PHYSICAL REVIEW X 10, 031014 (2020)

Electrical Probes of the Non-Abelian Spin Liquid in Kitaev Materials

David Aasen,1,2 Roger S. K. Mong,3,4 Benjamin M. Hunt,5,4 David Mandrus ,6,7 and Jason Alicea 8,9 1Microsoft Quantum, Microsoft Station Q, University of California, Santa Barbara, California 93106-6105 USA 2Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA 3Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA 4Pittsburgh Quantum Institute, Pittsburgh, Pennsylvania 15260, USA 5Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA 6Department of Materials Science and Engineering, University of Tennessee, Knoxville, Tennessee 37996, USA 7Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 8Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA 9Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA

(Received 5 March 2020; revised 12 May 2020; accepted 19 May 2020; published 17 July 2020)

Recent thermal-conductivity measurements evidence a magnetic-field-induced non-Abelian spin-liquid phase in the Kitaev material α-RuCl3. Although the platform is a good Mott insulator, we propose experiments that electrically probe the spin liquid’s hallmark chiral Majorana edge state and bulk anyons, including their exotic exchange statistics. We specifically introduce circuits that exploit interfaces between electrically active systems and Kitaev materials to “perfectly” convert electrons from the former into emergent fermions in the latter—thereby enabling variations of transport probes invented for topological superconductors and fractional quantum-Hall states. Along the way, we resolve puzzles in the literature concerning interacting Majorana fermions, and also develop an anyon-interferometry framework that incorporates nontrivial energy-partitioning effects. Our results illuminate a partial pathway toward topological quantum computation with Kitaev materials.

DOI: 10.1103/PhysRevX.10.031014 Subject Areas: Condensed Matter Physics

I. INTRODUCTION within the ground-state manifold—thus producing fault- tolerant qubit gates. And third, pairs of non-Abelian anyons The field of topological quantum computation pursues brought together in space can “fuse” to at least two different phases of matter supporting emergent particles known as types of particles; detecting the fusion outcome provides a “non-Abelian anyons” to ultimately realize scalable, intrinsi- means of qubit readout. cally fault-tolerant qubits [1,2]. This technological promise Fulfilling this potential requires, at an absolute minimum, derives from three deeply linked non-Abelian-anyon fea- synthesizing a non-Abelian host material and developing tures: First, nucleating well-separated non-Abelian anyons practical means of controlling and probing the constituent generates a ground-state degeneracy consisting of states that anyons. The observed fractional quantum-Hall phase at cannot be distinguished from one another by local measure- filling factor ν ¼ 5=2 [3], widely expected to realize the ments. Qubits encoded in this subspace enjoy built-in – protection from environmental noise by virtue of local non-Abelian Moore-Read state or cousins thereof [4 8], indistinguishability. Second, they obey non-Abelian braiding provided the first candidate topological-quantum-computing statistics. That is, adiabatically exchanging pairs of non- medium. Non-Abelian anyons in this setting carry electric 4 Abelian anyons effects “rigid” noncommutative rotations charge (e.g., e= ), and hence, can be manipulated via gating and probed using ingenious electrical interferometry schemes [9–11]. While experimental efforts in this direction continue [12], during the past decade intense experimental Published by the American Physical Society under the terms of activity has focused on “engineered” two-dimensional the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to (2D) and especially one-dimensional (1D) topological super- the author(s) and the published article’s title, journal citation, conductors [13,14] as alternative platforms. These exotic and DOI. superconductors can be assembled from heterostructures

2160-3308=20=10(3)=031014(37) 031014-1 Published by the American Physical Society AASEN, MONG, HUNT, MANDRUS, and ALICEA PHYS. REV. X 10, 031014 (2020)

(a) (b) (c) (d)

FIG. 1. (a) Lattice structure for the Kitaev honeycomb model. Spins exhibit bond-dependent nearest-neighbor interactions, with the x, y, and z components, respectively, coupling along green, red, and blue bonds. Vectors e1;2;3 point from the A sublattice (solid circles) to the B sublattice (open circles). (b) Sketch of the Majorana-fermion representation of the spin operators [Eq. (4)] used for the exact x;y;z solution. The br operators combine to form a Z2 gauge field, while the cr operators define itinerant Majorana fermions that hop between nearest-neighbor sites. Hermiticity of the cr’s allows the kinetic energy to be expressed as a sum over momenta in the right half of the Brilllouin zone (BZR, shaded region). (c) In the gapless spin-liquid phase, the fermionic spectrum features a single massless Dirac cone. (d) Breaking time-reversal symmetry via an applied magnetic field opens a gap at the Dirac point, generating a non-Abelian spin- liquid phase. involving ordinary, weakly correlated materials yet share When only the K term is present, the Hamiltonian reduces similar non-Abelian properties to the Moore-Read state (for to Kitaev’s famed exactly solvable honeycomb model [34]. reviews, see Refs. [15–25]). Specifically, the charged non- Here the ground state realizes a time-reversal-invariant Abelian excitations in the Moore-Read state are replaced by quantum spin liquid with gapless, emergent Majorana — non-Abelian defects i.e., domain walls and superconduct- fermions coupled to a Z2 gauge field. For this paper, it is — ing vortices that bind Majorana zero modes. In a topo- crucial to distinguish emergent Majorana fermions from logical superconductor, Majorana zero modes are equal physical Majorana fermions that appear as excitations at the superpositions of electrons and holes and thus carry no boundaries of two- and three-dimensional topological super- net charge. They do carry a physical-fermion-parity degree of conductors. The latter Majorana fermions are built from freedom, however, and are thus amenable to electronic ordinary electronic degrees of freedom, whereas the former probes including tunneling spectroscopy, interferometry, represent bona fide fractionalized quasiparticles born within Josephson measurements, etc.; see, e.g., Refs. [14,26–30]. a purely bosonic spin system. It follows that physical In fact, detailed blueprints exist for scalable topological- quantum-computation hardware based on 1D-topological- Majorana fermions can shuttle between the host topological superconductor arrays, relying largely on electrical tools for superconductor and a conventional electronic medium (e.g., operation [31]. a lead); conversely, emergent Majorana fermions live exclu- Still more recently, experiments suggest the emergence sively in the spin liquid. of yet another variant of the Moore-Read state, but in a Breaking time-reversal symmetry generates even more h fundamentally different physical setting from those above: striking physics: A nonzero Zeeman field gaps out the the honeycomb “Kitaev material” α-RuCl3 [32,33].As Majorana fermions, yielding a non-Abelian spin liquid background, consider a honeycomb lattice of spin-1=2 exhibiting a fully gapped bulk and a gapless, chiral moments governed by a Hamiltonian of the form Majorana fermion edge state that underlies quantized thermal Hall conductance [34]. This phase supports two X X nontrivial quasiparticle types: massive emergent Majorana − γ γ − h S H ¼ KSrSr0 · r þ: ð1Þ fermions and “Ising” non-Abelian anyons. The latter can be 0 r hrr i viewed as electrically neutral counterparts of the non- Abelian anyons in the Moore-Read state. Alternatively, they The first term encodes bond-dependent spin interactions, comprise deconfined cousins of non-Abelian defects in with γ ¼ x on the green bonds of Fig. 1(a), γ ¼ y on red topological superconductors that bind Majorana zero modes bonds, and γ ¼ z on blue bonds; note the strong frustration carrying an emergent rather than physical-fermion-parity arising from these competing spin couplings, which sup- degree of freedom. presses the tendency for conventional symmetry-breaking Jackeli and Khaliullin established that a class of strongly order. The second term in the Hamiltonian accounts for the spin-orbit-coupled Mott insulators can, quite remarkably, possible presence of a Zeeman field h, while the ellipsis be well modeled by Eq. (1) with inevitably present denotes additional allowed perturbations. corrections represented by the ellipsis being “small” [35].

031014-2 ELECTRICAL PROBES OF THE NON-ABELIAN SPIN … PHYS. REV. X 10, 031014 (2020)

Their pioneering result opened up the now experimentally their Mott-insulating character notwithstanding. We build off active field of Kitaev materials whose spins interact pre- of seminal theory works that highlight the possibility of dominantly via bond-dependent spin interactions of the type coherently converting physical fermions into emergent built into Kitaev’s honeycomb model [36–39]. All honey- deconfined quasiparticles in Abelian spin liquids [73] and comb-lattice Kitaev materials studied to date—α-RuCl3 non-Abelian quantum-Hall phases [74] to probe fractionali- included [40]—magnetically order at zero field. Evidently, zation [75]. We pursue a complementary approach that perturbations beyond the K term in Eq. (1), while nominally closely relates to the physics of “fermion condensation” small, destabilize the gapless quantum spin liquid [41] put on rigorous mathematical foundation in a similar setting (various experiments nevertheless report residual fraction- in Ref. [76]. Specifically, we introduce a series of circuits that alization signatures at “high” energies [42–52]). In α-RuCl3, interface electronically active systems—notably proximi- applying an approximately 10 T in-plane magnetic field tized ν ¼ 1 integer quantum-Hall states, though other destroys the zero-field magnetic ordering [53]. Numerous choices are possible—with Kitaev materials realizing a experiments are consistent with the fascinating possibility non-Abelian spin liquid. Strong interactions at their interface that the system then enters the non-Abelian spin-liquid phase can effectively “sew up” these very different subsystems, highlighted above [50,54–61]. Most strikingly, Kasahara leading to a striking and exceedingly useful phenomenon: A et al. [60] report thermal-Hall-conductance measurements physical electron injected at low energies on the electroni- that agree well with the quantized value expected from the cally active side converts with unit probability into an hallmark chiral Majorana edge mode. This experiment emergent fermion in the spin liquid. withstood some initial theoretical scrutiny [62,63], and Our circuits exploit this perfect conversion process to has very recently been extended in Ref. [64]. electrically reveal (via universal conductance signatures) Can one plausibly exploit α-RuCl3 (or perhaps some the spin liquid’s chiral Majorana edge state, bulk emergent related Kitaev materials) for topological quantum compu- fermions, and bulk Ising anyons, using variations of tation? This question is well motivated on at least two transport techniques developed for topological supercon- fronts. For one, the energy scales appear quite favorable. In ductors and fractional quantum-Hall states. Figures 8–10 Refs. [60,64], quantized thermal Hall conductance persists sketch the corresponding setups. The electrical conduct- up to temperatures of roughly 5 K, suggesting a spin-liquid ance of these circuits changes qualitatively upon perturbing bulk gap of similar magnitude—an encouraging figure the Kitaev material (again, an electrically inert element), compared to the gap expected in most other candidate non- e.g., to add or remove even a single bulk emergent fermion Abelian platforms [65]. Moreover, α-RuCl3 affords a great or Ising anyon; we argue that this feature makes our deal of materials-science flexibility [66–69]: It is exfoliat- predictions especially unambiguous. Moreover, the circuits able, amenable to nanofabrication, can be readily interfaced designed to detect individual bulk quasiparticles rely on with other materials, etc. interferometric signatures that further unambiguously Manipulating and probing the anyons as required for reveal the non-Abelian statistics of Ising anyons as well advanced applications nevertheless poses a major out- as the nontrivial mutual statistics between Ising anyons and standing challenge. In essence, the detailed road maps emergent fermions. developed for quantum-Hall and topological superconduc- These results collectively establish a partial road map tor platforms—which again heavily invoke electrical toward utilizing Kitaev materials for topological quantum tools—need to be largely rewritten for non-Abelian spin computation. En route to putting our predictions on firm liquids in Kitaev materials because they are electrically inert footing, we introduce some nontrivial technical innovations Mott insulators. Two subclasses of problems naturally arise as well. First, we resolve an outstanding puzzle in the here: (i) devising feasible techniques for creating, trans- literature concerning interacting Majorana fermions. porting, and fusing Ising anyons on demand in Kitaev Specifically, the interaction strength required to induce an materials and (ii) developing schemes for unambiguously instability in a self-dual Majorana chain has been found to detecting individual emergent fermions and Ising anyons as vary by orders of magnitude depending on subtle variations well as their nontrivial statistics. Vacancies and spin impu- in the microscopic interaction (for a recent review, see rities appear to be promising ingredients for item (i). At least Ref. [77]). We explain this peculiar behavior as arising from in the gapless spin-liquid phase of Kitaev’s honeycomb interaction-dependent renormalization of kinetic energy for model, both have been shown to trap Z2-flux excitations the Majorana chain. Second, we analyze anyon interferom- [70–72], which evolve into Ising anyons upon entering the etry in a new regime using a phenomenological picture non-Abelian phase. We leave detailed investigations of this combined with rigorous formalism that incorporates crucial issue for future work, and instead propose a series of energy-partitioning effects. The framework that we develop experiments that directly tackle item (ii). here could prove valuable in a variety of other contexts. Our primary innovation is that, counterintuitively, low- The remainder of the paper is organized as follows. We voltage electrical transport can be profitably employed to begin in Sec. II by reviewing the phenomenology of the probe the detailed structure of non-Abelian spin liquids, Kitaev honeycomb model. Section III explores interacting

031014-3 AASEN, MONG, HUNT, MANDRUS, and ALICEA PHYS. REV. X 10, 031014 (2020) X helical Majorana fermions from several perspectives, and K H ¼ iuˆ rr0 crcr0 : ð5Þ then Sec. IV bootstraps off of those results to describe how K 4 hrr0i a ν ¼ 1 quantum-Hall state can be sewn (in a precise sense) to a non-Abelian spin liquid with the aid of a super- ˆ ≡ γ γ ∈ 1 Above, we introduce link variables urr0 ibrbr0 that, conductor. In the next three sections, we introduce circuits crucially, commute with each other and with the “ ” that use this sewing to electrically interrogate a non- Hamiltonian. The link variables can thus be treated as Abelian spin liquid: Section V focuses on electrical classical parameters—thereby reducing the model to a free- detection of the chiral Majorana edge state, Sec. VI fermion problem in any fixed uˆ rr0 configuration [78]. introduces a circuit that probes bulk Ising anyons, and Physically, uˆ rr0 is a Z2 gauge field whose flux around Sec. VII introduces an interferometer that probes both bulk plaquette p is proportional to the conserved Wp operator in Ising anyons and emergent fermions, as well as non- Eq. (3) (hence, the absence of nontrivial dynamics). Abelian statistics. We conclude and highlight numerous The ground state of Eq. (5) arises in the sector with Z2 open questions in Sec. VIII. Several Appendixes provide gauge flux of π through every hexagonal plaquette [79]. Let additional details and supplementary results on our circuits us decompose the honeycomb lattice into A and B sub- as well as interacting Majorana-fermion models. e lattices, and also introduce vectors j¼1;2;3 that link the two sublattices; see Fig. 1(a). A convenient gauge encoding π ˆ 1 r II. KITAEV HONEYCOMB MODEL flux per plaquette is ur;rþej ¼þ for all on sublattice A. PHENOMENOLOGY Inserting this gauge choice into Eq. (5) yields a To set the stage, this section reviews the phenomenology Hamiltonian of the Kitaev honeycomb model [34], focusing in particular X X3 on universal properties of the non-Abelian spin-liquid ˜ K H ¼ icrcr e ð6Þ phase. We also establish various conventions here that K 4 þ j r∈A j¼1 are employed throughout. that describes the ground-state flux sector. One can view A. Gapless spin liquid Eq. (6) as an analog of graphene wherein Majorana fermions hop between nearest-neighbor honeycomb sites. We start with the “pure” Kitaev honeycomb model at To obtain the spectrum of H , we pass to momentum zero magnetic field: K space, employing conventions such that X − γ γ sffiffiffiffiffiffiffiffiffi HK ¼ KSrSr0 : ð2Þ 2 X rr0 ik·r h i cr∈A=B ¼ e cA=Bk Nu:c:k∈BZ γ sffiffiffiffiffiffiffiffiffi Once again, we have ¼ x, y, and z respectively on green, 2 X red, and blue bonds of Fig. 1(a). For any hexagonal ik·r −ik·r † ¼ ðe cA=Bk þ e cA=BkÞ; ð7Þ Nu:c: plaquette p, HK commutes with the operator k∈BZR

x y z x y z Wp ¼ S1S2S3S4S5S6; ð3Þ where Nu:c: is the number of unit cells. The momentum- † δ δ space operators so defined satisfy fcαk;cβk0 g¼ αβ k;k0 1 2 … 6 † where sites ; ; ; around plaquette p are labeled as in and cαk ¼ cα−k (reflecting Hermiticity of cr). In the second Fig. 1(a). The resulting extensive number of conserved line of Eq. (7), we use the latter property to express cr as a quantities ultimately enables an exact solution. To this end, sum over momenta in the right half of the Brillouin zone we reexpress the spins via (BZR), i.e., k with kx > 0 as shown in Fig. 1(b). Defining a † † † two-componentP spinor Ck ¼½cAkcBk and a function i − k e α α ; ξðkÞ¼−iðK=2Þ e i · j , Eq. (6) becomes Sr ¼ 2 br cr ð4Þ j   X α † 0 ξ ðkÞ br cr ˜ on the right side, and denote Majorana-fermion HK ¼ Ck Ck: ð8Þ ξðkÞ 0 operators that are Hermitian, square to the identity, and k∈BZR anticommute with one another. For an illustration, see Fig. 1(b). Remaining faithful to the original spin-1=2 The resulting single-particle energies are jξðkÞj2, and the Hilbert space requires enforcing the local constraint many-particle ground state populates all negative energy x y z brbrbrcr ¼þ1 at every site. levels. In the Majorana representation, the Hamiltonian This ground state realizes the gapless spin-liquid phase becomes of Kitaev’s honeycomb model. Specifically, Eq. (8)

031014-4 ELECTRICAL PROBES OF THE NON-ABELIAN SPIN … PHYS. REV. X 10, 031014 (2020) describes gapless (emergent) fermion excitations with a are henceforth dropped. The effective low-energy single massless Dirac cone centered at momentum Hamiltonian accordingly now reads Q ¼ð4π=3aÞxˆ,witha the lattice constant. See Fig. 1(c). Z k We now focus on these gapless excitations by writing ¼ H Ψ† − ∂ σy ∂ σx σz Ψ eff ¼ ½vbulkð i x þ i y Þþm ð12Þ Q þ q and retaining only modes with “small” q. Equation (8) r then reduces to the following effective Dirac Hamiltonian that captures low-energy fermionic excitations in the ground- with m ∝ jhj, and describes emergent fermions with a state flux sector: gapped Dirac spectrum illustrated in Fig. 1(d). [Without the Z generic perturbations that we implicitly include in Eq. (11), the Dirac gap would scale like hxhyhz instead of jhj.We H Ψ† σy − σx Ψ eff ¼ vbulk qðqx qy Þ q stress that this fine-tuned behavior is a pathology of Zq perturbating about the exactly solvable HK Hamiltonian Ψ† − ∂ σy ∂ σx Ψ ¼ vbulk ð i x þ i y Þ : ð9Þ as Ref. [80] discusses in detail.] r The resulting field-induced phase realizes a non-Abelian pffiffiffi spin liquid with Ising topological order. Although the bulk 3 4 Ψ ∝ Here, vbulk ¼ aK= , q CQþq, and in the last line we is fully gapped, the system’s boundary hosts a single Fourier transform back to real space. Furthermore, we emergent chiral Majorana mode with central charge employ ℏ ¼ 1 units, and continue to do so throughout c ¼ 1=2. (One can trace the edge state’s existence to the (for clarity, however, we express the conductance quantum quantized half-integer thermal Hall conductance that arises 2 as e =h). This gapless spin-liquid phase also admits gapped from gapping out a single Dirac cone; for related problems, Z H 2-flux excitations that are not captured by eff. see Refs. [81–84].) Low-energy edge excitations are Suppose that we now supplement Eq. (2) with generic described by the continuum Hamiltonian perturbations that preserve translation symmetry and time- Z T reversal symmetry , leading to a Hamiltonian of the form H − γ∂ γ edge ¼ ð ivedge x Þ; ð13Þ x H ¼ HK þ: ð10Þ where vedge is a nonuniversal velocity [85], x is a coordinate Despite the loss of exact solvability, one can address the along the boundary, and γðxÞ is a Majorana-fermion field. stability of the gapless spin liquid from the viewpoint of the (For clarity, we employ subscripts that distinguish edge and effective low-energy theory. The original spin operators bulk velocities, though later we abandon such a notation.) transform under T according to Sr → −Sr. Within the Here and below, we normalize continuum Majorana fields ground-state flux sector, T sends cr∈A → cr∈A and such that cr∈B → −cr∈B, and in turn transforms the low-energy z † t 1 Dirac field via Ψ → σ ðΨ Þ . The only translationally fγðxÞ; γðx0Þg ¼ δðx − x0Þ: ð14Þ invariant perturbation to Eq. (9) that can open an energy 2 † z gap is the mass term mΨ σ Ψ—which is odd under T and With this choice, the energy for an edge excitation with cannot appear provided time-reversal symmetry persists. momentum k is simply vk. Note that Eq. (13) exhibits a Consequently, the gapless spin liquid constitutes a stable global Z2 symmetry that sends γ → −γ, which as we see in symmetry-protected phase with some finite tolerance to the Sec. III A has important practical consequences for the ellipsis in Eq. (10). interfaces that we exploit later in this paper. The bulk of the non-Abelian spin liquid supports three B. Non-Abelian spin liquid gapped quasiparticle types. First, there are nonfractional- ized bosonic excitations—as in any phase of matter—that In this paper, we are primarily interested in the physics we call trivial particles labeled by 1. Second, the system resulting when time-reversal symmetry is explicitly broken hosts more exotic gapped emergent fermions (ψ particles) by an applied magnetic field. The field modifies Eq. (10) to captured by the effective Hamiltonian in Eq. (12). Third, X and most interestingly, gapped Z2-flux excitations bind − h S H ¼ HK · r þ: ð11Þ emergent Majorana zero modes and realize “Ising anyons” r (σ particles) with non-Abelian braiding statistics. These “ On symmetry grounds [80], the Zeeman term can be quasiparticle types obey the following nontrivial fusion ” expanded in terms of low-energy degrees of freedom as rules, h S ∼ βΨ†σzΨ β ∝ h · r þ. Here, j j is a nonuniversal ψ ⊗ ψ ≅ 1; σ ⊗ σ ≅ 1 ⊕ ψ; ψ ⊗ σ ≅ σ; ð15Þ constant that vanishes only for fine-tuned field orientations 64] ], while the ellipsis denotes additional symmetry- which roughly describe how they behave when brought allowed terms that are unimportant for our purposes and together in space. That is, two emergent fermions coalesce

031014-5 AASEN, MONG, HUNT, MANDRUS, and ALICEA PHYS. REV. X 10, 031014 (2020)

(a) (a) (b)

(b)

FIG. 2. Non-Abelian spin-liquid synopsis. (a) The boundary (c) hosts a gapless chiral Majorana mode, while the bulk supports two nontrivial gapped quasiparticle types: emergent neutral fermions ψ and Ising non-Abelian anyons σ. Fermions acquire a minus sign on crossing the wavy line (which represents a branch cut) emanating from σ. (b) Summary of quasiparticle braiding statistics. into a local boson, two Ising anyons can combine to yield either a local boson or an emergent fermion, and Ising anyons can freely absorb emergent fermions without “ ” FIG. 3. (a) Decoupled non-Abelian spin liquids hosting emer- changing their quasiparticle type. Viewed in reverse, a gent chiral Majorana fermions γ and γ at their interface. local boson can fractionalize into a pair of emergent R L (b) Strong interactions between γR and γL [described by fermions, an individual emergent fermion can further Eq. (17) with κ > 0] gaps out these modes—thus sewing the fractionalize into a pair of Ising anyons, and pairs of two phases into a single non-Abelian spin liquid. (c) Microscopic Ising anyons can be pulled out of the vacuum. Finally, ψ view of the spin-liquid interface viewed as two Kitaev honey- and σ particles exhibit not only nontrivial self-statistics, but comb models coupled via vertical bonds of strength Jinter > 0 also nontrivial mutual statistics: Taking a fermion all the (dashed lines); see Eq. (19). The interface is fully gapped, as in way around an Ising anyon, or vice versa, yields a statistical (b), when Jinter exceeds a critical value Jc, but otherwise hosts phase of −1. The above quasiparticle characteristics gapless Majorana modes, as in (a). become essential for the circuits developed in Secs. VI and VII. Figure 2 summarizes the bulk and edge content of hybridization between the subsystems can effectively sew the non-Abelian spin liquid. them together—producing a single, uninterrupted spin liquid. Our goal here is to understand this sewing-up III. PRIMER: INTERACTING HELICAL process, both from effective field theory and microscopic MAJORANA FERMIONS viewpoints. When the two layers decouple as in Fig. 3(a), their As an illuminating warm-up, next we explore gapless interface hosts helical Majorana modes whose kinetic nonchiral-Majorana fermions propagating in 1D with energy is described by the low-energy Hamiltonian strong interactions. We proceed in two stages: first exam- ining interfaces between non-Abelian spin liquids, and then Z turning to one-dimensional lattice models that harbor H0 ¼ ð−ivγR∂xγR þ ivγL∂xγLÞ: ð16Þ similar physics. Results obtained here carry over straight- x forwardly to the quantum-Hall–spin-liquid interfaces that Here, x is a coordinate along the interface, v is the edge- we introduce in Sec. IV and later exploit to electrically state velocity, and γ and γ respectively denote right- and detect chiral Majorana edge states and bulk anyons in R L left-moving Majorana-fermion fields. Upon turning on Kitaev materials (Secs. V through VII). interactions between the adjacent layers, the Hamiltonian becomes H ¼ H0 þ δH, where δH hybridizes the helical A. Sewing up non-Abelian spin liquids Majorana modes. Crucially, the form of δH is constrained Consider the setup from Fig. 3(a) consisting of two by the fact that γR and γL represent emergent fermions non-Abelian spin liquids realized in adjacent Kitaev mate- originating from disjoint spin liquids. In particular, rials. Physically, it is natural to anticipate that suitable only pairs of emergent fermions—which together form a

031014-6 ELECTRICAL PROBES OF THE NON-ABELIAN SPIN … PHYS. REV. X 10, 031014 (2020)

(a) 1.2 1.0 0.8 0.6 0.4 0.2 0.0 (b) –10 –2 –1 –0.5 0 0.3 0.5 1 2 10

γ γ − γ γ FIG. 4. Dimerization order parameter hi a−1 a i a aþ1i ver- sus U=t obtained from DMRG simulations of the model in Eq. (20) with t0 ¼ 0. This order parameter corresponds to the fermion mass in the continuum limit. For U=t ≳ 0.42799ð2Þ, our simulations capture a gapped, spontaneously dimerized phase for the chain, in agreement with Ref. [87]. At smaller U=t, the chain FIG. 5. (a) Snapshot of domain-wall excitations at the exactly instead realizes a critical state with central charge c ¼ 1=2.We solvable, spontaneously dimerized limit of Eq. (20) occurring for 0 0 2 1 additionally verify that this critical state extends to large negative t ¼ and U ¼ t= . Neighboring sites a, a þ enclosed by γ γ −1 U=t values. curved rectangles dimerize such that h a aþ1i¼ . Domain walls at which the dimerization pattern shifts by one site trap unpaired Majorana zero modes [nonenclosed dots in (a)]. In the boson—can tunnel across the interface. The simplest such low-energy limit, the dimerized chain is described at the mean- interaction is given by field level by Eq. (18), which includes a spontaneously generated mass m. Microscopic domain-wall excitations from (a) corre- Z spond in continuum language to excitations at which the mass m δH ¼ −κ ðγR∂xγRÞðγL∂xγLÞ; ð17Þ changes sign, as depicted in (b). Such domain walls constitute 1D x analogs of Ising anyons that arise at the spontaneously gapped spin-liquid interface of Fig. 3(b). where the two derivatives are necessitated by Fermi statistics. Notice that H exhibits two independent Z2 symmetries, one corresponding to γ → −γ and the other the two spin liquids have been sewn into one as sketched in R R Fig. 3(b). corresponding to γ → −γ . These symmetries can never L L Several consistency checks bolster the above picture. be broken explicitly by any physical perturbation, reflecting First, one can view the ground state jΨ i of H as a trial the fact that individual Majorana fermions γ and γ live MF MF R L wave function and the mass m as a variational parameter. In only within their respective spin liquids. Appendix A1, we optimize hΨ jH0 þ δHjΨ i with The coupling κ is formally irrelevant at the fixed MF MF respect to m for varying κ. This analysis indeed captures point described by the quadratic Hamiltonian H0. m κΛ2=v “Weak” κ thus has only perturbative effects, and most a nonzero mass provided the dimensionless ratio exceeds a critical value, where Λ is an ultraviolet momen- importantly does not gap out the helical Majorana modes. tum cutoff. (For κ < 0, the optimal mass always vanishes Evidently, sewing up the spin liquids requires strong within this treatment.) coupling. At “large” κ > 0, the system can lower its energy Second, the two nontrivial bulk anyons of the non-Abelian by condensing hiγ γ i ≠ 0—thereby spontaneously break- R L spin-liquid phase are encoded in the simple mean-field ing the two independent Z2 symmetries noted above (but Hamiltonian H describing the gapped interface [86]. preserving their product). For rough intuition, consider the MF Neutral fermions are clearly present as gapped excitations. term − κ=δx2 iγ x δx γ x δx iγ x γ x , which ð Þ½ Rð þ Þ Lð þ Þ½ Rð Þ Lð Þ Ising non-Abelian anyons form at domain walls in which the upon Taylor expanding in the microscopic length δx spontaneously chosen mass m changes sign; see Fig. 5(b) for generates the interaction from Eq. (17). The discrete form γ γ ≠ 0 κ an illustration. Unpaired Majorana zero modes bind to such above clearly reveals that hi R Li is favored provided domain walls, leading to the hallmark degeneracy associated is positive. In the condensed regime, the interface can be with Ising anyons. Furthermore, since the sign of the mass is modeled by an effective mean-field Hamiltonian arbitrary, separating the domain walls by arbitrary distances Z costs only finite energy—i.e., the Ising anyons are bona fide H − γ ∂ γ γ ∂ γ γ γ deconfined quasiparticles. Additional insights into the MF ¼ ð iv R x R þ iv L x L þ im R LÞ; ð18Þ x domain-wall structure can be gleaned from the lattice model that we discuss in Sec. III B. with m ∝ hiγRγLi a mass whose sign, importantly, is Third, the low-energy perspective that we present above chosen spontaneously. Equation (18) exhibits a fully seamlessly connects to microscopics. Let us add a spin-spin gapped spectrum, and thus describes a scenario where interaction

031014-7 AASEN, MONG, HUNT, MANDRUS, and ALICEA PHYS. REV. X 10, 031014 (2020) X δ − z z T H ¼ Jinter SrSr0 ð19Þ explicitly by any -preserving perturbation, similar to the ðr;r0Þ scenario encountered in Sec. III A. Upon restoring nonzero U, the leading term that that couples spins across bonds ðr; r0Þ [dashed lines in couples right and left movers corresponds to Eq. (17) with 0 κ ∝ Fig. 3(c)] that bridge the adjacent spin liquids. At Jinter ¼ , U [87]. Previous density-matrix renormalization-group one recovers decoupled spin liquids, and the interface hosts (DMRG) simulations of Eq. (20) (which we reproduce and gapless Majorana modes that are stable to weak perturba- extend to include U<0 in Fig. 4) indicate that for tions. On symmetry grounds, the boundary spin operators U ≳ 0.428t, criticality is destroyed in favor of a gapped, z z relate to continuum Majorana fields via Sr ∼ hSriþ dimerized phase that spontaneously breaks T symmetry z γ γ ≠ 0 α∶iγR∂xγR∶ on the lower edge of the interface and Sr ∼ [87]. In continuum language, here hi R Li condenses z hSri − α∶iγL∂xγL∶ on the upper edge. Here, angle brackets and the helical Majorana fermions are gapped via indicate ground-state expectation values, α is a nonuniver- generation of a mass m ∝ hiγRγLi with arbitrary sign. sal constant, and ∶∶ denotes normal ordering; the relative Appendix A2 analyzes Eq. (20) at t0 ¼ 0 using a varia- minus sign in the α pieces above reflects the opposite tional approach that predicts spontaneous dimerization for chirality for the two modes. Using this continuum expan- U ≳ 0.295t, in rough agreement with DMRG. sion, the microscopic interaction δH indeed generates the The gapped ground states at U ¼ t=2 are known exactly κ ∝ “ ” effective-Hamiltonian term in Eq. (17) with Jinter.At [87] and can be recovered by postulating perfect dime- — — O ≡ γ γ −1 γ γ 0 Jinter ¼ J corresponding to the strong-coupling limit rization with hi 2a 2aþ1i¼ and hi 2a−1 2ai¼ . the system forms a single, translationally invariant non- Decoupling the U term usingP this ansatz generates a mean- γ γ O Abelian spin liquid; here, all gapless modes at the interface field Hamiltonian HMF ¼ t a i 2a 2aþ1 for which ¼ have clearly been vanquished. It follows that the spin −1 in the ground state—indicating self-consistency. One liquids are sewn up provided Jinter exceeds a critical value can similarly show that the shifted dimerization with 0 γ γ −1 γ γ 0 Jc that satisfies

γ γ ∼ −1 a γ γ ; B. Insights from microscopic models hi a aþ1i const þð Þ hi R Li ð22Þ Complementary insights can be gleaned by examining a strictly 1D toy lattice model that also realizes interacting it follows that these two dimerizations correspond to helical Majorana fermions. Consider an infinite chain of opposite-sign masses in the continuum formulation. In Sec. III A, we observe that the interface between two physical (rather than emergent) Majorana fermions γa living on lattice sites a and governed by the microscopic sewn-up spin liquids [Fig. 3(b)] supports gapped emergent Hamiltonian fermionic excitations, and that domain walls at which the mass m changes sign correspond to Ising anyons hosting X 0 unpaired Majorana zero modes. The above mean-field H ¼ ðitγ γ 1 þit γ γ 2 −Uγ −2γ −1γ 1γ 2Þ; ð20Þ a aþ a aþ a a aþ aþ ansatz at U t=2, though operative in a physical-fermion a ¼ system, provides an intuitive picture for these fractionalized we take t>0 for concreteness throughout. References [87, quasiparticles [92]. The mean-field construction suggests 88] recently studied this model motivated in part by interest- that low-energy states can be labeled by domains — γ γ −1 ing connections to supersymmetry; see also Ref. [89]. exhibiting fixed dimerization either hi 2a−1 2ai¼ γ γ −1— Importantly, H preserves an (anomalous [90]) translation or hi 2a 2aþ1i¼ along with fermionic excitations γ → γ 0 0 symmetry T that transforms a aþ1.Att ¼ ,the within a given domain. Fermionic excitations arise from Hamiltonian further preserves an antiunitary chiral sym- flipping the sign of the dimerization expectation value at a a γ γ → 1 metry C that sends γa → ð−1Þ γa and i → −i [91]. particular bond, e.g., replacing h 2b−1 2bi þ for some We first specialize to t0 ¼ 0. In the U ¼ 0 limit the chain b. Figure 5(a) illustrates an excited configuration with is gapless. Here, one can capture the low-energy physics by domain walls separating the two dimerization patterns, writing while Fig. 5(b) shows the corresponding sign-changing mass profile in the continuum description. The domain a γa ∼ γL þð−1Þ γR; ð21Þ walls clearly harbor unpaired Majorana zero modes as a consequence of the dimerization shift. Pairs of domain where γR and γL again denote right- and left-moving walls share a pair of Majorana zero modes, and therefore Majorana fields, leading precisely to Eq. (16) with v ∝ t. nonlocally host a single complex fermionic mode. The Translation symmetry T sends γR → −γR (just as for one of occupancy of this complex fermion dictates whether the the Z2 symmetries present for the spin-liquid interface pair of domain walls fuse to the local vacuum or a fermion. examined above), while C swaps γR ↔ γL. A mass term Comparing these quantum states to those at the interface of imγRγL is odd under T, and thus can never be generated Fig. 3(b), we can identify domain walls in the 1D model

031014-8 ELECTRICAL PROBES OF THE NON-ABELIAN SPIN … PHYS. REV. X 10, 031014 (2020) with Ising anyons in the spin liquid, and the excitations to which they fuse as the local vacuum or the emergent fermion. More technically, we can also use this illustration to relate the ground-state degeneracy of Eq. (20) with open boundary conditions to the ground-state degeneracy of the spin liquid on a cylinder. For this purpose, we identify the FIG. 6. Schematic phase diagram of Eq. (20) as a function of 1D chain governed by Eq. (20) with the degrees of freedom 0 U=t and velocity anisotropy induced by t . Here, vR and vL, along a path that connects the upper and lower cylinder respectively, denote the velocities for right- and left-moving ends. We can then view U as a tuning potential that slices Majorana fermions at U ¼ 0; Eq. (23) specifies their ratio. We open or resews the cylinder as U passes below or above the assume 0 excited state). Conversely, if the chain begins in jσi, the IV. SEWING A NON-ABELIAN SPIN LIQUID TO domain walls shuttle unpaired Majorana zero modes to the AN ELECTRONIC QUANTUM HALL PHASE 1 ψ boundary and thus yield j i or j i. We have now seen two examples wherein strong 0 ≠ 0 0 Next, we restore t .AtU ¼ , the chain remains interactions catalyze a condensation transition with gapless, though the velocities v and v for left and right L R hiγRγLi ≠ 0. At the interface between two non-Abelian movers now differ due to the loss of C symmetry. Explicitly, spin liquids examined in Sec. III A, γR and γL both we have represent emergent fermions residing in initially separate fractionalized bosonic systems that are stitched together by − 2 0 vR t t the condensation. In the strictly 1D model from Sec. III B, ¼ 0 ; ð23Þ vL t þ 2t by contrast, both fields represent physical fermions. Next, we explore a system in which a very similar condensation which vanishes as t0 → t=2.(Fort0 >t=2, additional low- arises, but instead from the combination of a physical and energy modes appear; we consider only 0 ≤ t0 0. For instance, we find Uc=t ≈ 0.4302ð1Þ quantum-Hall edge couples to a conventional supercon- 0 0 as t =t → 0.5, compared to Uc=t ≈ 0.42799ð2Þ at t ¼ 0. ductor; here and in similar setups that we study in later Figure 6 summarizes the phase diagram extracted from sections, we assume fully gapped superconductivity, DMRG. though we briefly discuss the role of low-lying excitations A spontaneously dimerized gapped phase can also arise deriving from vortices and/or disorder in Sec. VIII. We first in the modified model obtained by replacing the four- present a qualitative picture for the physics that arises from γ γ γ γ fermion interaction in Eq. (20) with UR a−1 a aþ1 aþ2. interactions between these subsystems.

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(a) (b) Z H γ ∂ γ SL ¼ ðivL L x LÞð25Þ x

describes the spin liquid’s emergent chiral Majorana edge state with velocity vL. The second governs the proximitized ν ¼ 1 edge and takes the form Z H − ψ † ∂ ψ Δ ψ ∂ ψ QH ¼ ½ iuR R x R þ ði R x R þ H:c:Þ; ð26Þ x

where ψ R is a complex fermion operator that removes electrons from the edge state, u is the associated velocity, FIG. 7. (a) Interface between a non-Abelian spin liquid and a R and Δ is the proximity-induced pairing amplitude. Passing proximitized ν ¼ 1 integer quantum-Hall system. The bare ν ¼ 1 ψ γ γ0 edge state—indicated by double arrows—generically separates to a Majorana representation via R ¼ R þ i R, one can γ γ0 equivalently write [96] into two copropagating Majorana fermions R and R beneath the superconductor (SC) in the central region. (b) Strong interactions Z γ ’ γ can gap out R and the spin liquid s edge Majorana fermion L via H − γ ∂ γ − 0 γ0 ∂ γ0 ; QH ¼ ð ivR R x R ivR R x RÞ ð27Þ formation of a fermion condensate that partially sews the two x subsystems together. Physical Majorana fermions (black half- circles) and emergent Majorana fermions (red half-circles) are the velocities for the constituent copropagating Majorana then identified at the interface; see enlargement for an illustration. γ γ0 − 2Δ 0 2Δ fermions R and R are vR ¼ uR and vR ¼ uR þ , A striking consequence follows: An electron injected at low ν 1 respectively. energies into the ¼ edge splinters into a pair of Majorana The final term in Eq. (24) encodes interactions between fermions, one of which unavoidably enters the spin liquid. the spin-liquid and quantum-Hall edge states. Suppose that γR resides closest to γL as in Fig. 7. It is then reasonable to The quantum-Hall edge hosts a chiral mode that can be assume that interactions predominantly couple these fer- viewed as a pair of copropagating chiral Majorana fermions mions, so we take δH to be given precisely by Eq. (17). δ (hence, double arrows employed in our illustrations). Useful insight follows from reexpressingR H in terms of Beneath the superconductor, the loss of charge conserva- ψ δ − 1 κ γ ∂ γ ψ † ∂ ψ complex fermions R: H ¼ 4 xð L x LÞð R x R þ tion generically allows those copropagating Majorana ψ R∂xψ R þ H:c:Þ. Here we see that interactions transfer fermions to displace from one another as shown in electrically neutral dipoles as well as Cooper pairs from the Fig. 7(a). Microscopic interactions can backscatter only quantum-Hall edge to the spin liquid, consistent with our — — electrically neutral bosons e.g., energy between the preceding physical picture. quantum-Hall and spin-liquid edge states because the latter The full Hamiltonian in Eq. (24) reduces to the model edge modes reside in an electrical insulator that hosts only studied in the previous subsections, supplemented by a emergent fermions. More precisely, a charge-2ne excitation γ0 decoupled sector for the R chiral Majorana fermion. We (n is an integer) from the quantum-Hall edge can neutralize immediately conclude that strong interactions can condense by shedding its charge into the superconducting condensate hiγRγLi ≠ 0 and partially gap out the interface, again in line [95], and then backscatter into the spin-liquid edge mode. with the above physical picture. This conclusion holds even Sufficiently strong backscattering events of this nature if the velocities vL and vR—which bear no relation—differ partially sew up the quantum-Hall state and non-Abelian significantly; recall Sec. III B. spin liquid. That is, the emergent chiral Majorana fermion In the present context, the transition to a state with from the latter spin-liquid edge gaps out with “half” of the hiγRγLi ≠ 0 is sometimes referred to as fermion condensa- ν ¼ 1 edge mode, leaving a single physical chiral Majorana tion since the condensed object involves only one physical edge mode behind. As sketched in Fig. 7(b), an electron fermion (specifically, γR). A precise mathematical formu- injected at low energies into the “naked” part of the lation of this striking phenomenon can be found in Ref. [76]; quantum-Hall edge then splinters into a pair of Majorana see also Refs. [73,74,97] for related earlier applications. The fermions, one of which unavoidably enters the non-Abelian essential role played by proximity-induced superconductiv- spin liquid as an emergent fermion. ity also becomes clear from this vantage point. The con- For a more formal analysis, we write the effective γ γ ∼ ψ ψ † γ ≠ 0 densate hi R Li hið R þ RÞ Li clearly does not Hamiltonian for the interface as preserve U(1) charge conservation for the physical fermions. Without externally imposed superconductivity, interactions H H H δH ¼ SL þ QH þ : ð24Þ would thus need to spontaneously break U(1) in order to partially gap the interface, which cannot transpire due to the The first term quasi-1D nature of the interface. Finally, we note that

031014-10 ELECTRICAL PROBES OF THE NON-ABELIAN SPIN … PHYS. REV. X 10, 031014 (2020) essentially the same fermion condensation transition can (a) arise from interfacing a non-Abelian spin liquid with other electronic platforms, including conventional spinful 1D wires. We discuss alternative setups further in Sec. VIII.

V. ELECTRICAL DETECTION OF CHIRAL MAJORANA EDGE MODES IN NON-ABELIAN SPIN LIQUIDS As a first application of the phenomena that we develop in Sec. IV, we introduce a scheme for electrically detecting the spin liquid’s emergent chiral Majorana edge state. Figure 8(a) sketches the relevant circuit. Here, a pair of (b) ν ¼ 1 quantum-Hall systems flank a non-Abelian spin liquid. The interfaces are partially gapped by fermion condensation, facilitated by superconductors that are float- ing but exhibit negligible charging energy; note that the superconductors connect on the bottom end. A source on the far left is biased with voltage V, while a drain on the far right is grounded. We stress that no electrical current flows through spin liquid—which still realizes a good Mott insulator. Any current instead passes between the quan- tum-Hall systems via the intervening floating supercon- FIG. 8. (a) Circuit that electrically detects the spin liquid’s ductor in the form of Cooper pairs. Nevertheless, the spin emergent chiral Majorana edge state. Any electrical current that liquid is by no means a spectator: Electrons from the source flows between the source (S) and drain (D) necessarily passes propagate chirally along the ν ¼ 1 edge, then partially through the central superconductor (and not the spin liquid). convert into emergent chiral Majorana fermions in the spin Nevertheless, the perfect conversion between physical and liquid, and finally reenter the ν ¼ 1 edge as physical emergent fermions at the quantum-Hall–spin-liquid interfaces — fermions on the other end. This inevitable conversion dictates the conductance which at zero bias is quantized at 2 2 between physical and emergent fermions ultimately dic- G ¼ðe = hÞ. (b) Control circuit in which the spin liquid is tates the circuit’s electrical transport characteristics as we replaced with a trivial phase (e.g., a magnetically ordered state). Elimination of the physical-fermion ↔ emergent-fermion con- will see. version processes that underlie transport in (a) leads to vanishing We focus on vanishingly small temperature T and bias zero-bias conductance G ¼ 0 in this case. voltage V, where the conductance attains a universal quantized value of G ¼ðe2=2hÞ. To understand this value, first observe that the left and right halves constitute experiment—appears in Fig. 8(b). Most importantly, fer- 2 identical resistors in series; hence, G ¼ g= with g the mion condensation is now precluded, so that both ν ¼ 1 conductance for (say) the left half. One can deduce g as edge states must simply “turn around” at the interface with follows. An incident electron from the source splits up the central trivial region. In this case, the probability for into an emergent Majorana fermion in the spin liquid Andreev reflection vanishes at asymptotically low energies, and a physical Majorana fermion that reflects back to the yielding conductance G ¼ 0. source [see Fig. 8(a)]. The physical Majorana fermion is, The absence of Andreev processes can be most simply by definition, equal part electron and equal part hole, understood as a consequence of Fermi statistics: The induced 1 2 implying probability = for Andreev reflection. Thus, pairing term for the ν ¼ 1 edge state Δðiψ ∂ ψ þ H:c:Þ 1 2 2 R x R g ¼ 2 × ð e =hÞ, where the factor of 2 in the numerator necessarily contains a derivative and thus vanishes with the arises because each Andreev process injects a Cooper pair incident electron’s momentum. Alternatively, as the two into the superconductor. The overall conductance is then copropagating Majorana fermions traverse the superconduc- G ¼ðe2=2hÞ as advertised. Although we focus on V → 0 tor, they generally acquire different phase factors, in turn here, the conductance remains G ≈ ðe2=2hÞ for voltages “rotating” the incident electron in particle-hole space and below the gap scale of the fermion condensate. generating Andreev processes. The phase difference explic- What happens if the emergent chiral Majorana fermions itly reads δϕ ¼ðk − k0ÞL, where k and k0 denote the wave disappear entirely from the setup? One can readily arrange vectors of the two Majorana fermions as they pass through this scenario, e.g., by changing the magnetic field such that the superconducting region of length L. At finite incident the Kitaev material exits the non-Abelian spin-liquid phase. energy, the wave vectors differ, i.e., k ≠ k0, due to unequal The resulting circuit—which furnishes an essential control velocities for the Majorana fermions in that region; recall

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Eq. (27). As the incident-electron energy vanishes, however, stabilized by a large in-plane field because it is a spin- k; k0 → 0 and hence, δϕ → 0 [98]. An incoming electron polarized state [110]. Nearly transparent contacts have been must then exit the superconductor as an electron, yielding made between superconducting MoRe (Hc2 ∼ 10 T) and zero net current across the circuit in Fig. 8(b). In this control graphene in experiments designed specifically to study the scenario the conductance remains G ≈ 0 up to voltages interface between superconductors and quantum-Hall edge ∼ 1 0 − 0 δϕ ν 1 V ð =eLÞðvRvR=jvR vRjÞ,atwhich becomes states (including the ¼ state) [111,112]. Higher out-of- appreciable [96]. plane critical fields could be attained by using NbN or The contrast between the two circuits in Fig. 8 is NbTiN, both of which have been used to make contact to particularly striking given that they differ solely in the graphene [113,114] and have Hc2 exceeding 10 T, or even properties of an electrically inert element. Nontrivial by integrating a layered van der Waals superconductor such conductance quantization for Fig. 8(a) relies on emergent as FeSexTe1−x, which can have Hc2 in excess of 28 T at chiral Majorana fermions in the non-Abelian spin liquid T ¼ 0 [115]. together with fermion condensation, and thus constitutes an electrical signature of both phenomena. As an aside, local VI. ELECTRICAL DETECTION OF BULK ISING – heat capacity measurements at the quantum-Hall spin- NON-ABELIAN ANYONS liquid interface may serve as a complementary probe of fermion condensation. The local heat capacity per unit One can also employ electrical transport to detect π2 2 3 individual bulk Ising anyons in a non-Abelian spin length is cv ¼ð kBT= hvÞðcL þ cRÞ, where cL and cR denote the central charges for the “left” and “right” movers liquid—in fact using a slightly simpler circuit compared to Fig. 8. Consider now the setups shown in Fig. 9. There, at the interface. Decoupled edges yield cL þ cR ¼ 3=2, a single ν ¼ 1 quantum-Hall system is partially sewn to a while fermion condensation reduces the sum to cL þ cR ¼ 1=2 via partial gapping of the interface. Note, however, that spin liquid via fermion condensation mediated by a backing out the central charges from this approach requires grounded superconductor. A source at bias voltage V on (rough) knowledge of the edge velocity v. the left generates a current I that flows through the Figure 8(a) closely resembles the quantum anomalous superconductor to ground. More precisely, electrons ema- ν 1 Hall–superconductor heterostructures studied theoretically nating from the source (i) propagate along the lower ¼ in Refs. [30,99–104] and experimentally in Ref. [105], edge, then (ii) fragment into emergent and physical δϕ where precisely the same quantized conductance was Majorana fermions that acquire a phase difference upon proposed as a signature of physical chiral Majorana edges encircling the spin liquid, and finally (iii) recombine into states at the boundary of a two-dimensional topological either electrons, holes, or superpositions thereof depending δϕ superconductor. In that context, alternative quantization on . Recombination into a hole in this final step indicates mechanisms that do not invoke chiral Majorana modes absorption of a Cooper pair into the superconductor, thereby contributing to the current I. We are interested have also been introduced (e.g., disorder and dephasing, or → 0 if the superconductor behaves as a normal contact [106– in the conductance G ¼ dI=dV in the limit T, V (but see the next section for an extension to finite V). Just as we 108]; see also the critical discussion in Ref. [109]). If — — operative in our setups, such trivial mechanisms would—at see in Sec. V, the spin liquid although electrically inert most—depend weakly on the precise phase of matter plays a decisive role in electrical transport: One of two distinct universal quantized conductances emerges depend- realized in the Kitaev material, thus yielding similar trans- ing on the quasiparticle configuration in the spin liquid. port characteristics for both circuits in Fig. 8. Observing the Suppose first that the spin liquid’s interior is devoid of qualitatively different conductances predicted for Figs. 8(a) Ising non-Abelian anyons as in Fig. 9(a). Here, the phase and 8(b) would therefore strongly suggest against these difference acquired in stage (ii) is simply alternative interpretations. We now briefly discuss the feasibility of the experiment δϕ ¼ k L − k L ; ð28Þ shown in Fig. 8. We require a superconductor that can p p e e survive the high magnetic fields necessary to bring the left with k the momentum of the physical Majorana fermion and right sides into the quantum-Hall regime, and in p as it travels the distance L between the lower and upper particular into the ν ¼ 1 state. Let us consider for con- p ν ¼ 1 edge, and ke and Le the analogous quantities for the creteness a structure that consists of monolayer graphene in → 0 the quantum-Hall regime, coupled on both sides to emergent Majorana fermion. The limit kp, ke yields δϕ → 0, implying that at asymptotically low energies, α-RuCl3. The magnetic field necessary to bring α-RuCl3 into the gapped spin-liquid state is roughly 10 T applied incident electrons recombine into outgoing electrons with at 45° to the plane, resulting in a 7-T out-of-plane field and unit probability. To summarize, stages (i) through (iii) pro- 7-T in-plane field [60]. This field scale is advantageous for ceed according to the graphene ν ¼ 1 state, which is readily attainable in ψ → γ γ → ψ clean samples at fields much lower than 7 T, and is electron physical þ i emergent electron; ð29Þ

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(a) shown in Fig. 9(b) contains a single Ising anyon in the bulk. At asymptotically low incident-electron energies, the emergent Majorana fermion in stage (ii) acquires an addi- tional minus sign upon crossing the Ising anyon absorbed at the edge [i.e., at the termination of the wavy line in the inset of Fig. 9(b); see below for further details]. This all- important minus sign reflects the nontrivial mutual statistics between emergent fermions and Ising anyons in the spin liquid (recall Sec. II B). It follows that δϕ → π as kp, ke → 0, implying that at low energies, incident electrons recombine into outgoing holes with unit probability. Stages (b) (i) through (iii) can then be summarized as

ψ → γ γ electron physical þ i emergent → γ − γ → ψ ; physical i emergent hole ð30Þ

see Fig. 9(b). The perfect “Andreev conversion” of elec- trons into holes yields nontrivially quantized conduct- ance G ¼ð2e2=hÞ. More generally, if the fragmented emergent and physical Majorana fermions encircle nσ Ising anyons in the interior (c) of the spin liquid, the phase difference at low energies is δϕ ¼ πnσ, yielding zero-bias conductance

2e2 G ¼ mod ðnσ; 2Þ : ð31Þ h

The even-odd effect encoded in Eq. (31) represents a FIG. 9. (a),(b) Electrical detection of bulk Ising anyons via “smoking gun” electrical transport signature of bulk quantized zero-bias conductance G. Current flows from the Ising non-Abelian anyons. We stress that one can, at least source to the grounded superconductor. In (a), the spin liquid in principle, toggle between the two quantized conductan- contains no Ising anyons in the bulk. Here, electrons injected at ces by locally perturbing the spin liquid far from any zero energy along the lower ν ¼ 1 edge splinter into physical and electrically active circuit components. Trivial origins for emergent Majorana fermions, but simply recombine into elec- such exotic behavior would appear to require almost divine ν 1 trons at the upper ¼ edge. No current flows into the intervention. At present, however, it remains unclear how to superconductor and hence, G ¼ 0. In (b), a pair of bulk Ising feasibly manipulate Ising anyons so as to probe the even- anyons is pulled out of the vacuum, with one of those anyons dragged to the gapless edge. Nontrivial mutual statistics between odd effect in a systematic experiment. A worthwhile emergent fermions and the single remaining bulk Ising anyon preliminary study could instead rely on thermal fluctuations causes incident zero-energy electrons to recombine perfectly into and/or noise to stochastically drag Ising anyons on and off holes at the upper ν ¼ 1 edge. Each injected electron transmits a of the gapless spin-liquid edge (for related studies in a Cooper pair into the superconductor, yielding G ¼ð2e2=hÞ. quantum-Hall context, see, e.g., Refs. [117–120]). Such Enlargement: The Ising anyon dragged to the boundary couples processes would change nσ as a function of time, leading to to the chiral Majorana edge state with strength λ [see Eq. (32)]. telegraph noise with the conductance switching between (c) If random thermal processes toggle the system between G ¼ 0 and ð2e2=hÞ as sketched in Fig. 9(c) [121]. configurations (a) and (b), G exhibits telegraph noise as a The circuits in Figs. 9(a) and 9(b) can be viewed as function of time, switching stochastically between G ¼ 0 2 cousins of “Z2 interferometers” designed to electrically and ð2e =hÞ. probe physical Majorana fermions in proximitized topological-insulator surfaces [26,27]. In that context, as Fig. 9(a) illustrates. No current flows into the super- the conductance exhibits an analogous even-odd effect, conductor, and therefore, G ¼ 0. Note the similarity to the but in the number of superconducting ðh=2eÞ vortices physics encountered for the control circuit in Fig. 8(b). threaded through the device. Similar to Sec. V, fermion Next, imagine nucleating a pair of Ising non-Abelian condensation allows us to adapt such techniques developed anyons and then dragging one of those anyons to a gapless for exotic superconductors to probe non-Abelian quasipar- part of the spin-liquid edge [116]. The resulting setup ticles in a Mott insulator.

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As a technical aside, above we envision creating fermion living at the boundary acquires a statistical minus Fig. 9(b) by dragging an Ising anyon to the gapless sign upon encircling an Ising anyon, in turn influencing the boundary of the spin liquid. But if this Ising anyon resides electrical conductance, whereas encircling a neutral fer- some “small” distance from the edge [see enlargement in mion yields a trivial statistical phase. Here we study an Fig. 9(b)], how can one quantify whether the fragmented interferometer that enables emergent fermions injected physical and emergent Majorana fermions encircle only the from a lead (with the aid of fermion condensation) to bulk Ising anyon (corresponding to nσ ¼ 1), or also the splinter into unpaired Ising anyons—which exhibit non- Ising anyon near the boundary (corresponding to nσ ¼ 2)? trivial braiding statistics with both bulk Ising anyons and Following Ref. [116], this question can be addressed using neutral fermions, leading to conductance signatures of both a minimal model in which the gapless edge hybridizes with quasiparticle types. the Majorana zero mode γ localized to the adjacent Ising The device we consider appears in Fig. 10(a) and can be anyon. The Hamiltonian reads viewed as Fig. 9(a) with a constriction. At the constriction, Z the upper and lower spin-liquid edges couple via a Hamiltonian H ¼ ½−ivγR∂xγR þ iλγRγδðxÞ; ð32Þ x (a) where γR describes the emergent gapless Majorana fer- mion, and λ is the coupling strength to γ, assumed to reside at position x ¼ 0. Note that λ2=v defines an energy scale for the hybridization. Reference [116] showed that an incident Majorana fermion with energy E acquires a phase shift

2E þ iλ2=v eiϕðEÞ ¼ ð33Þ 2E − iλ2=v due to the λ coupling. The “high-” and “low-” energy limits of this result can be captured intuitively as follows. At incident energies E ≫ λ2=v, the gapless edge and the (b) adjacent Ising anyon essentially decouple; in this “high- energy” regime the physical and Majorana fermions should be viewed as encircling both Ising anyons in Fig. 9(b).No additional π phase shift arises, though finite, nonuniversal conductance generically emerges due to Andreev processes (which freeze out only at low energies). At E ≪ λ2=v, one can project onto Hamiltonian eigenstates by sending γRðxÞ → sgnðxÞγ˜RðxÞ, where γ˜RðxÞ is a slowly varying chiral Majorana fermion. In terms of γ˜R, the λ term in Eq. (32) disappearsR due to the sign change introduced H → − γ˜ ∂ γ˜ above, so that xð iv R x RÞ. In this precise sense, the adjacent Ising anyon has been absorbed by the gapless edge—its only trace is the π phase shift inherent in the FIG. 10. (a) Interferometer that electrically detects both bulk Ising anyons and bulk emergent fermions. The geometry can be definition of γ˜R. Hence, the physical and emergent Majorana fermions should now be viewed as encircling viewed as Fig. 9 with a constriction in the spin liquid, governed by H in Eq. (34). At the constriction, incident emergent only the bulk Ising anyon in Fig. 9(b). Our transport tun Majorana fermions can either tunnel across (tψ process) or analysis focuses on the asymptotic low-energy limit, where fractionalize into a pair of Ising anyons (tσ process)—one the latter scenario prevails. Both extremes captured above hopping across and the other encircling a bulk quasiparticle of are consistent with the general formula in Eq. (33). type a ¼ I, ψ,orσ. Nontrivial braiding statistics among the anyons in the spin liquid yields a-dependent electrical conduc- VII. INTERFEROMETRIC DETECTION OF tances [Eqs. (57) and (59)] that enable readout of the quasiparticle NEUTRAL FERMIONS, ISING ANYONS, AND type. Most notably, non-Abelian statistics between Ising anyons NON-ABELIAN STATISTICS vanquishes first-order conductance corrections from tσ events. (b) Illustration of the five paths taken by incident emergent The circuit introduced in Sec. VI reveals bulk Ising fermions up to first order in Htun. For details, see Secs. VII A anyons but is oblivious to the presence of bulk neutral and VII B. The ellipsis denotes higher-order contributions not fermions. This dichotomy arises because an emergent included here.

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H −iπhσ σ σ −iπhψ γ γ tun ¼ e tσ ðx2Þ ðx1Þþe tψ ðx2Þ ðx1Þð34Þ channel). Thus, in all five paths, emergent Majorana fermions incident from below necessarily exit the interfer- with σðxÞ the Ising conformal field theory (CFT) field and ometer as fermions. The conductance simply follows from tσ, tψ ≥ 0 real couplings. The tσ and tψ terms respectively the phase accumulated en route. We now separately shuttle Ising anyons and emergent fermions between examine each path from Fig. 10(b). positions x1 on the lower edge and x2 on the upper edge. Path (i). Consider an emergent Majorana fermion with (For a detailed discussion of the tσ term, see Ref. [122].) momentum ke that travels a distance Le the long way The phase factors in Eq. (34) involve the topological spin around the interferometer. The associated quantum ampli- hσ ¼ 1=16 of an Ising anyon and hψ ¼ 1=2 of a fermion, tude reads and are required for Hermiticity. Note that e−iπhψ ¼ −i ikeLe represents the usual imaginary coefficient accompanying a Ai ¼ e ; ð35Þ Majorana-fermion bilinear in the Hamiltonian; we employ which is simply the phase acquired by the fermion. this form simply to parallel the tσ term. Ising-anyon tunneling constitutes a relevant perturba- For the remaining cases, it is useful to express the iϕp tion to the fixed point describing decoupled edges, and in amplitude for path (p)asAp ¼ wpe ; here, wp encodes iϕ the asymptotic low-energy limit effectively chops the local physics at the constriction and e p captures the phase spin liquid in two at the constriction [122]. Fermion accumulated due to propagation along the boundary. Note tunneling, by contrast, is marginal. Throughout, we work that wii and wiii are proportional to tψ while wiv and wv are in a regime—to be quantified below—where both tunnel- proportional to tσ. ing terms can be regarded as weak. Incident fermions at the Path (ii). If the fermion propagates to the lower end of the lower edge then bypass the constriction and take “the long constriction and then tunnels across, the amplitude is ” way around with nearly unit probability, enabling a − H A ¼ w eikeðLe LaÞ; ð36Þ perturbative treatment of tun. ii ii Let us then examine Fig. 10(a) at temperatures T → 0 and finite bias voltages V below the gap scale for the where La is the perimeter of the region enclosing a as fermion condensate. We are interested in the conductance sketched in Fig. 10(a). GðVÞ when a quasiparticle of type a ¼ 1, σ,orψ resides in Path (iii). Let Lc be the distance between the constriction the right half of the interferometer. Figure 10(b) sketches and either end of the fermion condensate [see Fig. 10(a)]. The amplitude for path (iii) can then be written the five contributing processes up to first order in tψ and tσ. Path (i) corresponds to the dominant process whereby the A w eikeðLcþLaÞeikeðLaþLcÞ: incident edge emergent fermion bypasses the constriction iii ¼ iii ð37Þ and goes around quasiparticle a (as necessarily occurs in The first exponential represents the phase acquired upon Fig. 9). In path (ii), the incident fermion shortcuts across traveling to the upper part of the constriction, and the the constriction via the fermion-tunneling term tψ . In path second is the phase acquired by the fermion after tunneling (iii), the fermion travels to the upper side of the constriction across the constriction and returning to the fermion con- before similarly tunneling via tψ . Paths (iv) and (v) invoke densate. Noting that Le ¼ 2Lc þ La, we can simplify Ising-anyon tunneling tσ. In (iv), the incident fermion Eq. (37) to travels to the lower end of the constriction, then splinters into a pair of Ising anyons that recombine at the upper edge ik L L A ¼ w e eð eþ aÞ: ð38Þ into either a trivial particle or an emergent fermion depend- iii iii ing on a. And in (v), the incident fermion travels to the Path (iv). When the incident Majorana fermion splinters upper side of the constriction before similarly splintering in path (iv), its momentum ke can partition among the into Ising anyons. In what follows, we examine these resulting pair of Ising anyons in various ways that are processes within a phenomenological treatment that we compatible with energy conservation [126]. Suppose for eventually connect to more formal analyses given in now that the Ising anyon tunneling across the constriction Appendixes C and D (see also the analyses of related carries momentum k1, while the Ising anyon that takes the – interferometers, e.g., in Refs. [123 125]). We start with the long way around carries k2 ¼ k − k1. The amplitude for 1 ψ σ e case a ¼ or and then consider a ¼ . this event is

ik L ik1L ik2ðL þL Þ nψ A. Interferometer with a = 1 or ψ Aivðk1Þ¼wivðke;k1Þe e c e c e a c ð−1Þ − 1 ψ ikeLe ik1La nψ When a ¼ or , the splintered Ising anyons from paths ¼ wivðke;k1Þe e ð−1Þ : ð39Þ (iv) and (v) of Fig. 10(b) necessarily recombine into an outgoing emergent fermion at the upper edge (braiding σ Here, the factor wiv generically depends on both ke and k1 around either 1 or ψ preserves the Ising anyons’ fusion as a consequence of the relevance of the Ising-anyon

031014-15 AASEN, MONG, HUNT, MANDRUS, and ALICEA PHYS. REV. X 10, 031014 (2020)  tunneling term. The first exponential on the top line is the − ikeLe 1 ikeLa ikeLa phase acquired by the fermion as it travels to the con- A ¼ e þ½wiie þ wiiie striction; the second is the phase acquired by the Ising Z ke nψ −ik1L anyon that tunnels across the constriction and travels to the þð−1Þ dk1wivðke;k1Þe a 0 upper end of the fermion condensate, and the third is the Z  phase acquired by the Ising anyon that travels the long way ∞ ik1La þ dk1wvðke;k1Þe : ð42Þ around. In the last factor, nψ is the number of bulk neutral 0 fermions (mod 2) enclosed in the right half of the interferometer, i.e., nψ ¼ 0 if a ¼ 1, while nψ ¼ 1 if We can further constrain the form of the amplitude using a ¼ ψ. The all-important additional π phase that arises dimensional analysis. Since the scaling dimension of when nψ ¼ 1 reflects the nontrivial mutual statistics the Majorana fermion γðxÞ is 1=2, tψ has units of between Ising anyons and neutral fermions, and ultimately energy × length; similarly, the Ising field σðxÞ scaling allows the interferometer to detect the latter bulk quasi- dimension is 1=16, and so tσ has units of energy× particle type. ðlengthÞ1=8. Thus, we can write Events corresponding to distinct, physically permissible   k1 values must be integrated over since the wave function tψ wii;iii ¼ αii;iii; ð43Þ will consist of a weighted sum over all such energy ve partitionings. In particular, the pair of splintered Ising   anyons must both carry positive momentum and energy, tσ fiv;vðk1=keÞ w ðk ;k1Þ¼ ð44Þ yielding the inequality 0 ≤ k1 ≤ ke for path (iv). iv;v e 7=8 v k ke Path (v). Suppose now that an incident emergent e e Majorana fermion in path (v) similarly splinters into one with ve the emergent-fermion edge-state velocity, αii and Ising anyon carrying momentum k1 across the constriction αiii numerical factors, and fiv and fv dimensionless scaling k2 k − k1 and another that carries momentum ¼ e past the functions. Notice that the bracketed factors above are constriction. The corresponding amplitude is dimensionless. Determining the remaining unspecified quantities in A ikeðLcþLaÞ ik1ðLaþLcÞ ik2Lc −1 nψ Avðk1Þ¼wvðke;k1Þe e e ð Þ requires explicit calculations. For the fermion-tunneling

ikeLe ik1La nψ paths, Appendix C presents a standard Heisenberg-picture ¼ wvðke;k1Þe e ð−1Þ : ð40Þ analysis that yields The first three exponentials in the top line respectively 1 denote the phase acquired by the fermion prior to splinter- α −α − ii ¼ iii ¼ 2 : ð45Þ ing, the Ising anyon that tunnels across the constriction, and −1 nψ the Ising anyon that bypasses the constriction. The ð Þ Equations (43) through (45) then allow us to express the factor once again reflects the braiding statistics between amplitude as Ising anyons and neutral fermions.  Physically permissible k1 values must be integrated over, tψ ikeLe as in path (iv), but now the allowed range differs. Indeed, A ¼ e 1 þ i sinðkeLaÞ ve here the Ising anyon that tunnels across the constriction can  7=8 carry arbitrary positive momentum since the pair of Ising 2tσLa −1 nψ anyons that combines on the upper end of the interferom- þ ið Þ gðkeLaÞ ; ð46Þ ve eter always carries total momentum ke regardless of the magnitude of k1. For path (v), we thus have the inequality where we define 0 ≤ ∞ k1 < (neglecting an ultraviolet momentum cutoff Z Z  for simplicity). 1 1 ∞ −iuy iuy Upon summing over the five paths, the amplitude gðuÞ¼ 7 8 dy e fivðyÞþ dy e fvðyÞ : 2iu = 0 0 describing the arrival of the emergent Majorana fermion at the upper end of the fermion condensate is ð47Þ

Treating the Ising-anyon tunneling paths demands a more A ¼ A þ A þ A iZ ii iii Z sophisticated conformal field theory analysis carried out in ∞ ke Appendix D. There we show that þ dk1 Aivðk1Þþ dk1 Avðk1Þ: ð41Þ 0 0   π 1 1 u ;1 2 − 2 gðuÞ¼ 4 1F2 2 ; 4 u ð48Þ Inserting the above expressions for Ai through Av yields

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(a) −15=8 fiv;vðk1=keÞ ∼ ðk1=keÞ at small k1; the exponent 1.5 implies that the ke dependence of the weights defined in 1.0 Eq. (44) drops out at k1 → 0; i.e., in this regime, the Ising- anyon tunneling probability becomes independent of the 0.5 incident fermion momentum. Furthermore, fivðk1=keÞ ∼ −7=8 ð1 − k1=k Þ as k1 approaches k from below and 5 10 15 20 e e −7=8 fvðk1=keÞ ∼ ðk1=ke − 1Þ as k1 approaches ke from above. Notice that fvðk1=keÞ does not diverge as k1 (b) approaches ke from below—hence, for path (v), Ising anyons tunnel far more efficiently with momentum slightly

100 larger than ke compared to momentum just below ke. 50 It is instructive to examine some limits of the function gðkeLaÞ. At small arguments, one finds 10 5 π ≪ 1 ≈ ; gðkeLa Þ 4 keLa ð50Þ 1 0.5 i.e., the amplitude correction from Ising-anyon tunneling vanishes linearly with the fermion momentum. In this limit 0.0 0.5 1.0 1.5 of our perturbative analysis, one can use an operator product expansion (OPE) to fuse the Ising CFT fields in FIG. 11. (a) Plot of gðkeLaÞ [Eq. (48)] versus keLa, where ke Eq. (34), yielding denotes the incoming emergent-fermion momentum and La is the length defined in Fig. 10(a). This function determines the tσ 15 8 −iπhσ σ σ → = γ∂ γ correction to the fermion transmission amplitude [Eq. (46)], and e tσ ðx2Þ ðx1Þ const × tσLa x : ð51Þ thus also the conductance [Eq. (57)], for the interferometer in Fig. 10(a) when a ¼ I or ψ. (b) Magnitude of the scaling The term on the right side (which is a descendent of the functions fiv (red dashed line) and fv (blue line) obtained by identity) is irrelevant, which explains the linear vanishing inverting Eq. (47) as carried out in Appendix F. These scaling of the amplitude correction as keLa → 0. functions govern energy partitioning associated with Ising-anyon At keLa ≫ 1, one instead finds tunneling in paths (iv) and (v) summarized in Fig. 10(b) and discussed in Sec. VII A; recall Eq. (44). In the horizontal axis k1 cosðkeLaÞ is the momentum carried by an Ising anyon that tunnels across the gðkeLa ≫ 1Þ ≈ 1 − : ð52Þ constriction. The limiting scaling behavior displayed above keLa implies that Ising anyons tunnel primarily carrying momentum ≪ke and secondarily carrying momentum very near ke. Contrary to the purely oscillatory amplitude correction from fermion tunneling, the amplitude correction from Ising-anyon tunneling thus tends to an La-dependent with 1F2 a generalized hypergeometric function. Figure 11(a) constant as keLa → ∞, with subdominant oscillations that plots gðkeLaÞ versus keLa. The perturbative regime stipulated decay with a 1=ðkeLaÞ prefactor. The former feature earlier requires [128] reflects the propensity of Ising anyons to tunnel across the constriction with vanishingly small momentum. 7=8 tψ tσLa Oscillations in the latter piece come from Ising anyons ≪ 1 and ≪ 1 ð49Þ ve ve that tunnel with momentum near ke, while the decay arises because of the finite spread in the allowed momentum so that fermion and Ising-anyon tunneling across the con- carried. striction contribute only small corrections to the amplitude The tunneling Hamiltonian invoked in Eq. (34) could of in Eq. (46). course be amended in various ways, e.g., by allowing To make contact with our phenomenological picture, we fermions and Ising anyons to tunnel over a finite range of can invert Eq. (47) to extract the fivðk1=keÞ and fvðk1=keÞ positions between the upper and lower sides of the scaling functions that quantify the energy partitioning. constriction (as opposed to discrete points x1;2)orby Appendix F pursues this (nontrivial) exercise; for a sum- adding derivatives that effectively make the tunnel cou- mary, see Fig. 11(b). Both scaling functions exhibit a plings momentum dependent. It is thus important to address leading divergence at k1 → 0 and a subleading divergence which features of the amplitude correction in Eq. (46) are at k1 → ke. It follows that Ising anyons tunnel across the generic. The linear vanishing of both the tψ and tσ constriction primarily carrying “small” momentum and corrections with ke [cf. Eq. (50)] is certainly universal secondarily carrying momentum near ke. More explicitly, (though the prefactor of course is not). At ke → 0, one can

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exploit an OPE similar to Eq. (51) to reduce arbitrary Ga¼1;ψ ðVÞ fermion-tunneling and Ising-anyon tunneling terms to     e2 L L irrelevant descendants of the identity. The leading irrelevant ≈ 1 − cos eV p − e γ∂ γ h vp ve term is x , and the derivative ensures amplitude correc-    tions ∝k as k → 0. tψ L e e − p − Le þ La We further expect that the exponents governing the cos eV 2ve vp ve power-law divergences in the f scaling functions shown    iv;v t L − in Fig. 11(b) are universal. At k L ≫ 1, these divergences ψ p − Le La e a þ 2 cos eV determine the leading and subleading k L dependence of ve vp ve e a      the tσ correction specified by Eq. (52)—which would then 7=8 2tσLa L Lp L also be universal. Note, however, that the prefactors of the − ð−1Þnψ g eV a sin eV − e : v v v v two terms in Eq. (52) will depend on details of the e e p e tunneling Hamiltonian. The generic saturation of the tσ ð57Þ correction to an La-dependent constant at keLa → ∞ admits an intuitive explanation: Energy partitioning invar- iably suppresses oscillations as keLa increases, whereas In the first line, the oscillatory voltage dependence reflects processes for which Ising anyons tunnel across the con- the periodic revival and destruction of Andreev processes as striction carrying vanishingly small momentum naturally the phase difference accumulated by the physical and leave a ke-independent correction. [Again, the Ising-anyon emergent Majorana fermions varies in path (i). The next tunneling weights in Eq. (44) do not depend on the incident two lines encode corrections from fermion tunneling across fermion momentum in the k1 → 0 limit.] the constriction; hence, the dependence on the shifted path We are now ready to extract the conductance. The net lengths Le La. And by far most importantly, the correction phase acquired by an incident emergent Majorana fermion from Ising-anyon tunneling in the last line reveals the is given by presence of individual emergent fermions by virtue of the nψ dependence. ϕ A ei emergent ¼ : ð53Þ jAj B. Interferometer with a = σ Suppose now that a ¼ σ (which turns out to be far Moreover, the phase acquired by a physical Majorana simpler to analyze compared to the a ¼ 1 or ψ cases). We fermion with momentum k that travels the length L of p p assume that the bulk Ising anyon’s “partner” has been the fermion condensate is dragged to an adjacent gapless part of the edge in the right half of the interferometer. Path (i) acquires an additional π iϕ ik L e physical ¼ e p p ; ð54Þ phase relative to Eq. (35) due to the nontrivial mutual statistics between fermions and Ising anyons. By contrast, yielding a phase difference the phases from paths (ii) and (iii)—wherein the edge fermion encircles the bulk Ising anyon an even number of δϕ times—conform exactly to Eqs. (36) and (38), respectively a¼1;ψ ðke;kpÞ 7=8 [129]. In paths (iv) and (v), the incident emergent fermion tψ 2tσL nψ a splinters into two Ising anyons at the constriction, one of ¼ kpLp − keLe − sinðkeLaÞ − ð−1Þ gðkeLaÞ ve ve which now encircles a bulk Ising anyon. This braiding ð55Þ process changes the fusion channel for the splintered edge Ising anyons from ψ to 1. More physically, in paths (iv) and (v) the incident emergent fermion exits the interferometer to first order in tψ , tσ; cf. Eq. (28). Suppose next that an as a trivial boson, and hence, these paths no longer incident electron injected from the lead into the lower ν 1 ¼ contribute to the electrical conductance. edge of Fig. 10(a) carries energy E. The Majorana-fermion The amplitude from Eq. (46) accordingly becomes momenta are then ke ¼ E=ve and kp ¼ E=vp, where vp is the physical Majorana fermion’s edge velocity. As reviewed in Appendix G, the conductance at bias voltage V is   tψ ikeLe −1     A ¼ e þ i sinðkeLaÞ : ð58Þ 2 ve e 1 − δϕ eV eV Ga¼1;ψ ðVÞ¼ cos a¼1;ψ ; : ð56Þ h ve vp Following precisely the same steps outlined in the preced- Finally, expanding in tψ and tσ yields ing section, one obtains a conductance

031014-18 ELECTRICAL PROBES OF THE NON-ABELIAN SPIN … PHYS. REV. X 10, 031014 (2020)     2 (a) (b) e Lp Le G σðVÞ ≈ 1 þ cos eV − a¼ h v v   p e  tψ L L þ L − cos eV p − e a 2v v v e   p e  tψ L L − L þ cos eV p − e a : ð59Þ 2ve vp ve

In the limit tσ ¼ tψ ¼ 0, the interferometer effectively reduces to the setup in Figs. 9(a) and 9(b) that allow electrical detection of Ising anyons. Indeed, Eqs. (57) and FIG. 12. Variations on Fig. 7 in which the ν ¼ 1 quantum-Hall (59), respectively, yield G ¼ 0 and G ¼ð2e2=hÞ at V → 0, system is replaced by (a) a 2D spinless p þ ip superconductor and (b) a proximitized 1D wire. Both alternatives also enable in agreement with the analysis from Sec. VI. Allowing physical-fermion ↔ emergent-fermion conversion via fermion quasiparticle tunneling across the constriction additionally condensation. In (a), the physical chiral Majorana edge state of a reveals the non-Abelian statistics of Ising anyons as spinless 2D p þ ip superconductor gaps out with the spin liquid’s manifested by the disappearance of the oscillatory tσ emergent chiral Majorana edge mode—yielding a fully gapped correction in Eq. (57) when a bulk Ising anyon sits in interface between the two subsystems. Physical Majorana fer- the interferometer loop. Qualitatively similar physics mions from the p þ ip superconductor thus invariably enter the appears in quantum-Hall interferometers introduced in spin liquid as emergent Majorana fermions. In (b), right- and left- Refs. [9–11]. moving electron channels (double arrows) separate into Majorana modes beneath the proximitizing superconductor. Fermion con- densation arises when one of those Majorana modes gaps out VIII. DISCUSSION with the spin liquid’s edge state. (Among the three remaining modes beneath the superconductor, only one must be gapless.) A growing body of evidence supports the realization of a non-Abelian spin-liquid phase in the Kitaev material α-RuCl3 [50,54–61,64]. This remarkable development strongly motivates proposals for probing individual frac- transport characteristics of circuits employing such varia- tionalized excitations in honeycomb Kitaev materials, as tions would certainly be worthwhile. required for eventual topological-quantum-computing Moreover, developing a detailed microscopic under- applications. The Mott-insulating nature of the host plat- standing of the interaction mediating fermion condensation form renders the problem both interesting and nontrivial. [recall the κ term from Eq. (17)] remains an important open We introduce a strategy based, counterintuitively, on problem even for our quantum-Hall-based setups. We universal low-voltage electrical transport in novel circuits anticipate that spin-spin interactions between the quan- designed to perfectly convert electrons into emergent tum-Hall edge and spin-liquid edge (mediated, for instance, Majorana fermions born in the spin liquid, and vice versa. by an interplay between Coulomb repulsion and electron Similar techniques can be adapted to any bosonic topo- tunneling between the subsystems) constitute one natural logically ordered phase hosting gapless emergent-fermion microscopic mechanism that contributes to κ. In Sec. III A, edge states. we show on symmetry grounds that microscopic spin Perfect physical-fermion ↔ emergent-fermion conver- operators at the Kitaev-material boundary project onto the sion transpires when the non-Abelian spin liquid’s chiral kinetic energy operator in the chiral Majorana edge theory. Majorana edge state gaps out with a counterpropagating We expect that spin operators at the quantum-Hall boundary 1D electronic channel—forming a fermion condensate. similarly project onto the kinetic energy for the proximitized Throughout this paper, we consider proximitized ν ¼ 1 quantum-Hall edge—in turn yielding the desired κ term. integer quantum-Hall systems as the source of 1D electrons Working out detailed dependences on factors such as participating in fermion condensation. We adopt this choice Zeeman fields, structure and sign of the spin-spin interaction, due to the wide availability of ν ¼ 1 states and for and spin-orbit effects—and how they impact the all-impor- convenience in defining electrical transport quantities. tant sign of κ—would be very interesting. As a possible variation, one could replace the quantum- In Sec. V, we see that quantum-Hall–Kitaev-material– Hall system with a spinless 2D p þ ip superconductor, quantum-Hall circuits (Fig. 8) reveal the spin liquid’s which supports “half” of a ν ¼ 1 edge state and thus admits emergent chiral Majorana edge state via quantized zero- a fully gapped interface with the spin liquid. Alternatively, bias charge conductance. This purely electrical fingerprint one could employ proximitized 1D wires instead of 2D complements the well-known quantized thermal Hall sig- topological phases at the expense of introducing additional nature of a Majorana edge mode [34,60,64], and relies electronic channels. For illustrations, see Fig. 12. Exploring critically on the fermion condensation that underlies our

031014-19 AASEN, MONG, HUNT, MANDRUS, and ALICEA PHYS. REV. X 10, 031014 (2020) anyon-detection schemes. Section VI explores a somewhat that this work helps stimulate such efforts with the ultimate simpler quantum-Hall–Kitaev-material device (Fig. 9) aim of crafting a realistic road map toward topological designed to electrically detect bulk Ising non-Abelian quantum computation with Kitaev materials. anyons. In particular, the circuit admits zero-bias conduct- ance quantized at either G ¼ 0 or G ¼ 2e2=h depending on ACKNOWLEDGMENTS whether an even or odd number of Ising anyons resides in the bulk. This striking even-odd effect reflects the non- It is a pleasure to thank Chao-Ming Jian, Stevan Nadj- trivial mutual braiding statistics between Ising anyons and Perge, Achim Rosch, Ady Stern, and especially Paul emergent fermions. Taking the same circuit and adding a Fendley for stimulating conversations. This work is sup- constriction within the spin liquid leads to the interferom- ported by a postdoctoral fellowship from the Gordon and eter that we explore in Sec. VII (Fig. 10). At the Betty Moore Foundation, under the EPiQS initiative, Grant constriction, an emergent fermion can splinter into a pair No. GBMF4304; the Army Research Office under Grant of Ising anyons—one taking a shortcut across the pinch and No. W911NF-17-1-0323; the National Science Foundation the other continuing along the spin-liquid edge. The through Grants No. DMR-1723367 (J. A.) and No. DMR- electrical conductance is then sensitive to the presence 1848336 (R. M.); the Caltech Institute for Quantum of bulk Ising anyons and bulk emergent fermions since Information and Matter, a NSF Physics Frontiers Center Ising anyons exhibit nontrivial braiding statistics upon with support of the Gordon and Betty Moore Foundation encircling either quasiparticle type. Notably, non-Abelian through Grant No. GBMF1250; the Walter Burke Institute braiding statistics is manifested as a vanishing of certain for Theoretical Physics at Caltech, and the Gordon and conductance corrections when an odd number of bulk Ising Betty Moore Foundation’s EPiQS Initiative, Grant anyons sits in the interferometer [cf. Eqs. (57) and (59)]— No. GBMF8682 to J. A. B. M. H. acknowledges support similar to the physics encountered in fractional quantum- from the Department of Energy under the Early Career Hall architectures [9–11]. program (Grant No. DE-SC0018115). D. M. acknowledges Analyzing circuits with realistic imperfections poses support from the Gordon and Betty Moore Foundation another worthwhile direction for future investigation. For EPiQS Initiative, Grant No. GBMF9069. Finally, we instance, the superconductors in practice will likely contain acknowledge the 2018 Erice Conference on Majorana low-energy degrees of freedom due to disorder and/or Fermions and Topological Materials Science, where this vortices. Electrons from the ν ¼ 1 edge could directly hop work was initiated. onto these low-lying modes without encountering the fermion condensate or Andreev reflecting, thereby provid- ing a parallel conduction channel that modifies the con- APPENDIX A: VARIATIONAL ANALYSIS OF ductances predicted in this paper. Thermally excited INTERACTING MAJORANA FERMIONS quasiparticles in the spin liquid can also produce unwanted errors, e.g., due to processes wherein an edge emergent 1. Continuum model fermion splinters into Ising anyons that enclose a thermally Here we employ a variational approach to study the excited bulk quasiparticle residing near the boundary. For interacting continuum Hamiltonian H0 þ δH defined in our purposes, these corrections must be sufficiently small Eqs. (16) and (17). Specifically, we view the ground state of H that (i) a sharp contrast remains between the nontrivial and the free-fermion Hamiltonian MF from Eq. (18) as a trial control circuits in Fig. 8, and (ii) the quasiparticle-depen- wave function for the interacting problem, treating the mass dent conductances predicted for Figs. 9 and 10 remain m as a variational parameter. Note that sending γR → −γR discernible. leaves the interacting Hamiltonian invariant but sends For initial anyon-detection experiments, one could rely on m → −m. Without loss of generality, we therefore consider nature to thermally cycle between various bulk quasiparticle trial wave functions with m ≥ 0 below. configurations—leading to telegraph noise wherein the The analysis is most conveniently carried out in terms of conductance randomly cycles among the predicted values momentum-space Majorana fermions defined with Fourier- as a function of time [see, e.g., Fig. 9(c)]. Deterministic, real- transform conventions time anyon control is nevertheless clearly desirable. To this Z end, it is essential to develop practical means of trapping 1 dk γ ðxÞ¼pffiffiffi eikxγ ; ðA1Þ Ising anyons and emergent fermions in the spin liquid. We A 2 2π A;k anticipate that this subtle energetics issue can be profitably Z addressed by studying the Kitaev honeycomb model sup- pffiffiffi γ 2 dxe−ikxγ x plemented by generic symmetry-allowed perturbations. A;k ¼ Að ÞðA2Þ Finally, developing complementary methods of detecting individual bulk anyons that mitigate the experimental for A ¼ L or R. The corresponding commutation relations requirements of our scheme poses a key challenge. We hope are given by

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1 (a) fγAðxÞ; γBðyÞg ¼ δABδðx − yÞ; ðA3Þ 2 0.003

fγA;k; γB;qg¼2πδABδðk þ qÞ: ðA4Þ

Upon passing to momentum space, the free-fermion H Hamiltonian MF can be readily diagonalized, yielding pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0.000 single-particle excitation energies EðkÞ¼ ðvkÞ2þðm=2Þ2 and correlation functions –0.001   1 vk hγ γ i¼2πδðk þ qÞ 1 þ ; ðA5Þ 0.0 0.5 1.0 1.5 2.0 R;k R;q 2 EðkÞ   1 vk hγ γ i¼2πδðk þ qÞ 1 − ; ðA6Þ L;k L;q 2 EðkÞ (b) 4 i m hγR;kγL;qi¼2πδðk þ qÞ : ðA7Þ 4 EðkÞ 3

Here and in the remainder of this Appendix, expectation H 2 values are taken with respect to the ground state of MF. Equations (A5) through (A7) allow one to efficiently E ≡ H δH 1 evaluate the trial energy density trial h 0 þ i=L (L is the length of the interface). The results are conveniently expressed in terms of integrals 0 Z 0 50 100 150 200 2vΛ=m α ffiffiffiffiffiffiffiffiffiffiffiffiffiu Iα ¼ du p 2 ; ðA8Þ −2vΛ=m 1 þ u FIG. 13. (a) Variational energy difference ΔEtrial [Eq. (A14)] versus mean-field mass m for varying interaction strengths κ where Λ is a momentum cutoff for the interacting Majorana in the continuum model given by Eqs. (16) and (17). For fermions. The kinetic energy piece reads κΛ2 ≳ 102v, the variational energy is minimized with nonzero 2 m, signaling the spontaneous opening of a gap. (b) Optimal mass hH0i m as a function of interaction strength. In the normalizations ¼ − I2: ðA9Þ L 16πv adopted above, v is the velocity and Λ is the cutoff.

For the interactions, we first write 2 4 m κm 2 Z E ¼ − I2 − ðI2 þ I0I2Þ: ðA13Þ hδHi κ trial 16πv 1024π2v4 ¼ q2q4hγ γ γ γ i; ðA10Þ 4 R;q1 R;q2 L;q3 L;q4 L q1;q2;q3;q4 We now minimize Eq. (A13) with respect to m.For κ 0 0 κ 0 and then evaluate the expectation value in the integrand < , the minimization always yields m ¼ . With > , using Wick’s theorem: however, a nontrivial solution does arise beyond a critical interaction strength. It is convenient to examine hγR;q γR;q γL;q γL;q i¼hγR;q γR;q ihγL;q γL;q i 1 2 3 4 1 2 3 4 ΔE ≡ E − E − γ γ γ γ trial trial trialjm¼0; ðA14Þ h R;q1 L;q3 ih R;q2 L;q4 i γ γ γ γ þh R;q1 L;q4 ih R;q2 L;q3 i: ðA11Þ which quantifies the change in energy density due to a ΔE Λ2 nonzero mass m. Figure 13(a) plots trial=ðv Þ versus Some algebra yields m=ðvΛÞ for varying dimensionless interaction strengths κΛ2=v, while Fig. 13(b) displays the optimized mass as a δH κ 4 h i − m 2 function of κΛ2=v. A first-order jump in the optimal mass ¼ 2 4 ðI2 þ I0I2Þ: ðA12Þ L 1024π v appears at a critical interaction strength Summing the contributions above gives our trial energy κΛ2 ≈ 102 density, ð =vÞc : ðA15Þ

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Z The large numerical factor on the right-hand side naively π=2 dk cos k cosðkxÞ suggests that spontaneous mass generation requires implau- fc;x ¼ ; ðA22Þ −π 2 2π EðkÞ sibly large interactions. We stress that such a conclusion is = generally incorrect. In the next subsection, we see that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 4 ¯ 2 lattice models with microscopic interaction strength where EðkÞ¼ ð t sin kÞ þð m cos kÞ are the single- κ particle excitation energies for HMF. Umicroscopic yield ¼ cUmicroscopic, where c is a number 2 3 We can now straightforwardly obtain our trial energy in the range 10 − 10 . Thus, even modest microscopic density E H0 δH =N. The hopping contribution is interactions translate into “large” κ values that can exceed trial ¼h þ i the threshold in Eq. (A15), consistent with conclusions simply from more rigorous microscopic analyses reviewed in hH0i 2 Secs. III A and III B. ¼ −8t f 1: ðA23Þ N s; 2. One-dimensional lattice model The interaction term can be treated using the decomposition Next, we similarly study interacting Majorana fermions γ γ γ γ γ γ γ γ on an N-site chain with periodic boundary conditions h a aþm aþn aþpi¼h a aþmih aþn aþpi governed by a Hamiltonian − γ γ γ γ h a aþnih aþm aþpi γ γ γ γ H ¼ H0 þ ΔH; ðA16Þ þh a aþpih aþm aþni; ðA24Þ X γ γ yielding H0 ¼ it a aþ1; ðA17Þ a δ X h Hi −16 − δ − γ γ γ γ ¼ gðFmnp Fnmp þ FpmnÞ; ðA25Þ H ¼ g a aþm aþn aþp ðA18Þ N a where with t>0 and 0

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Note especially the large numerical prefactor in front of U in (a) connection with the discussion at the end of Appendix A1. 0.6 We now minimize Eq. (A29) with respect to m¯ . ΔE ¯ Figure 14(a) plots trial=t versus m=t for several U=t 0.4 values. The optimized m¯ as a function of U=t appears in

Fig. 14(b). Our variational analysis predicts spontaneous 0.2 dimerization—now via a continuous transition in contrast to the continuum model—for U ≳ 0.29t. This prediction agrees reasonably well with DMRG, which yields sponta- 0.0 neous dimerization for U ≳ 0.428t [87]. Interestingly, for 2 –0.2 U ¼ t= our variational ansatz actually becomes exact 0 2 4 6 8 since in this limit the interaction admits an exact self- consistent mean-field decoupling; recall the discussion in Sec. III B. Figure 14(c) shows the optimized variational (b) energy density together with the exact energy density 3.5 extracted from Ref. [87], which indeed agree at U ¼ t=2. 3.0 Finally, suppose that we instead take g ¼ −UR and m ¼ 1, n ¼ 2, p ¼ 3. This choice corresponds to a differ- 2.5 ent interaction in which four adjacent Majorana fermions 2.0 interact with strength UR. We consider only UR > 0 in our 1.5 variational analysis since negative UR values generate 1.0 additional gapless modes beyond those in Eqs. (16) and 0.5 (17) (see Appendix B for a synopsis); our trial wave 0.0 functions are thus not expected to adequately capture the 0.0 0.2 0.4 0.6 0.8 1.0 1.2 physics at UR < 0. The trial energy density for this model becomes (c) E −8 2 − 64 2 2 trial ¼ t fs;1 UR½t ðfs;1 þ fs;1fs;3Þ –0.50 ¯ 2 2 − –0.55 þ m ðfc;1 fc;1fc;3Þ: ðA31Þ –0.60 Upon connecting to the continuum energy density as we –0.65 outline above, one finds the relations –0.70 –0.75 v ¼ 4t; m ¼ 8m;¯ κ ¼ 256U : ðA32Þ R –0.80

Once again, a large prefactor appears in the expression for κ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 in agreement with the alternative derivation from Ref. [93]. Based on our continuum analysis, one might therefore expect that a spontaneous-dimerization transition sets in FIG. 14. Variational results for the lattice model given by 1 3 4 once U =t > 0 becomes of order unity, as arises for the Eqs. (A16) through (A18) with m ¼ , n ¼ , p ¼ ,and R ΔE alternative interaction U. Curiously, however, minimizing g ¼ U. (a) Variational energy difference trial [Eq. (A28)] ¯ versus mean-field dimerization order parameter m¯ for varying the lattice energy density in Eq. (A31) yields an optimal m ≳ 0 29 0 interaction strengths U.AtU . t, the variational energy is that vanishes for any UR > . Figure 15 illustrates this ¯ ΔE −1 ¯ minimized by nonzero m, indicating spontaneous dimerization conclusion by plotting trial=t versus tan ðm=tÞ for (as captured by DMRG, but at slightly larger interaction varying UR=t values. More rigorous DMRG simulations strengths U ≳ 0.428t [87]). (b) Optimal variational m¯ as a do capture a dimerization transition but only at extremely function of interaction strength. (c) Variational ground-state E E large UR=t. The following Appendix resolves the apparent energy trial (solid red line) and exact ground-state energy exact discrepancy between our continuum and lattice analyses (dashed blue line) [87] versus U=t. The two converge at and explains the curious suppression of the instability for U ¼ t=2, at which point the variational approach becomes the UR interaction. exact.

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14 Let us start from the noninteracting limit, UR ¼ 0. 12 Here, the chain is diagonalized by passing to momentum 10 space via 8 pffiffiffi 2 X ffiffiffiffi ika 6 γa ¼ p e γk; ðB4Þ N 4 k 2 with N the number of Majorana sites in the chain. In 0 our conventions, fγk; γk0 g¼δk;−k0 . Note also that self- 0.0 0.5 1.0 1.5 2.0 2.5 3.0 γ γ γ† Hermiticity of a implies that k ¼ −k; this relation allows a complete description of the chain using operators acting FIG. 15. Variational energetics for the lattice model given by in half of the Brillouin zone. In particular, one finds Eqs. (A16) through (A18) with m ¼ 1, n ¼ 2, p ¼ 3, and X − ϵ γ†γ g ¼ UR. The plot shows the variational energy difference H0 ¼ 0ðkÞ k k; ðB5Þ ΔE −1 ¯ ¯ −π 0 trial [Eq. (A28)] versus tan ðm=tÞ,withm the mean-field

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The first term in Eq. (B9) reflects renormalization of the 1.4 nearest-neighbor hopping amplitude t, while the second 1.2 represents an interaction-induced third-neighbor hopping. 1.0 The decomposition of interactions employed above is very 0.8 0.6 useful, at least for sufficiently small jURj=t, since the 0.4 expectation value of the energy for single-excitation states 0.2 can be read off immediately. Specifically, upon including the 0.0 bare component, the total kinetic energy for an excitation –1.0 –0.5 0.0 0.5 1.0 UR/t with momentum k becomes ϵðkÞ ≡ ϵ0ðkÞþδϵðkÞ.The dispersion for right- and left-moving Majorana fermions FIG. 16. Dimerization order parameter hiγ −1γ − iγ γ 1i ϵ a a a aþ in the continuum limit follows from expanding ðkÞ near versus UR=t obtained from DMRG simulations of the Hamil- k ¼ −π and 0; one finds ϵ ∼ vk with tonian in Eq. (B11). This Hamiltonian is the same as Eq. (B1) but with the interaction-induced kinetic energy renormalization 128 subtracted off. The subtraction reduces the critical interaction v ¼ 4t þ U ðB10Þ 3π R strength required for spontaneous dimerization by 3 orders of magnitude—from UR=t ≈ 250 down to UR=t ≈ 0.48. Outside of the renormalized velocity. the dome-shaped dimerized region, the system realizes a gapless 0 For UR < , interactions reduce v, and at a critical value state with central charge c ¼ 1=2 (at least over the UR=t window − 3π 32 ≈−0 295 1 2 ≳ 0 65 UR ¼ ð = Þt . t the velocity vanishes. Below shown). Reentrance of the c ¼ = critical state at UR=t . this value, the renormalized kinetic energy ϵðkÞ supports can also be understood from the subtraction, as we explain in additional pairs of gapless Majorana fermions—changing Appendix B. the central charge from c ¼ 1=2 to 3=2. Remarkably, the critical interaction strength U extracted from our treatment R Conspicuously, Fig. 16 also reveals a reentrant c ¼ 1=2 agrees quantitatively with DMRG predictions. We note that critical phase for U =t ¼ 0.64585ð3Þ. We can explain this the mean-field treatment performed in Ref. [94] recovers R feature as well by examining Eq. (B11), which can similar quantitative agreement. equivalently be written as For UR > 0, interactions instead enhance v. Recall from Sec. III A that in the low-energy description, the critical X X 1 2 0 γ γ γ γ γ γ phase with central charge c ¼ = becomes unstable to HR ¼ it a aþ1 þ UR a−1 a aþ1 aþ2 κΛ2 spontaneous mass generation when =v becomes of a  a  order unity, where Λ is a momentum cutoff and κ is the 2 7 − i γ γ γ γ coupling from Eq. (17). In the present context, we have π 3 a aþ1 þ a aþ3 : ðB12Þ κ ∝ UR. Upward renormalization of v clearly boosts the robustness of the c ¼ 1=2 phase against interactions, Up to moderate values of UR=t, it is natural to interpret the though we are unable to quantitatively obtain the critical second line—which is just δH expressed in real space— ∼ 250 KE value of UR at which DMRG finds an instability. as a correction to the bare kinetic energy on the first line. We can, nevertheless, stringently test the scenario However, at sufficiently large UR=t this “correction” over- above. If kinetic energy renormalization indeed pushes whelms the bare piece, suggesting the following alternative the spontaneous-dimerization transition to extremely viewpoint. Let us trivially rewrite H0 as large U values, then removing this renormalization R R    should reduce the critical UR by 3 orders of magnitude. X 4i 7 0 γ γ − γ γ γ γ Remarkably, we indeed find such a dramatic reduction. HR ¼ it a aþ1 π UR 3 a aþ1 þ a aþ3 Consider the modified Hamiltonian a X γ γ γ γ 0 − δ þ UR a−1 a aþ1 aþ2 HR ¼ HR HKE; ðB11Þ a  2 7 which is identical to Eq. (B1) except that the kinetic energy i γ γ γ γ þ π 3 a aþ1 þ a aþ3 ; ðB13Þ renormalization is subtracted off, thus yielding a UR- independent velocity. DMRG simulations find a transition “ ” to a gapped phase in this model at UR=t ≈ 0.48098ð1Þ, and view the top line as our new bare kinetic term. For comparable to the critical interaction strength obtained from UR=t > 3π=16 ≈ 0.59, all of the corresponding kinetic the alternative model in Eq. (20) at t0 ¼ 0. See Fig. 16. energy eigenvalues flip sign compared to the kinetic energy — Furthermore, at least over the range of UR shown in the from the t term alone thus completely changing the figure, the transition to a c ¼ 3=2 phase at UR < 0 has also character of the associated noninteracting ground state. been removed (as expected upon removal of the kinetic The second and third lines represent an interaction that has energy renormalization). no nontrivial matrix elements in the subspace with zero or

031014-25 AASEN, MONG, HUNT, MANDRUS, and ALICEA PHYS. REV. X 10, 031014 (2020) one single-particle excitations about that modified ground (a) state. [Note the relative sign between the final terms in Eqs. (B12) and (B13).] In this UR=t regime, the top line yields a velocity for right and left movers of v ¼ð256=3πÞUR − 4t. Once again, we end up with a kinetic energy scale that grows with UR, so that the dimerization instability is naturally suppressed beyond a critical value of UR=t as observed in DMRG. As an additional sanity check, one can extract kinetic energy renormalization arising from the four-fermion interaction in Eq. (20) via exactly the same procedure leading to Eq. (B7) above. The result takes the form in (b) Eq. (B8) where now 64 δϵ − 3 ðkÞ¼ 3π Uj sin kj: ðB14Þ

Near k ¼ 0 and −π, δϵðkÞ ∝ k3, indicating that velocity renormalization vanishes in this model. Thus, no such FIG. 17. (a) Geometry and coordinates used to explicitly suppression of the dimerization instability is expected in analyze the emergent-fermion transmission amplitude in the our scenario [131], and indeed, a transition occurs at the interferometer from Fig. 10. Appendixes C and D, respectively, ≈ 0 428 modest value U . t [87]. Reference [87] further treat the cases where only fermions tunnel (with coupling tψ ) and δ studied Eq. (B1) with H replaced by yet another inter- only Ising anyons tunnel (with coupling tσ) across the constric- action, tion. (b) Unfolded version of (a), not to scale. In terms of lengths X shown in Fig. 10(a),wehavex2 − x1 ¼ La and x3 − x0 ¼ Le. δ 0 γ γ γ γ − γ γ γ γ H ¼ Uy ð a−2 a aþ1 aþ2 a−2 a−1 a aþ2Þ: ðB15Þ a

In this case, we find that δϵðkÞ¼0—i.e., Uy produces no simplicity, we consider the case where no nontrivial kinetic energy renormalization at all. DMRG simulations quasiparticles reside in the bulk of the interferometer; find a transition at a similarly modest value U ≈ 0.45t i.e., we assume a ¼ I in Fig. 10. y ∂ γ [87]. Together, the results above strongly support our Evaluating the Heisenberg equation of motion t ¼ H H γ H explanation for the anomalously strong interaction strength i½ 0 þ tun; , with 0 the chiral Majorana kinetic required for dimerization in Eq. (B1), which has heretofore energy, one finds remained enigmatic. i Zooming out, we see from this discussion that micro- ∂ γ − ∂ γ −iπhψ δ − γ t ðx; tÞ¼ ve x ðx; tÞþ2 tψ e ½ ðx x1Þ ðx2;tÞ scopic details matter when dealing with instabilities arising from “strong” interactions that are irrelevant at weak − δðx − x2Þγðx1;tÞ: ðC1Þ coupling. The insights we obtain here can potentially be exploited to concoct new models that, depending on the As before, ve is the emergent-fermion edge velocity, while desired outcome, either enhance or suppress the effects of x1 and x2 respectively denote positions on the lower and such strong irrelevant interactions. upper sides of the constriction (see again Fig. 17). By solving the equation of motion, one can relate γðx3;tÞ to γðx0; 0Þ. The phase of interest follows from the equal-time APPENDIX C: ANALYSIS OF FERMION iϕ relation γðx3; 0Þ¼e emergent γðx0; 0Þ, so hereafter we focus TUNNELING ACROSS A CONSTRICTION on the solution at t ¼ 0. Here we study the interferometer in Fig. 10 in the limit Away from x ¼ x1;2, Eq. (C1) reduces to a standard þ where only fermions are allowed to tunnel across the chiral wave equation. Suppose that xj denotes a coordinate constriction. That is, we take the fermion-tunneling ampli- slightly larger than x , while x− denotes a coordinate ≠ 0 0 — j j tude tψ but set tσ ¼ in Eq. (34) in which case, we slightly smaller than x , and let k be the incident arrive at a free-fermion scattering problem that admits an j e ϕ emergent-fermion momentum. One immediately finds exact solution. Our goal is to deduce the phase ei emergent acquired by an emergent fermion that travels from position − ik ðx1−x0Þ γðx1 ; 0Þ¼e e γðx0; 0Þ; ðC2Þ x0 before the constriction to position x3 after the con- striction. Figure 17(a) illustrates the interferometer geom- − − γ 0 ikeðx2 x1Þγ þ 0 etry, while Fig. 17(b) shows an “unfolded” version. For ðx2 ; Þ¼e ðx1 ; Þ; ðC3Þ

031014-26 ELECTRICAL PROBES OF THE NON-ABELIAN SPIN … PHYS. REV. X 10, 031014 (2020)

1. Hamiltonian and conventions ik ðx3−x2Þ þ γðx3; 0Þ¼e e γðx2 ; 0Þ: ðC4Þ We set tψ ¼ 0 but take tσ ≠ 0 in Eq. (34), so that the full → Hamiltonian becomes (In the exponentials above, we replace xj xj since the difference is inconsequential.) Next, integrating Eq. (C1) H H H ¼ 0 þ tun; over a small region enclosing x1 and similarly for x2 yields H −iπhσ σ σ the linear relations tun ¼ e tσ ðx2Þ ðx1Þ: ðD1Þ

þ − þ − Precisely as in Appendix C, H0 describes a chiral right- 0 ¼ γðx ; 0Þ − γðx ; 0Þþt˜ψ ½γðx ; 0Þþγðx ; 0Þ; ðC5Þ 1 1 2 2 moving free Majorana fermion. In the tunneling term, hσ ¼ 1=16 is the conformal weight (spin) of the σ field and þ − ˜ þ − 0 ¼ γðx2 ; 0Þ − γðx2 ; 0Þ − tψ ½γðx1 ; 0Þþγðx1 ; 0Þ; ðC6Þ tσ ∈ R is the coupling coefficient. For the remainder of this Appendix we set the velocity ve ¼ 1. ˜ −iπhψ σ where tψ ¼ −iðtψ =4veÞe . Combining with Eq. (C3) We choose to be Hermitian and normalized such that iχ ik ðx2−x1Þ and defining e ¼ e e , one obtains 1 σ x0;t0 σ x; t : h ð Þ ð Þi0 ¼ 0 0 1=8 ðD2Þ −iχ 2 ðit − ix − it þ ixÞ 1 þ 2e t˜ψ þ t˜ψ γðxþ; 0Þ¼eiχ γðx−; 0Þ: ðC7Þ 2 1 2 iχ˜ ˜2 1 þ e tψ þ tψ The subscript of the correlation function h i0 indicates that the correlator is computed with respect to the free CFT Finally, Eqs. (C2), (C4), and (C7) together imply that Hamiltonian H0. [The choice of phase on the right side of iϕ γðx3; 0Þ¼e emergent γðx0; 0Þ with Eq. (D2) guarantees that the correlator hσð0; −iβÞσð0Þi ¼ Tr½σð0Þe−βHσð0Þ=Tre−βH is positive—a necessary condi- −iχ˜ ˜2 σ σ† H 1 þ 2e tψ þ tψ tion for ¼ .] It is straightforward to check that tun is iϕemergent ikeðx3−x0Þ e ¼ e iχ 2 : ðC8Þ 1 þ 2e t˜ψ þ t˜ψ indeed Hermitian. Likewise, we define the normalized Majorana field To make contact with Sec. VII A from the main text, we pffiffiffiffiffiffi ψ ¼ 4πγ; ðD3Þ now set x3 − x0 ¼ Le and x2 − x1 ¼ La. Additionally, we −iπhψ expand Eq. (C8) to first order in t˜ψ and use ie ¼ 1, such that leading to 1   0 0 hψðx ;tÞψðx; tÞi0 ¼ : ðD4Þ ϕ tψ − 0 − 0 − ei emergent ≈ eikeLe 1 þ ð−e ikeLa þ eikeLa Þ : ðC9Þ ðit ix it þ ixÞ 2ve

The three terms in brackets respectively correspond to paths (i), (ii), and (iii) discussed in Sec. VII A. From this explicit 2. Scattering states calculation, we can trace the relative minus sign between Next, we specify the formalism used to compute the the terms for paths (ii) and (iii) to the anticommutation transmission amplitude. We quantize the CFT along slices relations obeyed by the Majorana fermions. Comparing to at fixed positions x; wave functions are written on “position Eqs. (42) and (53), the weights wii and wiii are given by slices” that live for all of time. (One should contrast to the α −α −1 2 Eq. (43) with ii ¼ iii ¼ = , as quoted in Eq. (45). usual framework wherein states live on fixed time slices.) Such a reformulation is, in principle, applicable to all field theories, but is particularly convenient for CFTs due to the APPENDIX D: ANALYSIS OF ISING-ANYON symmetry between space and time coordinates [132].An TUNNELING ACROSS A CONSTRICTION interaction term may manifest itself in different ways In this Appendix, we continue to study the interferometer within this rotated frame. For example, a point defect in Fig. 10, but now allowing only Ising anyons to tunnel localized in space become a global (instantaneous) quan- across the constriction. The geometry and coordinates used tum quench. are again given in Fig. 17. We evaluate the transmission In the same spirit as Eq. (D1), we decompose the action amplitude describing propagation of an emergent fermion into its free part and an interaction part: from position to x0 to x3 perturbatively in Ising-anyon Z tunneling, assuming that the interferometer does not con- S ¼ P0dx þ Stun: ðD5Þ tain any nontrivial bulk quasiparticles as in Appendix C.

031014-27 AASEN, MONG, HUNT, MANDRUS, and ALICEA PHYS. REV. X 10, 031014 (2020)

The operator P0 generates spatial translations of the free Recall that we set the pesky velocity to unity; hence, here ω CFT, while Stun consists of a spatially nonlocal global term and below, corresponds to ke from Sec. VII A. (spanning all of time), Let Aðω; x0;x3Þ denote the amplitude for transmission of Z the Majorana fermion from position x0 to x3 at frequency ω. ∞ In terms of the field theory, A is the quantum amplitude S ¼ − dtH ðtÞ tun tun associated with the spatial strip ½x0;x3 with boundary −∞ Z ∞ conditions set by the incoming/outgoing states. Formally, −iπhσ ¼ −e tσ dtσðx2;tÞσðx1;tÞ: ðD6Þ the amplitude is defined via −∞

x3 “ ” iSjx For the problem at hand, the tunnel junction teleports ψ ω0 Pe 0 ψ ω ω; δ ω − ω0 h j j i particles between positions x1 and x2 and can be interpreted Að x0;x3Þ ð Þ¼ x3 ; ðD11Þ P iSjx0 as a wormhole allowing “time travel” between the two h e i positions. (This nonlocality makes it difficult to write the y corresponding momentum operator for the tunneling term.) where Sjx is the action [Eq. (D5)] restricted to the spatial We choose to work in the interaction picture, where interval ½x; y, and P denotes path ordering (of the position operators are related to those in the Schrödinger picture via coordinates). We expand A in powers of tσ,

−iP0x iP0x Oðx; tÞ¼e OðtÞe : ðD7Þ ð0Þ ð1Þ 2 ð2Þ A ¼ A þ tσA þ tσA þ …; ðD12Þ The state describing an incoming emergent fermion with positive frequency ω is written as so that AðnÞ captures the nth-order correction in the Z perturbative series. 1 ∞ −iωt The zeroth-order piece follows from free propagation of jψ ωi¼ dte ψðtÞj0i; ðD8Þ ψ 0 2π −∞ the scattering states, i.e., evolving ðx; tÞ with the tσ ¼ Hamiltonian: where j0i is the ground state of P0. This state exhibits the normalization ψ iP0ðx3−x0Þ ψ ð0Þ 0 h ω0 je j ωi0 A ðω; x3;x0Þδðω − ω Þ¼ iP0ðx3−x0Þ 0 he i0 hψ ω0 jψ ωi¼δðω − ω ÞðD9Þ iωðx3−x0Þ ¼ e hψ ω0 jψ ωi0: ðD13Þ and carries momentum ð0Þ We thus obtain the expected result A ðω; x3;x0Þ¼ ψ ω ψ ω − P0j ωi¼ j ωi: ðD10Þ ei ðx3 x0Þ.

The first-order correction to the amplitude is Z ∞ ð1Þ ω; δ ω − ω0 − ψ 0 H 0 − H ψ tσA ð x3;x0Þ ð Þ¼ ih ω0 ðx3Þj dt ½ tunðt Þ h tuni0j ωðx0Þi −∞ Z 0 ∞ −iπhσ 0 0 0 ¼ −itσe hψ ω0 ðx3Þj dt ½σðx2;tÞσðx1;tÞ − Δjψ ωðx0Þi : ðD14Þ −∞ 0

Δ −iπhσ Δ H Here, is a constant,R defined through tσe ¼h tuni0, chosen to cancel off the phase correction to the vacuum. We −iωt also let jψ ωðxÞi ¼ ðdt=2πÞe ψðx; tÞj0i denote a (temporal) plane wave at position x. Expanding the scattering states in terms of their integral definitions and the operators in the Schrödinger picture yields Z − −iπhσ 1 ie ω0 −ω − − ð Þ ω; δ ω − ω0 ið t2 t1Þ ψ iPðx3 x2Þ σ 0 iPLa σ 0 − Δ iPLa iPðx1 x0Þψ A ð x3;x0Þ ð Þ¼ 2 e h ðt2Þe ½ ðt Þe ðt Þ e e ðt1Þi0: ðD15Þ 2π 0 ð Þ t2;t1;t

iPðx3−x2Þ Above, we use x2 − x1 ¼ La. As the scattering states are eigenstates of the momentum operator, we can replace e 0 and eiPðx1−x0Þ with their respective eigenvalues eiω ðx3−x2Þ and eiωðx1−x0Þ. In addition, we can eliminate the factor δðω − ω0Þ on

031014-28 ELECTRICAL PROBES OF THE NON-ABELIAN SPIN … PHYS. REV. X 10, 031014 (2020) the left-hand side by integrating over ω on both sides; doing so fixes t1 ¼ 0 and eliminates one of the integrals on the right- hand side. With some simple substitution of variables, the first-order amplitude correction can now be written as

− π ω − − Z ie i hσ ei ðx3 x0 LaÞ ð1Þ iωt2 0 iPL 0 iPL A ðω; x3;x0Þ¼− e hψðt2Þ½σðt Þe a σðt Þ − Δe a ψð0Þi0: ðD16Þ 2π 0 t2;t

0 iPLa 0 −1=8 From Eq. (D2), one finds Δ ¼hσðt Þe σðt Þi0 ¼ð−iLaÞ .

3. Evaluation of the first-order correction ð1Þ 0 We now evaluate the first-order contribution A ðωÞ by first integrating over t and then integrating over t2. Let

0 0 iPL 0 iPL Iðt Þ¼hψðt2 − iϵÞ½σðt Þe a σðt Þ − Δe a ψðiϵÞi0 0 0 0 0 ¼hψðt2 − La − iϵÞσðt − LaÞσðt ÞψðiϵÞi0 − hσðt − LaÞσðt Þi0hψðt2 − La − iϵÞψðiϵÞi0 ðD17Þ be the regulated form the integrand in Eq. (D16) (without the eiωt2 factor). We take ϵ → 0þ at the end of the calculation. The correlation function I can be computed via standard CFT techniques [4,133]. For instance, one can evaluate hψðz1Þσðz2Þσðz3Þψðz4Þi using a conformal transformation that maps the plane to a cylinder, placing the σ fields at ∞ ψ ψ 0 0 t ¼ . The correlator then reduces to computing the two-point correlation function h ðx; tÞ ðx ;tÞicyl with the fermion field ψ having periodic boundary conditions. Undoing the conformal transformation yields the desired result: 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 −1=8 0 0 0 0 ð−iL Þ 6 ðϵ þ it2 − it Þðϵ þ it Þ ðϵ þ it2 − iL − it Þðϵ − iL þ it Þ 7 I 0 a 4 a a − 25 ðt Þ¼ 0 0 þ 0 0 : ðD18Þ 2ð2ϵ þ it2 − iLaÞ ðϵ þ it2 − iLa − it Þðϵ − iLa þ it Þ ðϵ þ it2 − it Þðϵ þ it Þ R I I 0 O 0 −4 0 Δ Notice that t0 is absolutely convergent since ðt Þ decays as ððt Þ Þ for large t . This convergence results from the subtraction (corresponding to the −2 term in brackets), which eliminates the leading contribution in Iðt0Þ. I 0 − − ϵ − ϵ ϵ ϵ The function ðt Þ has four branch points at t2 La iR, t2 i , i , and La þ i . Observe that the first two sit below the real axis, while the latter two sit above the real axis. The dt0 integral is to be evaluated with a branch cut connecting the two upper branch points, and a branch cut connecting the two lower points, such that I is analytic along the real line. To 0 0 simplify the terms in brackets, we introduce a shift of variables t ↦ t þðt2=2Þ: 2 3   sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 2 0 2 2 t2 1 6 ðt Þ þ a ðt Þ þ b 7 I t0 þ ¼ 4 þ − 25 ðD19Þ 2 1=8 0 2 2 0 2 2 2ð−iLaÞ ð2ϵ þ it2 − iLaÞ ðt Þ þ b ðt Þ þ a

with a ¼ ϵ þ it2=2, b ¼ ϵ þ it2=2 − iLa. At this point, we can utilize the integral identity in Eq. (E5) from Appendix E to write Z        ∞ t2 1 a − b a − b 0I 0 4 − dt t þ ¼ 1=8 × ða þ bÞ K E : −∞ 2 2ð−iL Þ ð2ϵ þ it2 − iL Þ a þ b a þ b a  a   2i1=8 L L ¼ K a − E a ; ðD20Þ 1=8 − − 2 ϵ − − 2 ϵ La t2 La i t2 La i where K and E are complete elliptic integrals of the first and second kind, respectively. Inserting this result into Eq. (D16) yields      − π ω − − Z 1 8 ie i hσ ei ðx3 x0 LaÞ ∞ 2i = L L ð1Þ ω; − iωt2 a − a A ð x3;x0Þ¼ lim dt2e 1=8 K E ϵ→0þ 2π −∞ t2 − L − 2iϵ t2 − L − 2iϵ Z  La  a  a ∞ 7 8 dy 1 1 = iωðx3−x0Þ iðωL Þy ¼ lim 2iLa e e a E − K : ðD21Þ ϵ0→0þ −∞ 2π y − iϵ0 y − iϵ0

031014-29 AASEN, MONG, HUNT, MANDRUS, and ALICEA PHYS. REV. X 10, 031014 (2020)   π 1 1 From the first to the second line, we substitute t2 ¼ u 2 0 gðu ≥ 0Þ¼ 1F2 ;1; 2 − u ðD27Þ Laðy þ 1Þ and introduce ϵ ∝ ϵ as the small parameter to be 4 2 4 taken to zero. Finally, we write the amplitude as plotted in Fig. 11(a). 7 8 ð1Þ = iωðx3−x0Þ Upon replacing ω → ke to match the notation from A ðω; x3;x0Þ¼2iLa e gðωLaÞ; ðD22Þ Sec. VII A and setting x3 − x0 ¼ Le, the zeroth- and first-order terms in the transmission amplitude are where gðuÞ is defined as a Fourier transform via 0 Z Að Þðk Þ¼eikeLe ; ðD28aÞ ∞ dy e gðuÞ¼ lim eiuyg˜ðy − iϵÞ; ðD23aÞ þ 2π 7 8 ϵ→0 −∞ ð1Þ ikeLe = tσA ðke; x3;x0Þ¼e × 2itσLa gðkeLaÞ: ðD28bÞ     1 1 Restoring ve factors recovers precisely the tσ correction g˜ðyÞ¼E − K : ðD23bÞ 0 y y quoted in Eqs. (46) and (48) in the nψ ¼ case.

States jψ ωi with negative frequencies do not exist in a APPENDIX E: A FEW FACTS REGARDING chiral CFT, so strictly speaking, the amplitude is ill-defined COMPLETE ELLIPTIC INTEGRALS ω 0 0 for < , and hence, gðu< Þ is ill-defined as well. Let KðkÞ and EðkÞ denote the complete elliptic integrals Nevertheless, it is convenient to now extend the domain of the first and second kind, respectively. For jkj < 1, they of gðuÞ defined in Eq. (D23a) to all real u. [We caution that are defined as one should not confuse this continuation with that adopted Z later in Eq. (F6), which serves a quite different purpose.] 1 2π θ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid The function gðuÞ extended in this way vanishes for u<0, KðkÞ¼ p ; ðE1aÞ 4 0 1 − k2 cos2 θ since g˜ðyÞ has no singularities in the lower half plane Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (Imy<0). In addition, limϵ→0þ Reg˜ðy − iϵÞ is symmetric in 1 2π θ 1 − 2 2 θ; y ϵ→0þ g˜ y − iϵ EðkÞ¼ d k cos ðE1bÞ , while lim Im ð Þ is antisymmetric. Together, 4 0 these properties allow us to write gðuÞ over the physical domain u ≥ 0 in terms of simply the imaginary part of g˜: beyond the unit circle they are defined via analytical Z continuation. Both are even [e.g., Kð−kÞ¼KðkÞ] and 2 ∞ ≤−1 ≥ 1 ≥ 0 ˜ have branch cutspffiffiffiffiffiffiffiffiffiffiffiffiffi along the real line at k and k . gðu Þ¼ dy sinðuyÞgiðyÞ; ðD24aÞ ¯ 1 − 2 0 1 π 0 Define x ¼ x .For

g˜iðyÞ¼− lim Img˜ðy − iϵÞ: ðD24bÞ 1 1 ϵ→0þ Kð þ iϵÞ − Kð − iϵÞ lim x x ¼ ixKðx¯Þ; ðE2Þ ϵ→0þ 2 1 −1 ˜ 1 For y> or y< , gðyÞ is purely real because =y lies 1 1   within the interval −1 < 1=y < 1 (cf. Appendix E). Eð þ iϵÞ − Eð − iϵÞ 1 ð Þ lim x x ¼ i xKðx¯Þ − Eðx¯Þ ; ðE3Þ Using Eqs. (E2) and (E3), the imaginary component g˜i ϵ→0þ 2 x can be written as along with the differential and integral identities 8 < 0; jyj > 1; pffiffiffiffiffiffiffiffiffiffiffiffiffi d ¯ x ¯ − ¯ g˜ ðyÞ¼ ðD25Þ EðxÞ¼ 2 ½KðxÞ EðxÞ; ðE4aÞ i : 1 1 − 2 ≤ 1 dx 1 − x y Eð y Þ; jyj ; d 1 ½Eðx¯Þ − Kðx¯Þ ¼ Eðx¯Þ; ðE4bÞ which is supported on the finite interval −1 ≤ y ≤ 1. From dx x Eqs. D3 we can now reexpress gðu ≥ 0Þ as Z Z 1 1 π2 ¯ x Z qffiffiffiffiffiffiffiffiffiffiffiffiffi dx EðxÞ¼ dx ¯ EðxÞ¼ 8 : ðE4cÞ 2 1 sinðuyÞ 0 0 x gðu ≥ 0Þ¼ dy E 1 − y2 : ðD26Þ π 0 y 1. An integral identity Evidently, g is a purely real function. As we show in Suppose that a, b are complex numbers such that Appendix E2, it can be written in terms of a generalized Rea>0 and Reb>0. Following hours of struggle with hypergeometric function, Mathematica, one can show that

031014-30 ELECTRICAL PROBES OF THE NON-ABELIAN SPIN … PHYS. REV. X 10, 031014 (2020) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z " # ∞ 2 2 2 2 which indeed matches the coefficients in the series expan- z þ a z þ b − 2 dz 2 2 þ 2 2 sion in Eq. (E7). −∞ z þ b z þ a      a − b a − b ¼ 4ða þ bÞ K − E : ðE5Þ APPENDIX F: EXTRACTION OF ISING-ANYON a þ b a þ b TUNNELING WEIGHTS To be precise, the integrand has branch points at ia and Here we formally invert Eq. (47) to extract the scaling ib; the integral is evaluated assuming branch cuts between functions fiv and fv that quantify energy partitioning in ia ↔ ib and −ia ↔ −ib with one pair above the real line Ising-anyon tunneling events. First we define and the other pair below. GðuÞ¼2igðuÞu7=8 ðF1Þ 2. A different integral identity Here we show that the integral and Z qffiffiffiffiffiffiffiffiffiffiffiffiffi 8 2 1 def sinðuyÞ 2 > f y ;y>0; gðuÞ¼ dy Eð 1 − y ÞðE6Þ < vð Þ π 0 y − −1 0 FðyÞ¼> fivð yÞ; can be expressed as a generalized hypergeometric function 0 y<−1;   π 1 1 ? u 2 so that Eq. (47) can be compactly expressed as gðuÞ¼ 1F2 ;1; 2 − u 4 2 4 Z   ∞ X∞ 1 3 2m−1 m iuy πu 1 ð2Þð2Þð 2 Þ 1 2 GðuÞ¼ dy e FðyÞ: ðF3Þ ¼ − u −1 4 m! m! m 1 ! 4 m¼0 ð þ Þ X∞ m Since GðuÞ is defined only for u>0, one cannot exploit πu ð−1Þ ð2m − 1Þ!! 2 ¼ u m: ðE7Þ standard plane-wave orthogonality to isolate FðyÞ.To 4 4mm!ðm þ 1Þ!ð2mÞ!! m¼0 proceed, we continue GðuÞ to u<0. For clarity, we denote the resulting function defined for all real u by GcðuÞ. To do so, we Taylor expand Eq. (E6) in powers of u and Care must be taken in defining this continuation to show that it takes the form of Eq. (E7). ensure that Fðy<−1Þ remains zero as demanded by our Notice that terms with even powers vanish in Eq. (E6). physical system. Consider the integral The coefficients for odd powers u2mþ1 are given by Z ∞ du 2 ð−1Þm FðyÞ ≡ e−iuye−ϵuGðuÞ; ðF4Þ c2 1 ¼ M ðE8Þ 0 2π mþ π ð2m þ 1Þ! m where we introduce a regulator with ϵ → 0þ for conver- with gence, which is necessary given that GðuÞ ∼ u7=8 as Z qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u → ∞. Evaluating this integral gives a result in terms 2m 2 Mm ¼ dy y Eð 1 − y Þ: ðE9Þ of a generalized hypergeometric function. Here we simply 0 note that

One can evaluate Mm as follows. Performing integration by jFðyÞj ¼ jFð−yÞj; ðF5aÞ parts twice and using Eqs. (E4a) and (E4b) yields Mm ¼ 2 2 ð2m þ 1Þ M − ð4m − 1ÞM −1, implying the recursion m m π 16 relation Fðy>1Þ¼−eið = ÞjFðy>1Þj; ðF5bÞ

ð2m þ 1Þð2m − 1Þ Fðy<−1Þ¼e−iðπ=16ÞjFðy<−1Þj: ðF5cÞ M ¼ M −1: ðE10Þ m 4mðm þ 1Þ m These crucial properties imply that the desired continuation 2 Since M0 ¼ðπ =8Þ from Eq. (E4c), we can deduce is given by 4m ! 1 ! π2 8 2 − 1 !! 2 1 !! that m ðm þ Þ Mm ¼ð = Þð m Þ ð m þ Þ .  Therefore, GðuÞ;u>0; GcðuÞ¼ ðF6Þ e−iðπ=8ÞGðjujÞ;u<0: π 2m − 1 !! −1 m ð Þ c2mþ1 ¼ ð Þ m ; ðE11Þ 4 4 m!ðm þ 1Þ!ð2mÞ!! It follows that

031014-31 AASEN, MONG, HUNT, MANDRUS, and ALICEA PHYS. REV. X 10, 031014 (2020) Z ∞ ϵ → 0þ du −iuy −ϵjuj −iðπ=8Þ In the bottom lines, we take the limit. We caution, e e GcðuÞ¼FðyÞþe Fð−yÞ¼FðyÞ; −∞ 2π however, that inserting the bottom lines of Eq. (F11) into Eq. (F3) would produce an unphysical infrared divergence; ðF7Þ hence, in the top line we explicitly display the regulariza- where again we introduce a regulator with ϵ → 0þ.By tion that circumvents this problem. Elsewhere, we are free ϵ → 0þ virtue of Eqs. (F5a) through (F5c), we see that Fðy<−1Þ to send as no such issues arise. The singularities at → 1þ ≫ 1 indeed vanishes. Taking the ϵ → 0þ limit allows us to y instead follow from the subleading Gðu Þ → explicitly deduce behavior; they can be captured by replacing GðuÞ −2iu−1=8ðcos uÞ in Eq. (F7), yielding ið9π=16Þ e fivðyÞ     1 2Γ 15 1 1 1 7 1 → 1þ ≈− −iðπ=16Þ ∓ 1 −7=8 ð 8 Þ 2 Fðy Þ ie 1 ðy Þ : ðF12Þ ¼ 3F2 − ; ; ;− ; y Γð Þ πy15=8 2 2 2 16 16 8 pffiffiffi 1   72 πΓð Þ 7 23 23 1 31 Finally, the asymptotic decay at y ≫ 1 encodes the − 16 F ; ; ; ; y2 25=8 9 3 3 2 16 16 16 2 16 ≪ 1 → 2 ×15Γð16Þ Gðu Þ behavior and follows from replacing GðuÞ ffiffiffi 15=8 15 8 2p 9   iðπ=2Þu in Eq. (F3): 2 = ×15 πΓ y 15 31 31 3 39 ð16Þ ; 2 þ 1 3 3F2 16;16;16 2;16 y ðF8Þ   7×23Γð16Þ π F y ≫ 1 ≈−ie−iðπ=16Þ y−23=8: F13 ð Þ 2Γ − 15 ð Þ and ð 8 Þ

ið9π=16Þ Figure 11(b) indicates the scaling behaviors cap- e fvðy<1Þ   tured above. 2 π Γ 15 1 1 1 7 1 cosð8Þ ð 8 Þ 2 ¼ 3F2 − ; ; ; − ; y πy15=8 2 2 2 16 16 pffiffiffi   APPENDIX G: CONDUCTANCE FROM 72 πΓð 1 Þ 7 23 23 1 31 − 16 F ; ; ; ; y2 MAJORANA PHASE ACCUMULATION 25=8 9 3 3 2 16 16 16 2 16 2 × 15Γð16Þ pffiffiffi   For completeness, we briefly review how electrical 215=8 × 152 πΓ 9 y 15 31 31 3 39 ð16Þ ; 2 conductance follows from the relative phases acquired þ 1 3 3F2 16 ; 16 ; 16 2 ; 16 y ; 7 × 23Γð16Þ by Majorana fermions propagating in the circuits from Figs. 9 and 10. We denote the part of the wave function ðF9Þ describing an electron incident at the lower edge of the

π 23   ν ¼ 1 quantum-Hall system by sinð ÞΓð Þ 1 23 31 1 ið9π=16Þ 1 8 8 ;1 e fvðy> Þ¼ 23=8 3F2 ; ; ; 2 : Z 2y 2 16 16 y x0 dx eiEx=uc†ðxÞj0i: ðG1Þ ðF10Þ −∞ 0 [Recall that fivðyÞ is defined for 0 0.] We thus complete the desired inversion. the edge velocity in the region without induced pairing, † Figure 11(b) from the main text plots the magnitude of the c ðxÞ adds the electron to position x of the edge, and x0 is scaling functions fiv and fv. the location at which the edge state meets the proximitizing Some limits of FðyÞ can be deduced from the asymp- superconductor. (In this Appendix, we use coordinates totics specified in Eqs. (50) and (52). The small-y singu- consistent with those in Fig. 17.) Employing a Majorana larities follow from the leading Gðu ≫ 1Þ behavior and can representation via c ¼ γ1 þ iγ2, Eq. (G1) equivalently be obtained by simply replacing GðuÞ → 2iu7=8 in becomes Eq. (F7). One finds Z x0   iEx=u γ − γ 0 15 π 2 3π 8 dx e ½ 1ðxÞ i 2ðxÞj i: ðG2Þ Γð Þ eið = Þ eið = Þ −∞ Fðjyj ≪ 1Þ ≈ 8 þ π ðϵ þ iyÞ15=8 ðϵ − iyÞ15=8 8 h i Beyond position x0, the constituent Majorana fermions < −iðπ=16Þ 2 π 15 −15=8 γ γ −ie π cosð8ÞΓð 8 Þ y ;y>0; 1 and 2 follow diverging paths that eventually recombine ≈ h i at position x3 at the upper edge of the ν ¼ 1 quantum-Hall : −iðπ=16Þ 2 15 −15=8 −ie π Γð 8 Þ jyj ;y<0: system. En route, they generally acquire different phase factors; hence, at the upper ν ¼ 1 edge, the outgoing part of ðF11Þ the wave function becomes

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Abelian and Non-Abelian Anyons in the Fractional Quan- and for such events tσ can not be regarded as weak. tum Hall Effect, arXiv:1112.3400. Consequently, finite length La is required to define a [121] We stress that observing this conductance dichotomy does perturbative regime, corresponding to the quoted inequal- not require random events that displace Ising anyons by 7=8 ity tσLa =ve ≪ 1. macroscopic distances. Rather, observing a jump to ap- [129] If the bulk Ising anyon’s “partner” was instead dragged 2 2 proximately ð e =hÞ conductance requires an Ising-anyon to the gapless part of the edge in the left interferometer pair to be separated by distances of order the Majorana half, paths (ii) and (iii) would acquire an additional π edge-state decay length such that one of the anyons phase. couples negligibly to the gapless edge mode. [130] A. Milsted, L. Seabra, I. C. Fulga, C. W. J. Beenakker, and [122] P. Fendley, M. P. A. Fisher, and C. Nayak, Edge States and E. Cobanera, Statistical Translation Invariance Protects a 5 2 Tunneling of Non-Abelian Quasiparticles in the = Topological Insulator from Interactions, Phys. Rev. B 92, Quantum Hall State and p þ ip Superconductors, Phys. 085139 (2015). Rev. B 75, 045317 (2007). [131] Actually, in the full microscopic model, δϵðkÞ reduces the [123] W. Bishara and C. Nayak, Edge States and Interferometers overall bandwidth even though the velocity remains fixed. in the Pfaffian and Anti-Pfaffian States of the ν 5 ¼ 2 This effect likely yields a slightly smaller critical U Quantum Hall System, Phys. Rev. B 77, 165302 (2008). compared to what would occur if the bandwidth was also [124] P. Bonderson, D. J. Clarke, C. Nayak, and K. Shtengel, fixed. Implementing Arbitrary Phase Gates with Ising Anyons, [132] For a chiral CFT, one can transform between the “position Phys. Rev. Lett. 104, 180505 (2010). slice” quantization and the “time slice” quantization via a [125] J. Nilsson and A. R. Akhmerov, Theory of Non-Abelian Wick rotation (real time to imaginary time), a 90° Euclid- Fabry-Perot Interferometry in Topological Insulators, ean rotation (exchange time and space), and then another Phys. Rev. B 81, 205110 (2010). Wick rotation (imaginary time back to real time). [126] For a discussion of energy partitioning in the Luttinger- 2 liquid context, see Ref. [127]. [133] C. Nayak and F. Wilczek, n-Quasihole States Realize 2n−1 [127] T. Karzig, G. Refael, L. I. Glazman, and F. von Oppen, -Dimensional Spinor Braiding Statistics in Paired Energy Partitioning of Tunneling Currents into Luttinger Quantum Hall States, Nucl. Phys. B479, 529 (1996). Liquids, Phys. Rev. Lett. 107, 176403 (2011). [134] Wolfram Research, Complete Elliptic Integral of the First [128] One might have naively guessed that Ising-anyon tunnel- Kind: Formula 08.02.04.0009, 2001 (accessed 2019), ing instead admits a perturbative treatment provided the http://functions.wolfram.com/08.02.04.0009.01. [135] Wolfram Research, Complete Elliptic Integral of the First incoming fermion momentum ke is sufficiently large that 7=8 Kind: Formula 08.02.17.0001, 2001 (accessed 2019), tσ=ðveke Þ ≪ 1. When this inequality holds, it would appear that one is probing the system at high energies for http://functions.wolfram.com/08.02.17.0001.01. [136] Wolfram Research, Complete Elliptic Integral of the which tσ has not yet flowed to strong coupling. The fallacy in this argument stems from energy partitioning. Regard- Second Kind: Formula 08.01.04.0009, 2001 (accessed 2019), http://functions.wolfram.com/08.01.04.0009.01. less of the magnitude of ke, at the constriction the incident fermion splinters into Ising anyons that share the incident [137] Wolfram Research, Complete Elliptic Integral of the energy in all permissible ways. In particular, the allowed Second Kind: Formula 08.01.17.0002, 2001 (accessed partitionings include cases where an Ising-anyon tunneling 2019), http://functions.wolfram.com/08.01.17.0002.01. across the constriction carries arbitrarily small momentum,

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