum eventually induce a phase transition in the charge sector. Hall trenches. Using complementary nonperturbative meth- Provided the spin sector remains gapped throughout, it ods, LGN showed that the self-dual limit corresponding to suffices to replace the cos θσ term in Eq. (90) by a constant u1 ¼ u2 realizes the same Z3 parafermion critical point as across the transition. The model then once again reduces to H Z the three-state quantum clock model [163]. LGN and hence exhibits a 3 parafermion critical point at To expose the connection to our quantum Hall setup, jtj¼jΔj. We stress that although it is difficult to make consider the Hamiltonian introduced in Sec. IVA for a rigorous statements about this nontrivial, strongly coupled single trench in a ν ¼ 2=3 fluid with backscattering and field theory, this outcome is nevertheless intuitively very Cooper pairing induced uniformly: natural given our results for criticality in spatially modu- Z lated trenches. X “ ” va 2 2 Our primary interest lies in stacking such critical 1D H ¼ ½ð∂xϕaÞ þð∂xθaÞ x 2π systems to access new exotic 2D phases. Physical inter- a¼ρ;σ chain perturbations can easily be constructed in terms of bosonized fields, as in Sec. V, although at the Z3 paraf- þ 4 cos θσ½t sinð3θρÞ − Δ sinð3ϕρÞ : (90) ermion critical point these fields no longer constitute the right low-energy degrees of freedom. An essential technical As before ϕρ=σ and θρ=σ represent fields for the charge and step is identifying the correspondence between bosonized spin sectors, while t and Δ denote the tunneling and pairing and CFT operators at criticality so that one can systemati- strengths. In writing the first line of H we have assumed a cally disentangle high- and low-energy physics. We will particularly simple form for edge density-density inter- now deduce this relationship for quasiparticle creation actions that can be described with velocities vρ=σ. Upon operators that are relevant for interchain processes in our comparison of Eqs. (42) and (89) one sees that the charge- ν ¼ 2=3 setup with uniform trenches. sector fields obey the same commutation relation as those To do so we first revisit the nonuniform system analyzed in the model studied by LGN. Furthermore, modulo the in Sec. IV. By combining Eqs. (48), (60a), and (60b),we spin-sector parts, the u1;2 perturbations in Eq. (88) have the obtain the following expansions valid at the parafermion same form as the tunneling and pairing terms above. These critical point:

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i½ϕ1↑ðxjÞþϕ1↓ðxjÞ j e ∼ að−1Þ ψ R þ bσRϵL þ; VIII. SUMMARY AND DISCUSSION

i½ϕ2↑ðxjÞþϕ2↓ðxjÞ iπ=3 j e ∼ e ½að−1Þ ψ L þ bσLϵRþ: The introduction to this paper provided a broad overview (92) of the main physical results derived here. Having now completed the rather lengthy analysis, we will begin this discussion with a complementary and slightly more tech- We remind the reader that the operators on the left-hand nical summary. side create charge-2e=3 quasiparticles on the top and Our setup begins with a spin-unpolarized ν ¼ 2=3 — bottom trench edges, at position xj in domain wall j. Abelian fractional quantum Hall state also known as — [ϕ1=2α relates to the charge- and spin-sector fields through the (112) state as the backbone of our heterostructure. Eqs. (41).] Moreover, on the right side a and b again denote The (112) state is a strongly correlated phase built from nonuniversal constants while the ellipses represent terms spin-up and spin-down electrons partially occupying their with subleading scaling dimensions. Connection with the lowest Landau level. At the boundary with the vacuum, its uniform trench can now be made upon coarse graining the edge structure consists of a charge mode (described by expressions above—specifically, by averaging over sums ϕ↑ þ ϕ↓) and a counterpropagating neutral mode and differences of quasiparticle operators at adjacent (described by ϕ↑ − ϕ↓). We first showed that a long domain walls in a given unit cell. (Each unit cell contains rectangular hole—a “trench”—in this fractional quantum two domains, as shown in Fig. 5.) The oscillating terms Hall system realizes a Z3 parafermion critical point when s clearly cancel for the sum, leaving coupled to an ordinary -wave superconductor. This nontrivial critical theory with central charge cL ¼ cR ¼ 4=5 is well known from earlier studies of the three-state quantum i½ϕ1↑ðxÞþϕ1↓ðxÞ e ∼ σRϵL þ; clock model, and moreover is important for characterizing

i½ϕ2↑ðxÞþϕ2↓ðxÞ iπ=3 edge states of the Z3 Read-Rezayi phase whose properties e ∼ e σLϵR þ; (93) we sought to emulate. We presented two related construc- tions. The first utilizes an alternating pattern of super- where x now denotes a continuous coordinate. One can conducting and nonsuperconducting regions in the trench, isolate the parafermion fields by instead averaging over as described in Sec. IV, to essentially engineer a nonlocal differences of quasiparticle operators at neighboring representation of the three-state clock model. The second, domain walls, which yields explored in Sec. VII, employs a “coarse-grained” variation wherein the trench couples uniformly to a superconductor throughout. Tuning to the Z3 parafermion critical point ∂ ei½ϕ1↑ðxÞþϕ1↓ðxÞ ∼ ψ ; x R þ follows by adjusting the coupling between domain walls i½ϕ2↑ðxÞþϕ2↓ðxÞ iπ=3 ∂xe ∼ e ψ L þ: (94) (in the case of modulated trenches) or electron tunneling across the trench (in the uniform-trench setup). In both scenarios, the neutral excitations are gapped out while the The extra derivatives on the left-hand side reflect the charge modes provide the low-energy degrees of freedom. fact that the parafermions acquire a relative minus sign One remarkable feature of our mapping is that we can under parity P compared to the fields on the right sides of identify the relation between “high-energy” operators and Eqs. (93) [116]. More generally, the coarse-graining chiral fields describing low-energy physics near criticality, procedure used here merely ensures that the quantum given by Eqs. (27) for the lattice construction and Eqs. (93) numbers carried by the bosonized and CFT operators agree and (94) for the continuum version. This key technical step with one another. enabled us to perform calculations parallel to those for the We are now in a position to recover the physics discussed coupled Majorana chains described in Sec. II—but at a in Sec. V from a system of spatially uniform critical nontrivial strongly interacting critical point. trenches. Equations (93) and (94) allow us to construct To construct a 2D non-Abelian phase reminiscent of the interchain quasiparticle hoppings that reproduce the λa;b Z3 Read-Rezayi state, we consider an array of these critical terms in Eq. (67). The effective low-energy Hamiltonians trenches in the ν ¼ 2=3 quantum Hall fluid, with neighbor- in the two closely related setups are then identical—and ing trenches coupled via charge-2e=3 quasiparticle hopping hence so are the resulting phase diagrams. In particular, as (see Fig. 6 for the lattice setup). With the correspondence Fig. 6(a) illustrates the uniform-trench system flows to the between quasiparticle operators and CFT fields in hand, we Fibonacci phase if the interchain coupling λa > 0 domi- find that the second-most-relevant interchain coupling nates. Determining the microscopic parameters (in terms of corresponds to a term that couples the right-moving the underlying electronic system) required to enter this parafermion field ψ RðyÞ from trench y with the adjacent phase remains an interesting open issue, although such a left mover ψ Lðy þ 1Þ from trench y þ 1. This perturbation state is in principle physically possible in either setup that gaps out each critical trench except for the first right mover we have explored. and the final left mover. The system then enters a stable 2D

011036-32 UNIVERSAL TOPOLOGICAL QUANTUM COMPUTATION ... PHYS. REV. X 4, 011036 (2014) chiral topological state, as shown in Sec. V, which we does not influence the Fibonacci phase’s universal topo- dubbed the Fibonacci phase. Since this phase exhibits a logical properties. The separation of superconductivity and bulk gap its topological properties are stable; therefore, it is topological properties here stands in stark contrast with the neither necessary to tune the individual chains exactly to case of a spinless p þ ip superconductor. There, an h=2e criticality nor to precisely set the most relevant interchain superconducting vortex binds a Majorana zero mode and coupling to zero. thus exhibits many characteristics of σ particles (i.e., Ising We uniquely established the universal topological prop- anyons), despite being logarithmically confined by order- erties of the Fibonacci phase by identifying its twofold parameter energetics. If superconductivity is destroyed by ground-state degeneracy on a torus (which implies two the condensation of double-strength h=e vortices, then the anyon species), fusion rules, and quantum dimensions via h=2e vortex becomes a bona fide deconfined σ particle in the topological entanglement entropy. The quasiparticle the resulting insulating phase. On the other hand, destroy- structure present here is elegant in its simplicity yet rich in ing superconductivity by condensing single-strength h=2e content, consisting of a trivial particle 1 and a Fibonacci vortices produces a trivial phase. The physics is completely anyon ε obeying the simple fusion rule ε × ε ∼ 1 þ ε.One different in the Fibonacci phase, where an h=2e vortex of the truly remarkable features of this state is that the braids trivially with an ε particle. (Here we assume that the ability to exchange Fibonacci anyons, and to distinguish the vortex does not “accidentally” trap a Fibonacci anyon.) Fibonacci anyon from the vacuum, is sufficient to perform Condensation of h=2e vortices therefore simply leaves the any desired quantum computation in a completely fault- pure Fibonacci phase with no residual order-parameter tolerant manner [148,149]. physics. It is interesting to note that richer physics arises The Fibonacci phase supports gapless edge excitations. upon condensing nh=2e vortices, which yields the When this state borders the parent Abelian quantum Hall Fibonacci phase tensored with a Zn gauge theory; this fluid from which it descends [as in Fig. 1(b)], the edge additional sector is, however, clearly independent of the modes are described by a chiral Z3 parafermion CFT with Fibonacci phase. central charge c ¼ 4=5—exactly as in the Z3 Read-Rezayi A number of similarities exists between our Fibonacci phase modulo the charge sector. The edge states arising at phase and previously constructed models that harbor the interface with the vacuum can be obtained upon Fibonacci anyons. We have already emphasized several shrinking the outer Abelian quantum Hall liquid, thereby parallels with the Z3 Read-Rezayi state. Teo and Kane’s hybridizing the parafermion and quantum Hall edge fields. coupled-wire construction of this non-Abelian quantum Thus, the Fibonacci-phase-to-vacuum edge is roughly the Hall phase is particularly close in spirit to this paper (and product of the Abelian quantum Hall edge and the indeed motivated many of the technical developments used parafermion theory. If the Fibonacci phase descends from here). The Z3 Read-Rezayi state, however, certainly rep- a bosonic analogue of the spin-unpolarized ν ¼ 2=3 state, resents a distinct state of matter with different universal i.e., the bosonic (221) state, then the boundary with the topological properties. For instance, there the fields ψ and σ vacuum exhibits edge modes described by the G2 Kac- (with appropriate bosonic factors) represent deconfined, Moody algebra at level one. This fully chiral edge theory electrically charged quasiparticles, whereas the Fibonacci has central charge c ¼ 14=5, contains two primary fields anyon ε provides the only nontrivial quasiparticle in the associated with the bulk excitations 1 and ε, and occurs also Fibonacci phase. Fibonacci anyons also occur in the exactly in the pure Fibonacci topological quantum field theory soluble lattice model of Levin and Wen [165]. Important discussed in Sec. VI. If instead the Fibonacci phase differences arise here, too: Their model is nonchiral and has emerges out of the fermionic (112) state, then the corre- the same topological properties as two opposite-chirality sponding edge is not fully chiral and does not, in general, copies of the Fibonacci phase constructed in this paper. admit a decomposition into independent left and right (See also the related works of Refs. [166,167] for loop-gas movers. However, we find that the edge theory may be models that may support such a nonchiral phase.) Finally, reconstructed such that it factorizes into two left-moving recent unpublished work by Qi et al. accessed a phase with c 2 fermions with central charge L ¼ and a right-moving Fibonacci anyons using Zn lattice operators as building sector identical to the bosonic case with central blocks, similar to those that arise in our spatially modulated charge cR ¼ 14=5. trenches [168]. It would thus be interesting to explore Because of the superconductivity in our setup, the possible connections with our study. Fibonacci phase admits gapless order-parameter phase We now turn to several other outstanding questions and fluctuations but is otherwise fully gapped away from the future directions raised by our results, placing particular edge. Nevertheless, its low-energy Hilbert space consists of emphasis on experimental issues. a tensor product of states for a topologically trivial super- Realizing non-Abelian anyons with universal braid conductor and those of a gapped topological phase. In this statistics in any setting carries great challenges yet corre- sense the superconductivity is peripheral: It provides an spondingly great rewards if those challenges can be over- essential ingredient in our microscopic construction but come. Our proposal is no exception. The price that one

011036-33 ROGER S. K. MONG et al. PHYS. REV. X 4, 011036 (2014) must pay to realize Fibonacci anyons as we envision here is As a final remark on experimental realizations, we stress that a fractional quantum Hall system must intimately that superconductivity may be altogether inessential—even contact an s-wave superconductor. For several reasons, at the microscopic level. To see why, it is useful to recall however, accessing the Fibonacci phase may be less that the superconductors in our construction simply provide daunting than it appears. First of all, Abelian fractional a mechanism for gapping the edge states opposite a trench quantum Hall states appear in many materials—and not just that is “incompatible” with the gapping favored by ordinary in buried quantum wells such as GaAs. Among several electronic backscattering. When balanced, these competing possible canvases, graphene stands out as particularly terms thus drive the system to a nontrivial critical point that promising due to the relative ease with which a proximity we bootstrapped off of to enter the Fibonacci phase. In effect can be introduced [169–171]. Graphene can also be beautiful theoretical studies, Refs. [58,83] showed that grown on metallic substrates [172]; if such a substrate similar incompatible gap-generating processes can arise in undergoes a superconducting transition, a strong proximity certain quantum Hall bilayers without Cooper pairing; for effect may result. instance, if one cuts a trench in the bilayer, electrons can Another point worth emphasizing is that weak magnetic backscatter by tunneling from “top to bottom” or “side to fields are not required, which is crucial given that our side.” It may thus be possible to realize the Fibonacci phase proposal relies on the fractional quantum Hall effect. This in a bilayer fractional quantum Hall setup by regulating the robustness stems from the fact that superconducting vor- interlayer and intralayer tunneling terms along trenches, tices in the Fibonacci phase need not carry topologically following Refs. [58,83]. Such an avenue would provide nontrivial particles. Assuming that Fibonacci anyons do not — another potentially promising route to Fibonacci anyons happen to energetically bind to vortex cores which again that is complementary to the superconductor-quantum Hall they need not—then any field strength up to the (type-II) heterostructures that we focused on here. superconductor’s upper critical field Hc2 should suffice. By Our construction naturally suggests other interesting contrast, in the case of a spinless p ip superconductor, þ generalizations as well. The ν ¼ 2=3 state is not the the density of vortices must remain low because they only spin-singlet fractional quantum Hall phase— necessarily support Majorana modes. Appreciable tunnel- another can occur, e.g., at ν ¼ 2=5. These systems ing between these modes, which will arise if the spacing may provide equally promising platforms for the between vortices becomes too small, therefore destabilizes Fibonacci phase or relatives thereof. Moreover, our the Ising phase. construction is by no means limited to fermionic We also reiterate that preparing precisely the somewhat quantum Hall phases. As we noted earlier the bosonic elaborate, fine-tuned setups explored here is certainly not (221) state, for instance, leads to nearly identical necessary for accessing the Fibonacci phase. Many of the physics (which is actually simpler in some respects). — features we invoked in our analysis including the multi- By following a similar route to that described here, it trench geometry and all of the fine-tuning that went with may be possible to build on these quantum Hall states — it served purely as a theoretical crutch that enabled us to to construct other non-Abelian topological phases, decisively show that our model supports this state and perhaps realizing Zk parafermions, SUð2Þk, or yet more identify its properties. The Fibonacci phase is stable to (at exotic phases. least) small perturbations, and the extent of its stability To conclude, we briefly discuss the longer-term remains a very interesting open question. It seems quite prospects of exploiting our model for quantum compu- possible that this stability regime extends across a large tation. Quantum information can be encoded in a many-ε swath of the parameter space for a quantum Hall state state using either a dense or sparse encoding. There are coupled to a superconductor. Hinting at this robustness is two states of three ε particles with total charge ε and also the fact that the Fibonacci phase that we have constructed is two states of four ε particles with total charge 1,and actually isotropic and translationally invariant in the long- either set can be used as a qubit. The unitary trans- wavelength limit. Hence, it is even possible that a com- formations generated by braiding are dense within the pletely “smeared” Abelian quantum Hall-superconductor projective unitary group on the many-anyon Hilbert heterostructure enters this phase even in the absence of space and, therefore, within the unitary group on the trenches. Although the methods used in this paper are not computational subspace [148,149]. However, this pre- applicable to this case, it may be possible to study such a supposes that we can create pairs of Fibonacci anyons at scenario by applying exact diagonalization or the density- will, and braid and detect them—which is challenging matrix renormalization group to small systems of electrons since they carry neither electric charge nor any flux. In in the lowest Landau level. Numerical studies along these this respect, the rather featureless ε particles are analo- lines are analogous to previous studies of the fractional gous to ψ particles in an Ising-anyon phase. This quantum Hall effect but with the added wrinkle that U(1) analogy suggests the following approach. Consider the charge-conservation symmetry is broken. This almost case of a single Ising or three-state clock model on a entirely untapped area seems ripe for discovery. ring. If we make one of the bond couplings equal to −∞,

011036-34 UNIVERSAL TOPOLOGICAL QUANTUM COMPUTATION ... PHYS. REV. X 4, 011036 (2014) then it breaks the ring into a line segment, and the spins Hamiltonian describing the hybridization of these modes is at the two ends are required to have opposite values. In given in Eq. (57). Below we identify the realization of the the Ising case, if one end is “spin-up,” the other is clock-model symmetries in this specific setup. The results therefore “spin-down,” and vice versa, forcing a ψ into apply straightforwardly to the multitrench case as well. the chain. However, this particle is not localized and can Note that we frequently make reference to the bosonized move freely. If we now couple many such chains, some fields and to the integer operators describing their pinning of which have ψ’s, then they can also move between induced by tunneling t or pairing Δ, defined in Sec. IV. chains and annihilate. However, we can in principle trap (i) In the limit where Δ ¼ t ¼ 0, the electron number on a ψ by reducing the gap at various locations. In the Z3 each side of the trench is separately conserved. This is clock case, if one end of a chain is A, then the other end reflected in independent global Uð1Þ symmetries that is “not-A.” (Here we are calling the three states A, B,and send θρ → θρ þ a1 and ϕρ → ϕρ þ a2 for arbitrary C.) This boundary condition forces an ε particle into a constants a1;2. Restoring Δ and t to nonzero values single chain. It is plausible that when the chains are breaks these continuous symmetries down to a pair of coupled through their parafermion operators, these ε discrete Z3 symmetries, which is immediately appar- particles will be able to move freely between chains. ent from Eq. (43). The remaining invariance under They could then similarly be trapped by locally sup- ϕρ → ϕρ þ 2π=3, which transforms nˆ j → 1 þ nˆ j, cor- pressing the gap, as in the Ising case. Showing that this responds to the clock-model symmetry Z3; similarly, scenario is correct or designing an alternate protocol for the transformation θρ → θρ þ 2π=3 sends mˆ j → 1 þ mˆ j dual manipulating Fibonacci anyons poses an important and corresponds to Z3 . challenge for future work. (ii) The symmetry Tx corresponds to a simple translation mˆ → mˆ nˆ → nˆ along the trench that shifts j jþ1 and j jþ1. ACKNOWLEDGMENTS (iii) In the clock model, parity P corresponds to a reflection We are grateful to Parsa Bonderson, Alexey Gorshkov, that interchanges the generalized Majorana operators Victor Gurarie, Roni Ilan, Lesik Motrunich, Hirosi Ooguri, αRj and αLj. Since the analogous operators defined in John Preskill, Miles Stoudenmire, and Krysta Svore for Eqs. (49) involve quasiparticles from opposite sides of illuminating conversations. This work was supported in the trench, here the equivalent of P corresponds to a π part by the NSF under Grant No. PHYS-1066293 and the rotation in the plane of the quantum Hall system. We hospitality of the Aspen Center for Physics, where the idea seek an implementation of this rotation that leaves the ρ S for this work was conceived. We also acknowledge funding total charge and spin densities þ and þ invariant, from the NSF through Grants No. DMR-1341822 (D. J. C. changes the sign of the density differences ρ− and S−, and J. A.), No. DMR-MPS1006549 (P. F.), No. DMR- and preserves the bosonized form of δH in Eq. (43). 0748925 (K. S.), and No. DMR-1101912 (M. F.); the The following satisfies all of these properties: Alfred P. Sloan Foundation (J. A.); the Sherman θρðxÞ → −θρð−xÞ − π=3, ϕρðxÞ → ϕρð−xÞþ4π=3, Fairchild Foundation (R. M.); the DARPA QuEST program θσðxÞ → −θσð−xÞ, and ϕσðxÞ → ϕσð−xÞ.(Wehave (C. N. and K. S.); the AFOSR under Grant No. FA9550-10- included the factor of 4π=3 in the transformation of φρ 1-0524 (C. N.); the Israel Science Foundation (Y. O.); the so that the generalized Majorana operators in our Paul and Tina Gardner Fund for the Weizmann-TAMU quantum Hall problem transform as in the clock model Collaboration (Y. O.); the U.S.–Israel Binational Science under P. This factor transforms all electron operators Foundation (A. S.); the Minerva Foundation (A. S.); trivially and thus corresponds to an unimportant global Microsoft Research Station Q; and the Caltech Institute gauge transformation.) Taking the rotation about the for Quantum Information and Matter, an NSF Physics midpoint of a pairing-gapped section, the integer ˆ ˆ Frontiers Center with support of the Gordon and Betty operators transform as M → −M, mˆ j → −mˆ −j−1, ˆ Moore Foundation. and nˆ j → nˆ −j þ M þ 2 under this operation. (iv) Charge conjugation C arises from a particle-hole † APPENDIX A: SYMMETRIES IN THE transformation on the electron operators ψ 1α → ψ 1α † QUANTUM HALL SETUP and ψ 2α → −ψ 2α, which leaves the perturbations in Eq. (40) invariant. In bosonized language, this oper- The quantum clock model reviewed in Sec. III exhibits a dual ation maps θρ → −θρ − π=3, ϕρ → −ϕρ þ π=3, number of symmetries, preserving Z3 and Z3 trans- θσ → −θσ, and ϕσ → −ϕσ. The integer operators in formations, translations Tx, parity P, charge conjugation C, Mˆ → −Mˆ mˆ → −mˆ nˆ → −nˆ and a time-reversal transformation T . In this Appendix we turn transform as , j j,and j j C illustrate that each of these symmetries exhibits a physical under . Note that it is easy to imagine adding perturba- analogue in the quantum Hall architectures discussed in tions that violate this symmetry in the original edge Secs. IVand V. To this end consider the geometry of Fig. 5, Hamiltonian (e.g., spin flips acting on one side of the in which a single trench hosted by a ν ¼ 2=3 system yields trench); however, such perturbations project trivially into a chain of coupled Z3 generalized Majorana operators; the theground-statemanifold.Hence,oneshouldviewC asan

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emergent symmetry valid in the low-energy subspace in assume throughout this Appendix that the bulk properties of which we are interested. the parent and descendant phases remain intact. (v) Finally, for the equivalent of the clock-model sym- Ignoring chirality for the moment, we find only three metry T we need to identify an antiunitary trans- such bosonic combinations (i.e., triplets with integer formation exhibited by our ν ¼ 2=3 setup that squares conformal spin and quantum dimension d ¼ 1). They to unity in the ground-state subspace and swaps the are ð1; 1;ZÞ, ðξ; η; 1Þ, and ðξ; η;ZÞ. The right- and left- αRj and αLj operators. Physical electronic time removing conformal dimensions of these fields are respec- T R versal ph composed with a reflection y about the tively given by (0, 1), (1, 0), and (1, 1). Consequently, the length of the trench (which can be a symmetry for first two fields form chiral bosons and so cannot condense electrons in a magnetic field) has precisely these without an accompanying bulk phase transition in the properties—i.e., T ¼ T Ry. This operation trans- — ph y parent or nucleated liquid which again we preclude here. forms the electron operators as ψ 1α → iσαβψ 2β and ξ; η;Z Z y The last field ð Þ represents a nonchiral 2 boson, and ψ 2α → iσαβψ 1β and sends the bosonized fields to when condensed results in the Mð6; 5Þ minimal model on θρ → θρ, ϕρ → −ϕρ þ π=3, θσ → −θσ, and the edge, as we now argue. ϕσ → ϕσ þ π. The integer operators correspondingly To see this result note that one can divide the 40 fields of T Mˆ → −Mˆ mˆ → mˆ Mˆ transform under as , j j þ , and SUð2Þ3 ⊗ SUð2Þ1 ⊗ SUð2Þ4 into sets of fields Ai and Bi nˆ → −nˆ j j. Notice that whereas this composite oper- (with i ¼ 1; :::::; 20) related by fusion with the Z2 boson ation squares to −1 when acting on the original ðξ; η;ZÞ which we now assume condenses. That is, electron operators, in the projected subspace T R 2 1 ð ph yÞ ¼þ as desired. Ai × ðξ; η;ZÞ ∼ Bi;

Bi × ξ; η;Z ∼ Ai: (B1) APPENDIX B: Mð6;5Þ EDGE STRUCTURE ð Þ VIA BOSON CONDENSATION This identification reduces the number of fields from 40 to This Appendix deals with the setup shown in the left side 20—still more than are present in the Mð6; 5Þ minimal of Fig. 10, in which a parent state described by an SUð2Þ4 model. There is, however, an additional criterion that one TQFT hosts a descendant SUð2Þ3 ⊗ SUð2Þ1 phase [91]; needs to consider. Namely, only when the conformal spins see Tables I and II for summaries of the field content in of Ai and Bi match (mod 1) can a well-defined spin be each region. Our specific goal is to substantiate the claim assigned to the new field Ai ≡ Bi following the condensa- made in Sec. VI that the Z and ðξ; ηÞ bosons supported in tion of ðξ; η;ZÞ; otherwise, those fields become confined. the bulk of the parent and descendant states, respectively, One can readily verify that there are ten pairs of fields Ai are equivalent at their interface. [We are again using and Bi for which the conformal spins agree in the above ⊗ notation where fields from SUð2Þ3 SUð2Þ1 are labeled sense, and these deconfined fields correspond to the ten A; B A B ð Þ, with in SUð2Þ3 and in SUð2Þ1.] To meet this fields of the Mð6; 5Þ minimal model. objective we will describe how one can recover, via edge This picture of Mð6; 5Þ as an SUð2Þ ⊗ SUð2Þ ⊗ M 6; 5 3 1 boson condensation, the ð Þ minimal model describ- SUð2Þ4 edge theory with ðξ; η;ZÞ condensed is very useful. ing gapless modes at the interface between the parent and In particular, since ð1; 1;ZÞ × ðξ; η;ZÞ ∼ ðξ; η; 1Þ, it fol- descendant phases. As we will see, this viewpoint makes lows that the Z and ðξ; ηÞ bosons native to the parent and Z ξ; η the identification of the and ð Þ bosons immediately descendant phases are indeed identified at their interface, obvious. which is what we set out to show. First, observe that the gapless modes bordering SUð2Þ4 and SUð2Þ3 ⊗ SUð2Þ1 topological liquids are naively captured by an SUð2Þ3 ⊗ SUð2Þ1 ⊗ SUð2Þ4 CFT, where the overline indicates a reversed chirality. For concreteness we will assume that the sector with an overline describes [1] J. M. Leinaas and J. Myrheim, On the Theory of Identical left movers while others correspond to right movers. Particles, Nuovo Cimento Soc. Ital. Fis. 37B, 1 (1977). Adopting similar notation as above, we describe fields [2] F. Wilczek, Magnetic Flux, Angular Momentum, and from the product edge theory as triplets of fields from the Statistics, Phys. Rev. Lett. 48, 1144 (1982). constituent sectors, e.g., ðε; η;XÞ. (Note that this Appendix [3] F. A. Bais, Flux Metamorphosis, Nucl. Phys. B170,32 will employ the same symbols for primary fields at the (1980). [4] G. A. Goldin, R. Menikoff, and D. H. Sharp, Representa- interface and bulk anyons to facilitate the connection with — tions of a Local Current Algebra in Non-simply Connected Sec. VI.) In total, 40 such triplets exist far more than the Space and the Aharonov-Bohm Effect, J. Math. Phys. M 6; 5 ten fields found in ð Þ. Any nonchiral boson in this (N.Y.) 22, 1664 (1981). edge theory can, however, condense at the interface, thereby [5] G. A. Goldin, R. Menikoff, and D. H. Sharp, Comments on reducing the number of distinct deconfined fields. To avoid “General Theory for Quantum Statistics in Two possible confusion, we stress that in contrast to Sec. VI we Dimensions,” Phys. Rev. Lett. 54, 603 (1985).

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Tunneling of Non-Abelian Quasiparticles in the ν ¼ 5=2 [164] We use rescaled fields compared to Ref. [163] to highlight Quantum Hall State and p þ ip Superconductors, Phys. the relationship with our ν ¼ 2=3 problem. Rev. B 75, 045317 (2007). [165] M. A. Levin and X.-G. Wen, String-Net Condensation: [152] E. Grosfeld and A. Stern, Observing Majorana Bound Physical Mechanism for Topological Phases, Phys. Rev. B States of Josephson Vortices in Topological Superconduc- 71, 045110 (2005). tors, Proc. Natl. Acad. Sci. U.S.A. 108, 11 810 (2011). [166] L. Fidkowski, M. Freedman, C. Nayak, K. Walker, and Z. [153] Superconductivity technically does not allow for continu- Wang, From String Nets to Non-Abelions, Commun. Math. ous ramping of the flux, but this barrier can be easily Phys. 287, 805 (2009). avoided. For the purpose of this thought experiment, one [167] P. Fendley, S. V. Isakov, and M. Troyer, Fibonacci can imagine temporarily snaking the flux so that it threads Topological Order from Quantum Nets, Phys. Rev. Lett. the ν ¼ 2=3 regions but avoids passing through the 110, 260408 (2013). trenches. Once a value of h=2e is reached, the flux can [168] X.-L. Qi, H. Jiang, M. Barkeshli, and R. Thomale (to be then be moved entirely within the cylinder. published). [154] In fact, modularity requires that the set of all anyons [169] H. B. Heersche, P. Jarillo-Herrero, J. B. Oostinga, L. M. K. decomposes into pairs ðA; AεÞ, where the Fibonacci anyon Vandersypen, and A. F. Morpurgo, Bipolar Supercurrent completely factorizes within the fusion rules. That is, Aε × in Graphene, Nature (London) 446, 56 (2007). B ∼ ðA × BÞε and Aε × Bε ∼ ðA × BÞð1 þ εÞ. [170] X. Du, I. Skachko, and E. Y. Andrei, Josephson Current [155] A. B. Zamolodchikov, Infinite Additional Symmetries in and Multiple Andreev Reflections in Graphene SNS Two-Dimensional Conformal Quantum Field Theory, Junctions, Phys. Rev. B 77, 184507 (2008). Theor. Math. Phys. 65, 1205 (1985). [171] C. Ojeda-Aristizabal, M. Ferrier, S. Guéron, and [156] It is important to note that 1 and ϵ~ are the only H. Bouchiat, Tuning the Proximity Effect in a primary fields when considering the full G2 Kac-Moody Superconductor-Graphene-Superconductor Junction, algebra. The ϵ~ tower ðhϵ~ ¼ 2=5Þ contains a multiplet Phys. Rev. B 79, 165436 (2009). of seven fields, all with scaling dimension 2=5, transforming [172] X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, in one of the fundamental representations of G2.However, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. these seven fields are related to each other Colombo, and R. S. 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