Numerical investigation of gapped edge states in fractional quantum Hall-superconductor heterostructures
NNumerical investigation of gapped edge states in fractional quantumHall-superconductor heterostructures
C´ecile Repellin
Max-Planck-Institut f¨ur Physik komplexer Systeme, 01187 Dresden, Germany
Ashley M. Cook and Titus Neupert
Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
Nicolas Regnault
Laboratoire Pierre Aigrain, D´epartement de physique de l’ENS,´Ecole normale sup´erieure, PSL Research University,Universit´e Paris Diderot, Sorbonne Paris Cit´e, Sorbonne Universit´es,UPMC Univ. Paris 06, CNRS, 75005 Paris, France
Fractional quantum Hall-superconductor heterostructures may provide a platform towards non-abelian topological modes beyond Majoranas. However their quantitative theoretical study remainsextremely challenging. We propose and implement a numerical setup for studying edge states offractional quantum Hall droplets with a superconducting instability. The fully gapped edges carrya topological degree of freedom that can encode quantum information protected against local per-turbations. We simulate such a system numerically using exact diagonalization by restricting thecalculation to the quasihole-subspace of a (time-reversal symmetric) bilayer fractional quantumHall system of Laughlin ν = 1 / π Josephson effect, evidencing theirtopological nature and the Cooper pairing of fractionalized quasiparticles. The versatility and effi-ciency of our setup make it a well suited method to tackle wider questions of edge phases and phasetransitions in fractional quantum Hall systems.
I. INTRODUCTION
The fractional quantum Hall [1] (FQH) effect harborsa variety of exotic topologically ordered quantum phases,from the well understood Laughlin states all the way tostates [2, 3] with non-Abelian quasiparticles such as Ma-jorana fermions and Fibonacci anyons. The experimentalexploration of these systems faces two challenges : First,the desired more exotic topological orders, which couldbe used, e.g., for universal quantum computation [4], canonly be accessed under extreme experimental conditions,as they are protected by very small energy gaps. Thus,despite intense efforts, definite experimental confirmationof the non-Abelian nature of a FQH phase is still lacking.Second, the topological information is encoded in dege-nerate ground states or the state of quasiparticles, and istherefore intrinsically hard to measure and manipulate.To overcome both of these obstacles, several recent stu-dies proposed focusing on more conventional FQH states,such as the Laughlin ν = 1 / ν = 1 / Z Read-Rezayi state. Barkeshli subsequently poin-ted out that topological information is also stored in apair of counter-propagating ν = 1 / e/ e/ e ). Thisnonlocal observable defined along the (closed) edge dis-tinguishes three topologically degenerate ground statesof the edge. By appropriately coupling several gappededges, one can in principle manipulate their topologicalground state [17]. Another approach to engineer parafer-mion excitations from Abelian topological order relies onlattice defects and was recently implemented numericallyin Refs. 18 and 19. The FQH edge states are also a conve-nient system to study the bulk-boundary correspondencein topologically ordered systems. Unlike noninteractingsymmetry protected phases in two spatial dimensions,interacting integer and fractional quantum Hall statescan support several distinct edge phases with differentuniversal properties but the same symmetries [20, 21].While effective models (e.g., using a bosonized descrip-tion of the edge [22, 23]) have permitted striking predic-tions at the edge of topologically ordered systems, openquestions remain which can only be addressed by a mi-croscopic approach. First, in the context of the bulk-boundary correspondence, which boundary phase is fa-vored by certain microscopic interactions remains largelyunexplored (especially when non-abelian liquids are used a r X i v : . [ c ond - m a t . s t r- e l ] M a y e i ' N orb , n N orb , sc / N orb , sc / )) m · · ·· · · µ m ' e i ' a ) b ) c ) " " = ⌘ Figure
1. Schematics of the physical geometry and the oneused for the numerical investigation. a) Fractional topologicalinsulator heterostructure in which carriers with spin up anddown (red and blue) form a fractional quantum Hall statewith opposite chirality. Proximity to superconducting reser-voirs (yellow) induces a superconducting gap in their edgechannels. To study the Josephson effect, relative phase ϕ bet-ween the left and right superconducting order parameter isincluded. b) When imposing periodic boundary conditionsalong the edges, resulting in a cylinder geometry, each edgecarries a topological degree of freedom. The boundary condi-tions can be twisted by inserting a flux φ into the cylinder forspin up electrons and − φ for spin down electrons. In the Lan-dau gauge orbitals are localized along the cylinder, where weconsider N orb , n and N orb , sc normal and superconducting or-bitals, respectively. The typical separation between orbitals is π(cid:96) B L y , where L y is the cylinder perimeter. The droplet is confi-ned by a linear potential µ m . c) With the counter-propagatingedges gapped out, the bilayer FQH state on the cylinder is to-pologically equivalent to a single layer FQH state on a torus,where the fluxes φ and ϕ run through its two noncontractiblecycles and can be used to explore its topological ground statedegeneracy. It is thus topologically equivalent to the groundstate degeneracy of the gapped edge modes. as the building blocks). As for localized edge modes, thebraiding of non-abelian excitations relies on the possibi-lity to tunnel quasiparticles through the bulk while kee- ping the edge gap open[10]. This hypothesis relies on thehierarchy of energy and length scales in the system. Nu-merical simulations are necessary to achieve such quan-titative analysis, and have the potential to identify chal-lenges that could have been obscured by effective analyti-cal models. They seem indispensable as experiments areundertaking the first steps to realize the ideas outlinedabove [24, 25].Here, for the first time, we undertake an extensive nu-merical calculation of a FQH system coupled to a su-perconductor using exact diagonalization. More expli-citly, we consider a bilayer FQH system, with magne-tic field perpendicular to the layers, where the orienta-tion of the field for one layer is opposite to that for theother layer. This is equivalent to a time-reversal symme-tric fractional topological insulator [26]. This construc-tion permits gapping out of the edge states with singletinterlayer superconducting pairing. Our calculations areperformed on a cylinder geometry in which the bilayer-FQH droplet has two edges. To make numerics feasible,we restrict our study to the subspace of zero energy bulkand edge excitations of the Laughlin ν = 1 / π -periodic Josephson effect. II. HAMILTONIAN AND EFFECTIVE HILBERTSPACE
In the Landau gauge, a single-particle basis that spansthe lowest Landau level on a cylinder of circumference L y is given by φ m ( x, y ) = 1 L y (cid:96) B √ π e i mπyLy e − (cid:96) B (cid:18) x − πm(cid:96) BLy (cid:19) , (1)where (cid:96) B is the magnetic length which we will set to unityin the rest of the manuscript. We truncate the single-particle Hilbert space of the cylinder by allowing for m to take integer values in the range − N Φ / ≤ m ≤ + N Φ / N Φ . Note that m is either an integer orhalf integer depending on the parity of N Φ . Here, m playsthe role of both the y -momentum of the wave functionand at the same time determines the location of the wavefunction along the x direction. This coupling of momen-tum and position is enforced by the lowest Landau levelprojection.We now consider a bilayer system, where the two layersare distinguished by a spin index ↑ , ↓ and the Hall effectin one layer is opposite in chirality to the Hall effect in theother layer. This is the case in so-called fractional topolo-gical insulators, where the labels ↑ , ↓ may correspond tothe physical spin and the spin-dependent magnetic fieldis akin to the spin-orbit coupling. An alternative scena-rio more relevant to traditional FQH experiments is onein which ↑ , ↓ label the carriers in two adjacent quantumwells, one being electron-like and the other hole-like. Ahomogeneous magnetic field gives rise to edge states inboth quantum wells and the direction of propagation inone quantum well is opposite to that in the other quan-tum well. Our numerical study applies equally well toeach of these physical realizations, but we choose to des-cribe our results using the terminology of the fractionaltopological insulator realization. In either case the singleparticle eigenstates are ψ ↑ m ( x, y ) = φ m ( x, y ) ,ψ ↓ m ( x, y ) = φ ∗ m ( x, y ) = φ − m ( − x, y ) , (2)with − N Φ / ≤ m ≤ + N Φ /
2. This ensures that the sys-tem is invariant under time-reversal symmetry T = K i σ y for spinful fermions, where K is complex conjugation and σ y is the second Pauli matrix acting in spin space. Notethat none of the topological features we are interested inare protected by T . In fact, in the electron-hole bilayerrealization of the system, T is not the physical symme-try of the system, but an artificial symmetry of the modelthat may be broken in a microscopic realization.In the Fock space spanned by the single-particle states ψ ↑ m and ψ ↓ m , we consider a Hamiltonian of the formˆ H = ˆ H − body , ↑ + ˆ H − body , ↓ + s (cid:88) m = − s µ m (cid:16) ˆ c † m, ↑ ˆ c m, ↑ + ˆ c † m, ↓ ˆ c m, ↓ (cid:17) + 12 C (cid:16) ˆ N − N (cid:17) + ∆ s (cid:88) m = − s f m (cid:16) ˆ c † m, ↑ ˆ c † m, ↓ + h . c . (cid:17) , (3)where H − body ,σ is the two-body interaction for fermionswith spin σ = ↑ , ↓ and ˆ c † m,σ creates an electron in state ψ σm . For H − body ,σ we use the pseudo-potential Hamil-tonian with V being the only non-zero pseudo-potential coefficient. The coefficients µ m describe a confining po-tential that is rotationally symmetric around the cylinderaxis. The operator ˆ N = ˆ N ↑ + ˆ N ↓ measures the total num-ber of particles. A schematic representation of this setupis sketched in Fig. 1.Due to the presence of mean-field superconductivity,the particle number is not conserved. To tune the sys-tem into a regime with finite particle number density,the charging energy of strength 1 / (2 C ) has been added tothe Hamiltonian. The two parameters C and N permittuning of the average number of particles in the system.Finally, ∆ is the overall strength of the superconduc-ting coupling while f m is the (dimensionless) variationof the superconducting order parameter along the cylin-der, assuming a superconducting pairing potential thatis rotationally symmetric along the cylinder axis. We willfurther assume a superconducting pairing potential thatis nonzero only at the edge of the FQH droplet. In theelectron-hole bilayer realization of the system, the termproportional to ∆ takes the role of a charge conservingbackscattering term between the layers.The Laughlin state edge and bulk quasihole excitationsare the exact zero energy states of the model interactionˆ H − body ,σ at filling ν = 1 /
3. Their corresponding wa-vefunctions have an analytical expression on the cylin-der geometry [32]. Being Jack polynomials, they can beconveniently decomposed into the occupation basis [33].In Ref. 34, a careful and detailed numerical study of thisstate and its edge excitations was performed on the cylin-der geometry using a confinement similar to the µ m termof Eq. (3). In particular, the low energy spectrum (i.e.,below the bulk energy gap) for a finite size quantum Halldroplet has the characteristic shape shown in Fig. 2 a).To make progress in the numerical evaluation of Hamil-tonian (3), we send the gap of the FQH state to infinity,i.e., we set V → ∞ , by projecting the Hamiltonian tothe zero-energy subspace of H − body , ↑ + H − body , ↓ . Thisis the space of Laughlin quasiholes in each layer. Thedensest state(s) in this subspace are the Laughlin FQHstates with a filling fraction of 1 / / C and N of the charging energyterm are used to control (i) the average number of par-ticles in the droplet and (ii) the energy difference to sec-tors with nearby particle numbers. We choose these pa-rameters by diagonalizing the system in the absence ofsuperconductivity. Note that N is not equal to the par-ticle number in the non-superconducting ground state,because the µ m term also contributes an energy cost thatdepends on the particle number. We choose 1 / C and N such that the ground state has a desired particle number˜ N and the lowest energy state in the sector with ˜ N + 2particles is degenerate with the lowest energy state in thesector with ˜ N − + + ++ +++ +++++ +++++++ +++++++++++ +++++++++++++ ++++++++++++++++++ ++++++++++++++++++++++ ++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++ ++++++++++++++++++++++ ++++++++++++++++++ +++++++++++++ +++++++++++ +++++++ +++++ +++ ++ + +- - - E Ea ) b )12 12 K y K y gapspread S z = 0 S z = 1 ++ E E c ) K y E ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ +++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ E K y E + + Figure
2. Spectra of Hamiltonian Eq. (3) on a cylinderwith circumference L y /(cid:96) B = 15 . N Φ = 21 flux quanta,1 / (2 C ) = 0 . N = 30, and a linear symmetric che-mical potential µ m as shown in Fig. 1 b). a) Spectrum of asingle layer of the cylinder with 6 fermions. Energies are shif-ted such that the ground state energy is zero. Shaded areasare a guide for the eye showing the ‘arcs’ in the relative mo-mentum K y arising from the edge states of the Hall droplet.The inset is a zoom on the low lying state within the graybox, showing the energy difference δE between the first ex-cited state and the ground state. b) and c) are spectra forthe full double layer with b) ∆ = 0 while in c) ∆ = 2 . N orb , sc = 16 superconducting orbitals. Blue figures givethe number of states in the circle. In c) the superconductinggap in the edge states is apparent from the three topologi-cal ‘ground states’ moving below the bottom of the next arc.Orange states have total spin S z = 0, gray states S z = 1, andstates with higher spin only appear above this energy window.Both spectra symmetrically extend to K y → − K y . the ˜ N and the ˜ N ± S z = ( ˆ N ↑ − ˆ N ↓ ) (which also encodes the fermion parity)and the relative (angular) momentum K y = (cid:80) i ∈↑ m i − (cid:80) j ∈↓ m j , where m i and m j are the quantum numbersof the occupied ↑ -spin and ↓ -spin states, respectively, asdefined in Eq. (2). In the electron-hole bilayer realizationof the system, ˆ S z is the particle number operator. III. RESULTSA. Spectral evidence for the topological edge states
We present the spectral features associated with themodel defined by Hamiltonian (3) and sketched in Fig. 1 :three low-energy states protected by an energy gap. Wechoose a symmetric confining potential µ m = | m | that islinear in the orbital space index with a slope of 1 (whichwill serve as the unit of energy throughout this paper),as shown in Fig. 1 b).
1. Decoupled layers
We first study the spectrum of two decoupled layers,i.e., in the absence of superconductivity when ∆ = 0,which is shown in Fig. 2 b). It can be understood asa combination of two independent spectra of a confinedLaughlin ν = 1 / µ m = | m | , the Laughlin stateon the cylinder has a characteristic spectral feature as afunction of K y : focusing on a single layer with N/ N even), the ground state has K y = 0.The lowest lying excitations appear in the momentumsector K y = ± N/ δE denotes the energy differencebetween these states and the ground state. Further low-lying states are located in the sectors K y = nN/ n ∈ Z .The lowest lying states in the other momentum sectorsare higher in energy, giving rise to an arc-like structurein the spectrum, as observed in Fig. 2 a) . These arcs arehighlighted by the shaded region in Fig. 2. The loweststates at momenta K y = nN/ n ∈ Z are those wherethe droplet has been rigidly moved by n orbitals, givingrise to an extra energy cost of about nν due to the hi-gher chemical potential of the now occupied orbitals incomparison to the emptied ones. Since the center of massof the wave function is moved by n orbitals, the changein center of mass momentum is K y = nN/
2. The lowestexcitations in other momentum sectors are local edge ex-citations or combinations thereof. Since they increase thesize of the Hall droplet, they cost more energy than therigidly moved Laughlin state.Within these considerations, we can understand thespectrum of Fig. 2 b) as a finite-size representation ofa collection of gapless FQH edge states. In particularwe can understand the low-energy structure as super-positions of the states in the two layers ↑ , ↓ . We de-note by | σ, (cid:105) , | σ, ± N/ (cid:105) the three lowest states in eachof the σ = ↑ , ↓ sectors which occur at momenta 0 and ± N/
2. The state | ↑ , (cid:105) ⊗ | ↓ , (cid:105) is then the nondegene-rate ground state labelled by a blue ‘1’ in Fig. 2 b). Thestates | ↑ , ± N/ (cid:105) ⊗ | ↓ , (cid:105) and | ↑ , (cid:105) ⊗ | ↓ , ± N/ (cid:105) are fourdegenerate states at momenta K y = ± N/ δE abovethe ground state. The states | ↑ , N/ (cid:105) ⊗ | ↓ , N/ (cid:105) and | ↑ , − N/ (cid:105) ⊗ | ↓ , − N/ (cid:105) are degenerate at K y = 0 and max. charge imbalance q ⇡ N o r b , n ` B / L y ⇡ N o r b , n ` B / L y ⇡ N o r b , n ` B / L y ⇡ N o r b , n ` B / L y a ) b ) c ) d ) gap spread-to-gap ratio s avoidance-to-spread ratio r Josephson quality factor r Figure
3. Characterization of the low-energy spectrum of Hamiltonian Eq. (3) as a function of 2 πN orb , n (cid:96) B /L y and the strengthof the pairing potential ∆ . 2 πN orb , n (cid:96) B /L y approximates the physical distance between the superconducting regions and canbe tuned by varying L y . The charging energy is optimized for each L y using the procedure defined in Supplementary Note II.Other parameters are identical to those of Fig. 2. a) Gap between the three lowest states in the K y = 0 sector and the nextexcited states. Gray color indicates that the three lowest states do not have K y = 0. b) Spread of the three lowest states inthe K y = 0 sector as indicated in Fig. 2 c) divided by the gap. Gray color indicates the region in which the ratio exceeds 1or where the three lowest states do not have K y = 0. c) The largest eigenvalue q of the charge imbalance operator defined inEq. 4 in the space of the tree lowest states (at K y = 0). Gray color is used if the three lowest states do not have K y = 0. d)The difference r between the energy of the second lowest eigenstate at Josephson phases ϕ = 0 and ϕ = π , normalized by thespread and each time measured with respect to the ground state energy, as defined in Eq. 6. The closer r gets to 1, the bettercan the 6 π Josephson effect be observed. The gray color has the same meaning as in c). labelled by a blue ‘2’ in Fig. 2 b). They occur at exactly2 δE above the ground state.
2. Gapping the three-fold ground state
We now compare the low-energy structure of the sys-tem with zero [Fig. 2 b)] and non-zero superconductingpairing in the outer orbitals [Fig. 2 c)]. The states thatwere at 2 δE in the former system moved substantially be-low the ones formerly at δE . Thus, the spectrum cannotbe decomposed into that of two independent layers any-more. Superconductivity coupled the layers. Closer ins-pection also reveals a tiny but nonvanishing lifting of thedegeneracy between the two lowest-lying excited statesat K y = 0. We interpret the three lowest states in the K y = 0 sector as the quasi-degenerate topological statesof the edges and the gap above them as the superconduc-ting gap induced in the counter-propagating FQH edgemodes.The three-fold ground state degeneracy of the gappededge states can be understood as follows. By introdu-cing a gap, the superconducting coupling turns the bi-layer quantum Hall state with edges into a single-layerquantum Hall state on a manifold without boundary. Assketched in Fig. 1 c), this manifold is topologically equi-valent to a torus, where the space between the two layersbecomes the interior of the torus. This is in line with theopposite sign of the Hall conductivity in the two layers,because the normal to the torus surface is also rever-sed. On the torus, a Laughlin state at filling ν = 1 / φ and the Josephson phase ϕ , respectively.More relevant to the physics of the bilayer heterostruc-ture is an interpretation of the ground state degeneracy interms of Cooper-paired Laughlin quasiparticles. Due tothe mean-field superconducting order parameter, the par-ticle number is only defined modulo 2. Assuming that thelow-energy Laughlin quasiparticles (of charge e/
3) thatcomprise the edge mode are Cooper-paired, this leavesthree nonequivalent configurations for the charge of oneedge : 0, 2 e/
3, and 4 e/ e/ e/
3, or vice versa.We thus expect a total of three nearly degenerate topo-logical ground states from this consideration as well, inline with our numerical observation.Beyond the two special cases shown in Fig. 2 b) and c),we have performed an extensive study of the spectralproperties when varying the system parameters. Someresults are given in Fig. 3 for the largest system size thatcan be reached. Another system size is discussed in Sup-plementary Note I. We fix the total number of supercon-ducting orbitals N orb , sc , equally split between the twoends of the cylinder. The selected value is a compromisebetween fully covering the edge modes and a large en-ough non-superconducting region of N orb , n consecutiveorbitals where an incompressible liquid can develop [asdepicted in Fig. 1 b)]. For each perimeter L y the char-ging energy parameter 1 / C is optimized as discussedin Supplementary Note II ( N being fixed for the fulldiagram to N = 30). Instead of using L y for the verti-cal axis, we have plotted the data as a function of theapproximate width of the normal region, i.e., π(cid:96) B N orb , n L y .Such a quantity is more natural when comparing differentsystem sizes.In Fig. 3 a), we show the energy gap above the three lo-west energy states. We set the gap to zero if these threestates do not have K y = 0. We also provide s , the ra-tio between the energy spread of the three lowest energystates and the gap as previously defined. We cap s to oneor set it to one if the gap is zero or the three lowest energystates do not have the expected quantum numbers. To beable to claim that we have a low energy manifold madeof these three states separated by a gap from the higherenergy excitations, we need s <
1. The smaller s is, thecloser to an exact degenerate manifold we are. As can beobserved in Fig. 3 b), we have a large region where s issmall, beyond a critical value of ∆ depending on L y . B. Charge distribution
In this section we study the charge distribution bet-ween the left and right halves of the system. In a physicalrealization of the system, two scenarios should be distin-guished. If the two edges are coupled to the same super-conductor, which implies that they are phase coherent, noquantized charge can be associated with one edge alone.In contrast, if the two edges are gapped by independentsuperconducting reservoirs, they carry independent frac-tionally quantized charges. However, in this latter casethe charging energy is expected to lift the ground statedegeneracy and the states are not topological.The observable that measures the charge disproportio-nation between the two superconducting edges is givenby ˆ Q R − ˆ Q L = (cid:88) σ = ↑ , ↓ (cid:88) m (cid:90) d x (cid:90) L y d y sgn( x ) × | ψ σm ( x, y ) | ˆ c † m,σ ˆ c m,σ , (4)where the origin of the x axis coincides with the centerof the m = 0 orbital. It measures the charge differencebetween the left ( x <
0) and right ( x >
0) half of the sys-tem. We compute the expectation value of ˆ Q R − ˆ Q L in themanifold formed by the three lowest states in the K y = 0sector, yielding a 3 × ± q and 0, where q is an a priori unspecified realnumber.Figure 4 shows the evolution of q with the strengthof the superconducting pairing ∆ . For ∆ = 0, we canunderstand the nearly quantized value q ≈ / K y = 0 states are compri-sed of one state for which both up- and down-spin dro-plets are centered around m = 0 and a pair of states in L y Δ Q qq L y L y = 7 . L y = 10 . = 0 L y = 15 . Figure
4. Largest eigenvalue q of the operator ˆ Q L − ˆ Q R ,defined in Eq. (4), that measures the charge imbalance bet-ween the left half and the right half of the cylinder depicted inFig. 1 b) in real space, computed in the manifold of the threelowest energy states with momentum K y = 0. The data wasobtained with Hamiltonian Eq. (3) using the same parametersas in Fig. 2, except for L y /(cid:96) B = 7 . πN orb , n (cid:96) B /L y (cid:39) . L y /(cid:96) B = 10 . πN orb , n (cid:96) B /L y (cid:39) .
6) and L y /(cid:96) B = 15 . πN orb , n (cid:96) B /L y (cid:39) .
5) as indicated.Right of the respective dashed lines, the spread-to-gap ratioshown in Fig. 3 b) is less than one. Moreover the charge im-balance nearly vanishes in the parameter regime of optimalspread-to-spread ratio. The inset shows q as a function of L y for ∆ = 0. which they are both centered around m = ±
1. The cen-ter of charge of the former is located exactly at x = 0,while the latter two states have an excess of charge ± q/ x = 0 (i.e., x >
0) in each layer, and the op-posite deficit left of x = 0 (i.e., x < Q R − ˆ Q L is thus diagonal in this basis, with respec-tive eigenvalues 0 , ± q . In the thermodynamic limit, the x > ± /
3, sum-ming up to an expectation value q = 4 / Q R − ˆ Q L .In the absence of superconductivity, the relevant lengthscales are the perimeter L y and the width of the dro-plet 2 πN orb (cid:96) B /L y (where N orb = 3 ˜ N / − . (cid:96) B for the ν = 1 / q ≈ / L y to respect this criterion (obtained for L y /(cid:96) B = 10 . πN orb (cid:96) B /L y (cid:39) . (cid:54) = 0,we observe a rapid decrease in q , reaching (and passing)zero near the value of ∆ that leads to an optimal spread-to-gap ratio of the nearly degenerate ground states. Thiscan be observed by comparing Fig. 3 b) and Fig. 3 c). Inthe latter, we present q as a function of ∆ and the phy-sical distance between the superconducting edges. Two / ⇡ / ⇡E Ea ) b ) Figure
5. Evolution of the energy levels under spin-dependent flux insertion for Hamiltonian Eq. (3) with thesame parameters as in Fig. 2 except for L y = 7 . /(cid:96) B and∆ = 1 .
2. The spin-dependent flux insertion moves particlesin the background of the linear onsite potential, giving riseto the overall φ -dependent energy shift of the eigenstates. a)Evolution of the low energy spectrum. Red are the four loweststates in the K y = 0 sector, black are the lowest states in eachof the other K y = 0 sectors. Thus, not all states in the grayregion are shown. b) Close-up of the evolution of the threelowest states corresponding to the topological edge degrees offreedom, showing how the three states are permuted (up tosmall anticrossings) as φ is changed by 2 π . trends can be observed : (i) In the lower right cornerof this parameter space, where the edges are in closestspatial proximity, q is the smallest. (ii) In contrast, thelargest values of q are found in the upper left cornerof this parameter space. However, in this limit of small L y , corresponding to a thin cylinder, the charge distri-bution strongly varies with x even in the center of thedroplet. This yields contributions to the expectation va-lue of ˆ Q R − ˆ Q L from the center of the droplet x = 0,such that the operator does not allow for measurementof edge properties only.Given the limited system sizes we can study numeri-cally, it is hard to infer the behavior of the system inthe thermodynamic limit from this computation. We do,however, present data for one other system size in Sup-plementary Note I which shares the qualitative featuresdiscussed above with Fig. 3. In summary, we observedthat the charge imbalance of the three lowest states inthe K y = 0 sector evolves from being nearly quantized to4 e/ C. Spin-dependent flux insertion
To confirm the topological nature of the observed de-generate ground states, we perform a numerical chargepumping experiment. The adiabatic insertion of a magne- tic flux φ along the cylinder axis is equivalent to changingthe boundary conditions of the electronic wave functionsfrom periodic to twisted by an angle 2 πφ/φ , where φ is the flux quantum. In a Landau level, as φ is increa-sed from 0 to φ , all single-particle orbitals are shiftedby one unit of the quantum number m → m + 1. To seethis, notice that changing the boundary conditions fromperiodic to twisted amounts to replacing m by m + φ/φ in Eq. (1). In a Laughlin 1/3 state, every orbital has anaverage occupation 1/3, so that, in the thermodynamiclimit, a fractional charge e/ | Ψ (cid:105) , | Ψ + (cid:105) , | Ψ − (cid:105) by their charge imbalance 0 and ± q , into one another.As we will demonstrate, we can use charge pumpingto permute these ground states. Since the two layers ofour system are time-reversed partners with opposite Hallconductivities, we have to insert flux with opposite orien-tation for the ↑ -spin and ↓ -spin particles. Only then is anet charge pumped from one edge of the system to theother. We will refer to this as spin-dependent flux inser-tion [see Fig. 1 b)].Suppose we start with a state | Ψ (cid:105) that has charge 0on both edges. As unit φ spin-dependent flux is adiaba-tically inserted, charge is transferred from the left to theright edge, so that the resulting state is | Ψ + (cid:105) . The otherground states are expected to transform into one anotheranalogously : | Ψ + (cid:105) → | Ψ − (cid:105) , | Ψ − (cid:105) → | Ψ (cid:105) . Thus, after in-sertion of a quantum of spin-dependent flux, we expectto obtain a permutation of the three ground states. Thisexpectation is independent of the presence of a quantiza-tion of q , because the spectrum has to be invariant under φ → φ + φ . The observation of the state permutationunder spin-dependent flux insertion could, however, beobstructed by large avoided crossings in the evolution ofthe energy levels. It is important to stress that the spec-trum remains gapped (above the three ground states)during the entire process of spin-dependent flux inser-tion. This gap is provided by the superconducting orderparameter that couples states of different particle num-ber on each edge. Without the superconductivity, chargepumping would still occur between the gapless edges, butthe adiabatic process would simply accumulate charge atone edge and deplete the other, mapping the eigenstatesto others with ever higher energy with each quantum ofspin-dependent flux inserted.To implement the spin-dependent flux insertion, we ob-serve that the substitution m to m + φ/φ in Eq. (1) isequivalent to substituting m by m + φ/φ in µ m and f m for an infinitely long cylinder. In a finite cylinder, suchan approach is still valid for the low-energy subspace aslong as the number of orbitals is larger than the numberof orbitals typically covered by the incompressible liquid.In our case, this is roughly given by the number of or-bitals needed by a single Laughlin state with ˜ N /
N / − N Φ + 1 orbitals, m = − N Φ / , · · · , N Φ / m = − N Φ / , · · · , N Φ / µ m and f m , which allows their argument totake real values and substitute in the Hamiltonian (3) µ m → µ m + φ/φ (5a)and f m → f m + φ/φ . (5b)When tuning φ , the potential experiences a kink around m = 0 which would result in a kink in the energy spec-trum. We have thus replaced the absolute value around0 by a quartic polynomial interpolation that ensures thepotential and its derivative are continuous. Similarly for f , we use a linear interpolation for any orbital at theboundary between a superconducting ( f = 1) and a nor-mal ( f = 0) region.The low-energy spectrum of the resulting φ -dependentHamiltonian is plotted in Fig. 5. Up to a small avoidance,the three ground states permute as anticipated, while thespectral gap above them stays intact in the process. Theoverall evolution of all energy levels with a minimum at φ = φ / D. π Josephson effect
As a second piece of evidence that the heterostruc-ture realizes the topological superconducting edges, wecalculate the evolution of the energy levels that corres-ponds to the 6 π Josephson effect. In order to do so, therelative complex phase between the mean-field supercon-ducting order parameters on the left and the right edge, ϕ , is varied. The Josephson effect requires quasiparticletunneling processes between the superconducting edges.Necessarily, the spectrum of the heterostructure displaysa 2 π periodicity in ϕ . However, in the thermodynamic li-mit, in which the three states are degenerate, the groundstate of the system does not return to itself when ϕ isadvanced by 2 π . Rather, it evolves into another degene-rate ground state and only after ϕ is advanced by 6 π does the system return to its initial state. The reason forthis behavior is that the elementary excitations of thesuperconducting edge are Cooper paired quasiparticlesof charge 2 e/
3, delocalized along the cylinder perimeter,which tunnel across the bulk gap.
E E'/ ⇡ '/ ⇡a ) b ) Figure
6. Evolution of the energy levels of the 6 π Josephsoneffect for Hamiltonian Eq. (3) with the same parameters as inFig. 2 and ∆ = 2 .
13, varying the phase difference ϕ betweenthe left and the right superconducting edge. a) Evolution ofthe low energy spectrum. Red are the four lowest states inthe K y = 0 sector, black are the lowest states in each ofthe other K y = 0 sectors. Thus, not all states in the grayregion are shown. b) Close-up of the evolution of the threelowest states corresponding to the topological edge degrees offreedom, showing a 6 π periodicity with a small avoidance ofthe crossings between the states. To observe the 6 π Josephson effect numerically in ourfinite-size setup, the energy scale associated with the tun-neling must be larger than the finite size splitting bet-ween the ground states. For the system sizes accessibleto exact diagonalization calculations, this is not generi-cally the case, even when the spread-to-gap ratio shownin Fig. 3 b) is large. Since the tunneling amplitude is ex-ponentially small in the distance between the edges, weexpect a favorable regime for large cylinder circumference L y (at fixed number of non-superconducting orbitals), sothat the physical distance ∝ L − y between the edges issmall. Figure 6 shows the spectral evolution as a functionof ϕ in this regime. We observe that the three low-lyingstates are indeed permuted as ϕ advances by 2 π up toa residual small avoidance of the crossings between thestates.To investigate in which region of phase space this typeof spectral evolution can be found, we plot the ratio r oflargest avoided crossing over energy spread of the groundstate manifold, i.e. the quantity r := max [( E (0) − E (0)) , ( E ( π ) − E ( π ))] E ( π ) − E (0) , (6)where E ( ϕ ), E ( ϕ ), E ( ϕ ) are the energies of the threelowest states as a function of flux ϕ . When the avoidancein the evolution of the energy levels vanishes, such that E (0) = E (0) and E ( π ) = E ( π ), then r → π Josephson effect becomes clearly observable. In theopposite limit where the low-energy spectrum is essen-tially independent of ϕ E i (0) = E i ( π ), i = 0 , ,
2, wehave r → width of normal region energy gapspread/gapavoidance- to-spread ratio I. b ac ks ca tt e r i n g d o m i na t e d II. π J o s e p h - s on r e g i m e III. s e p a r a t e d e dg e s = 2 . L x,n /` B Figure
7. Phase diagram along the ∆ = 2 . L x,n = π(cid:96) B N orb , n L y . The vertical axis is either the energy gap (red),the spread over gap ratio (dashed black), or the avoidance-to-spread ratio for the Josephson effect (dashed blue). Weclearly discriminate three different regimes : I for small L x,n ,the backscatteting dominated regime with the breakdown ofthe gapped edge modes ; II the intermediate region corres-ponding to the 6 π Josephson regime ;
III at large L x,n , theseparated gapped edge modes. small to overcome the finite-size induced energy splittingbetween the three low-lying states. Figure 3 d) shows r as a function of the strength of the superconductingpairing potential and the physical distance between theedges. Indeed, we find that the 6 π Josephson effect isbest observed when both the spread-to-gap ratio and thedistance between the edges is smallest.
IV. DISCUSSION
We numerically studied a heterostructure of a bilayersystem of FQH Laughlin states with counter-propagatingedge modes that are gapped out by a mean-field super-conducting order parameter. The system in the cylindergeometry with two gapped edges realizes a nonlocal to-pological qutrit.Our calculations were performed using exact diagona-lization and by restricting the computation to the qua-sihole subspace of the Laughlin state in each layer. Des-pite this simplification, the system size is still limited.Nevertheless, we have been able to demonstrate four keyfeatures : (i) the edges develop a spectral gap inducedby the superconducting coupling, (ii) the expected num-ber of three nearly degenerate ground states without anycharge imbalance between the two halves of the system,(iii) that charge pumping can permute the ground states,and (iv) that the system exhibits a 6 π -periodic Joseph-son effect. For each signature, we discussed the suitableparameter regime.While the details of the phase diagram are affected byimportant finite-size effects, similar features can be iden- tified in all studied systems for ∆ > ∼ .
5. Extrapolatingfrom these features, we propose a physical summary anda highlight of our quantitative results on the 6 π Joseph-son effect in Fig. 7. We focus on a given value of thepairing parameter ∆ = 2 . L x,n is deduced from thenumber of orbitals without superconducting coupling asdiscussed in Sec. III A 2. As seen in Fig. 7, the value of L x,n determines the behavior of the system among threeregimes. When L x,n > ∼ . l B , there is a large gap andan approximate threefold degeneracy, leading to well de-fined gapped edge modes. But the tunneling required forthe 6 π Josephson effect is exponentially suppressed. Inthe intermediate regime 2 . l B < ∼ L x,n < ∼ . l B , the 6 π Josephson effect is clear and a robust gap remains. Notethat the optimal value of L x,n is roughly twice the cor-relation length[35] ξ (cid:39) . l B of the Laughlin ν = 1 / L x,n is small, i.e.,a distance lower than the correlation length ξ , the indu-ced gap at the edges collapses. Thus our results validatethe hypothesis of the previous effective approaches andprovide an estimate of the characteristic dimensions forfuture experimental implementations.Our work provides the first quantitative study of frac-tional edge modes coupled to superconducting leads ina fully microscopic model. Future works, most probablyrelying on the density matrix renormalization group cal-culations, should be able to rely on our setup and resultsto provide new insights. In particular, it could overcomethe size limitation and address the potential new emer-ging phases when substituting the Laughlin state withany richer topological order. ACKNOWLEDGMENTS
We are very grateful to Maissam Barkeshli for seve-ral inspiring discussions during the course of this workand for helpful comments on the manuscript. We thankC. Nayak, B. A. Bernevig, C. Mora and B. Estiennefor fruitful discussions. C. R. was partially supportedby the Marie Curie programme under ED Grant agree-ment 751859. N. R. was supported by Grant No. ANR-16-CE30-0025. A. M. C. also wishes to thank the As-pen Center for Physics, which is supported by NationalScience Foundation grant PHY-1066293, and the KavliInstitute for Theoretical Physics, which is supported bythe National Science Foundation under Grant No. NSFPHY-1125915, for hosting during some stages of thiswork.T. N. acknowledges support from the European Re-search Council (ERC) under the European Union’s Hori-zon 2020 research and innovation programm (ERC-StG-Neupert-757867-PARATOP).0
AUTHOR CONTRIBUTIONS
CR and NR developed the numerical code. CR, AC,and NR performed the numerical calculations. All au-thors contributed to the analysis of the numerical dataand to the writing of the manuscript.
Supplementary Material
Annexe A: Results for a different system size
In the article, we have focused on a single system. Thissystem was the largest one for which a complete studycould be performed. We have considered system sizes,with different numbers of superconducting orbitals andvarious number of particles ¯ N . Here we provide additio-nal numerical results for a smaller system size of N φ = 20in Fig. 8. The charging energy parameters are optimizedsuch that the ground state in the absence of supercon-ducting coupling has now a number of particles ¯ N = 10.When compared to Fig. 3 for N Φ = 21 and ¯ N = 12, thissmaller system has proportionally more superconductingorbitals ( N orb , sc = 16) while preserving an equivalentlynormal region size. Note that a similar system for ¯ N = 12would require N φ = 23 and N orb , sc = 18 which is unfor-tunately numerically out of reach at the moment.As a consequence of having more superconducting or-bitals, we now observe a much larger region where the6 π Josephson effect can be observed [Fig. 8 d)]. It stillrequires the two physical edges to be close but it appearsfor a wider range of superconducting coupling strength.Having a smaller system is twofold has nevertheless somedrawback. Indeed, the gap (Fig. 8 a) and the spread overgap ratio (Fig. 8 b) have similarities with the bigger sys-tem discussed in the main text. This is especially valid inthe range where the normal region has a size lower than4 l B . Note that plotting the numerical data as a functionof the normal region width helps a lot to compare dif-ferent system sizes. But the agreement is getting lowerwhen we increase the normal region size, with a muchweaker gap and a worse s .We also provide a phase diagram for the spin-dependent flux insertion in Fig. 9 both for the system sizeaddressed here [Fig. 9a)] and for the system size discussedin the main text [Fig. 9b)]. We again use the ratio r (de-fined in Eq. (6)) with flux ϕ replaced by spin-dependentflux φ . A small value of r indicates the regime wherethe spin-dependent flux insertion is clearly observable.As can be seen in Fig. 9, we find a large region that doesnot even require the two edges to be too far from eachother (albeit not too close) and where there is a clearmixing of the three states under the spin-dependent fluxinsertion. Annexe B: Parameter determination for thecharging energy
Here we explain how we determine the parameters en-tering the charging energy term, C and N . We demandthat the ground state is in the sector with some targetnumber of particles ¯ N and that the lowest energy eigens-tates of Hamiltonian (3) for ∆ = 0 in the particle num-ber sectors ¯ N − N +2 are degenerate. Denoting the1 max. charge imbalance q ⇡ N o r b , n ` B / L y ⇡ N o r b , n ` B / L y ⇡ N o r b , n ` B / L y ⇡ N o r b , n ` B / L y a ) b ) c ) d ) gap Josephson quality factor r spread-to-gap ratio s avoidance-to-spread ratio r Figure
8. Characterization of the low-energy spectrum of Hamiltonian Eq. (3) as a function of 2 πN orb , n (cid:96) B /L y , which is thephysical distance between the superconducting regions and changed by varying L y , and the strength of the pairing potential∆ , with parameters N φ = 20 and N orb , sc = 16, as well as a mean number of particles ¯ N = 10. a) Gap between the three loweststates in the K y = 0 sector and the next excited states. Gray color indicates that the three lowest states do not have K y = 0.b) Spread of the three lowest states in the K y = 0 sector as indicated in Fig. 2 c) divided by the gap. Gray color indicates theregion in which the ratio exceeds 1 or where the three lowest states do not have K y = 0. c) Maximal eigenvalue of the operatormeasuring the charge imbalance between the left and right half of the system in the space of the tree lowest states. Gray coloris used if the three lowest states do not have K y = 0. d) The difference between the energy of the second lowest eigenstate atJosephson phases ϕ = 0 and ϕ = π , normalized by the spread and each time measured with respect to the ground state energy.The closer this quantity gets to 1, the better can the 6 π Josephson effect be observed. The gray color has the same meaning asin c) ⇡ N o r b , n ` B / L y ⇡ N o r b , n ` B / L y a ) b ) avoidance-to-spread ratio r avoidance-to-spread ratio r spin-dependent fluxspin-dependent flux N = 21 N = 22 Figure
9. Avoidance-to-spread ratio r for spin-dependentflux, defined in Sec. A, for Hamiltonian Eq. (3) as a functionof 2 πN orb , n (cid:96) B /L y , which is the physical distance between thesuperconducting regions and changed by varying L y , and thestrength of the pairing potential ∆ , a) with parameters N φ =21 and N orb , sc = 17, as well as a mean number of particles¯ N = 10 b) with parameters N φ = 22, N orb , sc = 17 and amean number of particles ¯ N = 12. Gray color is used if thethree lowest states do not have K y = 0. The dark diagonalline in b) is an artifact of the spread being extremely smallalong this line. energy of the chemical potential term of the Hamiltonian as E (0) N for N particles, this yields the condition E (0)¯ N +2 + 12 C ( ¯ N + 2 − N ) = E (0)¯ N − + 12 C ( ¯ N − − N ) . (B1)We solve this to determine the charging energy as12 C = E (0)¯ N +2 − E (0)¯ N − N − ¯ N ) . 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