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Controlling non-Abelian statistics of Majorana fermions in semiconductor nanowires

Jay D. Sau,1 David J. Clarke,2 and Sumanta Tewari3

1Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics,University of Maryland, College Park, Maryland 20742-4111, USA

2Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA3Department of Physics and Astronomy, Clemson University, Clemson, SC 29634

Under appropriate external conditions a semiconductor nanowire in proximity to ans-wave superconduc-tor can be in a topological superconducting (TS) phase. Thisphase supports localized zero-energy Majoranafermions at the ends of the wire. However, the non-Abelian exchange statistics of Majorana fermions is difficultverify because of the one-dimensional topology of such wires. In this paper we propose a scheme to transportMajorana fermions between the ends of different wires usingtunneling, which is shown to be controllable bygate voltages. Such tunneling-generated hops of Majorana fermions can be used to exchange the Majoranafermions. The exchange process thus obtained is described by a non-Abelian braid operator that is uniquelydetermined by the well-controlled microscopic tunneling parameters.

PACS numbers: 03.67.Lx, 03.65.Vf, 03.67.Pp, 05.30.Pr

I. INTRODUCTION

Majorana fermions (MFs) have been the subject of in-tense recent study in part due to their potential application intopological quantum computation (TQC).1–7 Unlike ordinaryfermionic or bosonic operators for which the particle creationoperators are the hermitian conjugate of the annihilation op-erators, MF operators,γ, are self-adjoint (γ† = γ). In thissense, MFs are their own anti-particle and the realization ofsuch excitations would be the first example of such particleswhich had been proposed more than seventy-five years ago.8

MFs are of interest for TQC because despite having no inter-nal degrees of freedom individually, a pair of MFs, sayγ1 andγ2, have two distinct possible states (fusion channels). Thesestates, which may be thought of as the two possible occupa-tion states of the complex fermionic operatorc† = γ1+iγ2

2 , areenergetically degenerate to a degree exponential in the sepa-ration of the Majorana fermions, and correspond to the eigen-states of the combined operatorıγ1γ2, with eigenvalues±1.Such a topological protected degeneracy has yet to be seen innature, and the observation of such would be a major break-through in physics. The central idea of TQC is to use thetopologically degenerate states of a pair of MFs as a 2-leveltopological qubit which would in principle be protected fromdecoherence. The manipulation of the information containedin the topological qubits requires the use of topological braidoperations which consists of moving the MFs around one an-other.

In the past few years, topological superconductors havebecome promising candidates for realizing MFs.9–16 Re-cently, it has been proposed that a semiconductor thin filmwith Rashba-type spin-orbit (SO) coupling together withproximity-induced superconductivity and Zeeman splittingwould be a suitable platform for realizing a Majorana-fermion-carrying topological superconducting (TS) state.17–20

Thes-wave superconducting pairing potential can be inducedin the semiconductor system by placing it in proximity to aconventional superconductor such as aluminum. The Zee-man splitting can similarly be induced, in principle, by prox-imity to a magnetic-insulator.20 The one-dimensional ver-

sion of this system, i.e. a semiconducting nanowire withproximity-induceds-wave superconductivity, has also beenshown to host MF as zero-energy modes at the ends of thewire under appropriate conditions.21,22 The one-dimensionalnanowire geometry has the specific advantage that the Zeemansplitting VZ in the nanowire is not required to be proximity-induced from a magnetic insulator, but instead can be intro-duced by a magnetic field parallel to the nanowire.22 Such aparallel magnetic field would not introduce unwanted orbitaleffects such as vortices if a thin-film superconductor is usedto generate the superconducting proximity-effect. The pro-posed semiconducting structures exist in a TS phase and sup-ports MFs at its ends when thes-wave superconducting pairpotential∆, Zeeman splittingVZ , and the chemical poten-tial µ satisfy the conditionV 2

Z > ∆2 + µ2.17,20,22 Thus thechemical potentialµ, which can be controlled by an externalgate potential, can be used to tune a nanowire from the TSphase to a non-topological (NTS) phase. In fact, thes-waveproximity effect on a InAs quantum wire, which also has asizable SO coupling, may have already been demonstrated inexperiments.23 These semiconductor based proposals for real-izing a TQC platform can take advantage of the considerablyadvanced semiconductor fabrication technology. Therefore, itseems that a Majorana-carrying TS state in a semiconductorquantum wire may be within experimental reach.

Until recently, motivated by experiments, most discussionsof observing non-Abelian statistics using MFs has been re-stricted to 2D systems. In 2D non-Abelian systems, the quan-tum information associated with MFs can be manipulated ina topologically protected manner by exchanging the Majo-rana bound states (e.g., by adiabatically moving vortices inp+ip superconductors).24,25 The protection of the topologicaldegeneracy associated with MFs requires the MFs to remainspatially separated at all steps of the exchange. Therefore, atfirst glance, it appears that it is impossible to exchange theMFs at the ends of a 1D wire, since any such attempt wouldnecessarily lead to the spatial overlap of MFs at some stageof the overlap process. A solution to this problem has beenprovided by by Aliceaet al.26, who have shown that connect-ing up a system of nanowires into a network allows one to

2

exchange MFs. In their proposal, the two MFs,γ1 andγ2, atthe ends of a wire can be exchanged by introducing an addi-tional nanowireB. The additional nanowire,B, allows oneto temporarily move one of the MFs, sayγ1, away from theoriginal nanowire,A, so that the other MFγ2 can be movedacross the wireA without colliding withγ2. The MFγ1 canthen be returned from the wireB back to it’s original wireA.While this scheme solves the basic problem of non-Abelianstatistics in1D, it requires the transport of a MF across a tri-junction between two topological nanowiresA andB, whichis potentially a more complex topological object than the sim-ple topological nanowire. The continuous transport of MFsthrough such a tri-junction is potentially dependent on detailsof the junction that may be difficult to control.27 Moreover,from a theoretical point of view, the explicit determination ofthe non-Abelian statistics in this geometry in terms of the mi-croscopics of the junction is somewhat complicated.

In this paper we propose an alternative scheme to transportMFs at the ends of one dimensional semiconductor nanowireswhere the ends of the nanowires remain fixed, but the tunnel-ing amplitudes between the end MFs are varied. Bringing theend MFs closer together allows one to create a non-vanishingHamiltonian of the MFs which can generate effecting MFhopping from one end site to another. Using this picture ofa dynamically changing tunneling Hamiltonian, we will beable to derive a simple explicit expression for the non-Abelianstatistics transformation of the MFs in terms of tunneling ma-trix elements.

II. OUTLINE AND SUMMARY OF RESULTS

As mentioned in the previous paragraph, MFs are strictlyzero-energy modes with an associated topological degeneracyonly in the limit when they are separated by a distance thatis large compared to the decay length of the MFs. The twostates of the MFsγ1 andγ2 can be described in terms of the2 possible occupation states of the Dirac fermionc† = γ1 +iγ2. These 2 states correspond to the eigenvalues0 and1 ofthe number operatorn = c†c = 1+iγ1γ2

2 . In general, theHamiltonian for a pair of MFs with a non-negligible splittingproduces a splitting between the 2 energy states and can bewritten as

Htunneling = iζ12(x)γ1γ2 (1)

whereζ12(x) is the tunneling matrix element for the MFswhich depends on the separationx between the MFsγ1 andγ2.25 The energy splitting between then = 0 andn = 1 statesis given by|ζ1,2(x)|. Therefore, the topological degeneracyof MFs emerges only in the limitx ≫ ξ when the MF overlapmatrix-elementζ12(x) vanishes because of the localization ofthe MF wave-functions. Hereξ is the localization length ofthe MFs. The tunneling of MFs at the ends of different wires,whose ends are placed close together, is entirely analogoustothe tunneling of electrons between two quantum dots whichcan be controlled by raising and lowering the barrier betweenthe dots. Similarly, tunneling amplitudes between MFs on dif-ferent semiconductor nanowires can be controlled simply by

adding a gate controllable tunnel barrier between the MFs.30

Gate voltages can also induce tunneling between MFs at theends of the same TS segment by tuning the nanowire close toa TS-NTS phase transition.20 Bringing the nanowire close tothe TS-NTS transition, decreases the gap of the system whichin turn increases the localization lengthξ of MFs in the wireand allows the tunneling between the initially localized MFsat the ends of TS segments. The quantitative details of how thetunneling is controlled in topological nanowires is discussedin the appendix.

Tunneling of ordinary fermions such as electrons can beused to move electrons from one quantum dot to another ina system of quantum dots. In this paper, we will show thatthe same principle applies to MFs, and repeated use of thetunneling Hamiltonian in Eq. 1 can be used to exchange MFsγ1 andγ2 in a system of TS nanowires that hosts such MFsat its ends. The unitary time-evolution operatorU associ-ated with the exchange mapsγ1 → Uγ1U

† = λγ2 andγ2 → Uγ2U

† = −λγ1 whereλ can be directly computedfrom the tunneling matrix elementsζi,j involved in movingthe MFγ1 to the starting position of the MFγ2. In the low-energy subspace of MFs, the time-evolution operationU , thatdescribes the exchange process has the usual form of a braidmatrix24

U = eπ4λγ1γ2 . (2)

While there are only two possible answersλ = ±1 for thebraid-matrix, it is critical to be able to determine the factorλ for a given braid since this is what distinguishes ’clock-wise’ from ’counter-clock-wise’ exchanges. For the specificgeometry discussed in the appendix with wires placed in asuperconducting film together with an in-plane magnetic fieldat 45 degrees to the wires, the sign of the braid matrix,λ,is determined by the sign of the Rashba spin-orbit couplingconstantα.

The braiding scheme we will discuss is potentially re-lated to measurement-only schemes for braiding of topolog-ical quasiparticles.28,29However, it is not clear how the anyonmodel postulates assumed in the measurement-only theory ap-ply to the superconducting nanowire systems described bymean-field BCS theory. For example, the identification ofthe tunneling matrix element between MFs in Eq. 1 with thetopological charge measurement in Ref. 29 becomes subtlein cases where the sign of the tunnelingζ12(x) oscillates insign with the separationx. On the other hand the approachin this paper is based only on the MF tunneling Hamilto-nian Eq. 1, which can be derived microscopically from BCSHamiltonians.31 The tunneling matrix elementsζij(x) them-selves depend on the details of the nanowire system such asthe spin-orbit coupling, the orientation of the wire and theZeeman splitting. Therefore, we first consider exchange ofMFs around a specific triangular loop geometry in terms of thetunnel matrix elements between the various MFs and then inthe appendix, we show how the microscopic tunneling param-eters may be calculated in one specific geometry. As a resultof our calculation, we find that for general values of the tun-neling, the parameterλ in the braid matrixU has the simpleform λ = sgn(ζ12)χ whereχ is the junction chirality of the

3

FIG. 1: (Color online)(a) Combination of SC, Zeeman and gatepo-tentials leads to nanowire segments in TS and NTS phases. Thegates1 and3 are adjusted such that the nanowire is in TS phase,while 2 is adjusted so that the wire is in NTS phase corresponding tothe schematic in panel (b). (b) Nanowire segments in TS phaseareshown as blue (solid) lines and NTS segments are shown as red (dot-ted) lines. Orange (light) and blue (dark) circles indicateunpairedand paired MFs at TS-NTS interfaces, respectively. MFs are pairedby tunneling across the TS or NTS segments denoted by light blueoval. Decreasing the tunneling amplitude betweenγ2 andγ3 and si-multaneously increasing the tunneling amplitude betweenγ1 andγ2can effectively transfer MFγ1 → γ3.

triangular loop that we will define as the product of tunnelingsaround the loop. Finally, we would like to note, that while themotivation of exchanging MFs is to be able to manipulate theinformation contained in topological qubits constructed fromMFs in an effort to perform TQC, it is well-known that braid-ing by itself is insufficient for TQC.6 However, MF exchangesare still crucual as one of the most direct tests of non-Abelianstatistics and probably also for any future TQC schemes usingMFs.

III. MF TRANSPORT

To understand how MF transport in a system of nanowirescan be induced by tunneling, consider the simple system ofnanowires shown in Fig. 1(a) consisting of three semicon-ducting nanowire segments. Two of these segments (shownby solid blue lines in the schematic in Fig. 1(b)) are in theTS phase, while the wire shown with the red dashed line isin the NTS phase and serves as the tunnel barrier connect-ing MFs. Each end of the wires in the TS phase supports aMF (shown as discs). In the initial state (shown in the upperpanel of Fig. 1(b)), the gate voltage of the NTS segment ischosen to allow a finite tunneling amplitude (shown by lightblue oval) across it. This pairs up the MFsγ2 andγ3 into fi-nite energy states with a gap. Thus the operatorsγ2,3 becomegapped MFs (shown as dark blue discs) and cannot be used tostore quantum information as can be done with the true zero-energy MFs (shown as light orange discs). The transfer of theMF from position1 to 3 is achieved by adiabatically deacti-vating the tunneling in the NTS segment2 − 3 and activatingthe tunneling in the TS segment1− 2.

The process of transferring the MF from position1 to 3

shown in Fig. 1(b) is described by the time-dependent tun-neling Hamiltonian that is derived by extending the tunnelingHamiltonian in Eq. 1 and can be written as

H = [ζ12α(t)γ1γ2 + ζ23(1− α(t))γ2γ3] (3)

where ζ12 and ζ23 are the activated tunneling amplitudesacross the segments1 − 2 and2 − 3 respectively. Over thetransfer processα(t) varies adiabatically fromα(0) = 0to α(t1) = 1. It is convenient to understand the braid-ing procedure for MF operators in the Heisenberg repre-sentationγj(t) = U †(t)γjU(t) where theU is the uni-tary time-evolution operatorU(t) = Te−i

∫t

0H(τ)dτ (which

is a time-ordered exponential). The operatorsγj(t) can becomputed from the Heisenberg equation of motionγj(t) =

i[H(H)(t), γj(t)]. The Hamiltonian (Eq. 3) describing theevolution ofγj(t) can be written compactly in terms of aneffective B-field(Bj(t)) as

H(H)(t) =∑

a,b,c=1,2,3

ǫabcBa(t)γb(t)γc(t) (4)

whereǫabc is the anti-symmetric Levi-Civita tensor. The time-dependentB-field given by

B(t) = (1− α(t))ζ23(1, 0, 0) + α(t)ζ1,2(0, 0, 1). (5)

The Heisenberg equation of motion forγa(t) takes the form

γa = 2ǫabcBb(t)γc(t). (6)

This equation of motion is identical to that of the spin opera-torsσa(t) of a spin-1/2 particle in a time-dependent magneticfield B(t) (with a HamiltonianH(H)(t) = −B(t) · σ(t)).Furthermore, the initial condition on the operatorγ(t) = γ1corresponds to the spin-operatorσ(t) = σ1(0) in an initialeffective magnetic fieldB(0) = ζ23(1, 0, 0) that is alignedor anti-aligned withσ1. Thus, after a time-evolution underan adiabatically varying magnetic field, the spin (and corre-spondingly the MF) remains aligned or anti-aligned with thefinal magnetic fieldB(t1) = ζ12(0, 0, 1) at timet = t1. Thisleads to the expression

γ3(t1) = sgn(ζ12ζ23)γ1(0). (7)

Thus the transfer of the tunneling amplitude from the segment2− 3 to the segment1− 2, leads to transport of the MF fromposition1 to position3. The hopping of MFs between sites de-scribed by Eq. 7 is identical to the motion of regular fermionicoperators under that action of tunneling. We will representthisprocess by the MF trajectory

12−→ 3. (8)

The result in Eq. 7 is consistent with a somewhat differentapproach suggested by Kitaev.32

IV. MFS AS DEFECTS IN DIMER LATTICES

In the previous section, we saw that to transport a singleMF from one position to another it was necessary to make use

4

FIG. 2: (Color online)(a) Configuration of TS nanowires in a squaredimer lattice with 2 isolated MFs. MFs are effectively boundto de-fects (unpaired sites) on the dimer lattice. Tunneling between differ-ent TS segments fuses TS wires into isolated effective ’topologicalwires’ shown in green (gray) blocks with MF end modes. (b) MFtransfer processes analogous to Fig. 1 can extend effectiveTS wiresand move MFs in dimer lattice in a way analogous to Alicea et al.26.(c) and (d): Similar transfer processes can be used to switchthe MFend modes between different effective ’topological wires’. All op-erations of the MF square lattice either exchange, contractor switchends of different ’topological wires’.

of another pair of MFs which were coupled by a weak tunnel-ing so that they were not strictly zero-energy MFs. The singleMF being exchanged had to be a true zero-energy MF withno tunneling, while the pair of MFs with tunneling betweenthem may be thought of as a gapped MF dimer. Exchangesof true zero-energy MFs requires a generalization of this pic-ture to include several isolated MFs which are not coupledto any other MF by tunneling. As is clear from Fig. 1(b),such a process also requires a supply of gapped pairs of MFs(i.e. MF dimers). In this paper, we will consider a system ofnanowires with end MFs most of which are paired up by tun-neling into MF dimers as shown in Fig. 2(a). If all MF sites arecompletely paired up, then the system has no true zero-energyMFs and no topological degeneracy or non-Abelian statistics.Therefore we consider a system, where in addition to the MFdimers, there are a few isolated MFs that are unpaired by tun-neling. If one considers a regular lattice of nanowires so thethe end MFs live on the vertices of a square lattice (Fig. 2), asystem of dimerized MFs forms a dimer covering of the lat-tice, while isolated MFs are associated with defects (unpairedsites) in the MF dimer lattice. As seen by comparing Fig. 2(a)and (b), the transport of MFs on the dimer-lattice is obtainedby changing the dimerization pattern on the dimer lattice anal-ogous to Fig. 1(b).

The MFs on the dimer lattice can also be thought of as theend points of topological wires. To see this we define effective’topological wires’, shown by green (gray) boxes in Fig. 2, byconsidering TS wire segments coupled by tunneling as a sin-gle topological wire. This identification, relates our proposalin a direct way to the proposal of Alicea et al.26 However, thedetails of the physical implementation remain different andthe dimer implementation presented in this paper will allowus

to directly use Eq. 7, to determine the form of the braid-matrixin Eq. 2. The continuous processes required by Alicea et. al.for exchanging MFs were extending and contracting topolog-ical wire segments together with an operation that we will re-fer to as exchanging the ends of different topological wires.This process required bringing together a pair of topologicalwires in a tri-junction and effectively takes a pair of topologi-cal wires with end pointsγ1,2 andγ3,4 and creates a new pairof wires with end pointsγ1,3 andγ2,4. All these processescan be accomplished in MF dimer lattice by repeated applica-tion of the process shown in Fig. 1 associated with Eq. 7. Theanalogue of the extension and contraction process in a dimerlattice is shown in Fig. 2(a) and (b), while the end switchingprocess is shown in Fig. 2(c) and (d).

V. NON-ABELIAN STATISTICS OF MFS IN NANOWIRES

In this section, we show explicitly, that exchange of MFsin any dimer lattice can always be described by an equationof the form of Eq. 2. Unpaired MFs can be exchanged viadiscrete tunneling operations of the form shown in Fig. 1(b).Since the physical positions of the MFs are exchanged by thecorrect sequence of MF transfers, the resulting transformationof the MFs at the end of the transformationt = tfinal has thegeneral form

γ1(tfinal) = λγ2(0)

γ2(tfinal) = λγ1(0). (9)

However, consistency with non-Abelian statistics also requireus to prove thatλλ = −1. If λλ = −1, the exchange trans-formation can be represented by the operatorU of the formEq. 2.

To showλλ = −1, let us label the unpaired MFs, which areto be exchanged as1 and2 and take all other MFs as paired,(2n − 1, 2n) for n = 2, . . . , N . The positions of the MFsfollowing each step (labelled by the indexp) of the exchangeprocess, which permutes the positions of the MFs, can be rep-resented by the functionπp(j), wherej = 1, . . . , 2N is theMF index. After each stepp, the MF coordinates are updatedfrom πp−1 to πp according to the relation

πp(j) = πp−1(Cp(j)) (10)

whereCp is a cyclic (clockwise or anticlockwise) permutationof the MFsap, 2np− 1 and2np corresponding to Eq. 7. Herewe chooseap to be one of the unpaired MFs,1 or 2, andnp >1 such that MF dimer(2np − 1, 2np) is paired. The equationof motion for the unpaired Majorana operators correspondingto Eq. 7 is

γπp+1(ap)(tp+1) = λpγπp(ap)(tp). (11)

where

λp = sgn(ζπp+1(2np−1)πp+1(2np)ζπp(2np−1)πp(2np)). (12)

The total signλλ picked up by the unpaired MFs is the prod-uct λλ =

∏

p λp. To calculate this product we define a se-quenceQp = sgn(

∏

n>1 ζπp(2n−1),πp(2n)). From Eq. 12 it

5

follows thatQp+1 = λpQp so that

Qfinal = λλQp=0. (13)

Note that since each cyclic permutationCp contains evennumber of exchanges (i.e is an even permutation), the per-mutationsπp at each step (including the final permutationπfinal), which is a product ofCps, is also an even permu-tation.

Since the Hamiltonian is required to return to its initialconfiguration, the MFs at positions(2n − 1, 2n) must bepaired by tunneling forn > 1. This requires thatπfinal iscomposed of a pair exchange of the positions of MF dimers(2n− 1, 2n) ↔ (2n′ − 1, 2n′) together with possible internalflips (2n − 1 ↔ 2n) of the dimers. Since,πfinal is an evenpermutation, and dimer exchanges are even permutations, thenumber of internally flipped dimers(2n− 1 ↔ 2n) in πfinal

is even. Moreover, the unpaired pair of MFs(1, 2) is flippedin πfinal. Thus an odd number of the paired MF dimers(2n− 1 ↔ 2n), must be flipped forn > 1. Each such dimerflip changes the sign ofQfinal, sinceζ2n−1,2n = −ζ2n,2n−1

for n > 1. This leads to the relationQfinal = −Qp=0 =

λλQp=0, proving the consistency condition for non-Abelianstatistics i.e.λλ = −1.

VI. EXCHANGE AROUND A TRIANGULAR LOOP

A specific realization of the non-Abelian statistics innanowire systems is provided by a triangular loop geometryshown in Figs. 3 and 4. The triangular loop consists of oneend (A2,B2 andC2) of each of three TS segments (A, B andC) connected by NTS segments to form a triangle. The otherends are labeledA1, B1 andC1. The MFs to be exchanged,referred as1 and2, are assumed to be localized at 2 of these 6ends of TS segments. Each of the steps for the MF exchange(shown in Figs. 3 and 4 ) consists of moving exactly one MFfrom one position to the other (shown by dotted arrows) byadiabatically turning off the tunneling in some wire segmentand increasing it in an adjoining segment as discussed before.

The procedure to exchange the MFs1 and2 at the endsof different TS segment through the tri-junction takes placein four steps shown in Fig. 3. The signs associated with theexchangeλ andλ can be determined by following the trajec-tories of the MFs1 and2 and applying Eqs. 7 and 9. FromFig. 3, it is clear that the sequence of positions followed bythe MFs1 and2 are

MF 1: A2C2−→(3)

B2

MF 2: B2C2−→(2)

C1C2−→

(4)≡(1)A2 (14)

respectively. Here we show only the MF that is moved in eachstep, which is numbered in Fig. 4 as(j = 1, . . . , 4) (markedbelow the arrows in Eq.14). The MF motion is shown usingthe notation defined in Eq. 8 so that the sign can be calculatedusing Eq. 7. Applying Eq. 7, the parametersλ andλ simplify

FIG. 3: (Color online) MFs1 and2 at the ends of different TS seg-ments are exchanged. This is achieved by switching tunnelings onand off on TS and NTS segments in 4 steps going from a state shownin one panel to the next panel. Dotted arrow shows motion of MFfrom the previous panel.The labelling for the sitesA1,A2, B1,B2,C1 andC2 is shown in panel (1).

FIG. 4: (Color online) MFs1 and2 at the ends of the TS segmenton the left leg are exchanged in seven steps similar to Fig. 3.Step(7) transfers state shown in panel (6) back to panel (1) with the effectthat the Majoranas 1 and 2 are interchanged.

to

λ = −λ = sgn(ζA2B2)χ (15)

whereχ = sgn(ζA2B2ζB2C2

ζC2A2) is defined to be the chi-

rality of the tri-junction.27

Similarly MFs at the ends of the same TS segment can beexchanged using six steps shown in Fig. 4. From Fig. 4, it isclear that the sequence of positions followed by the MFs1 and2 are

MF 1: A1A2−→(3)

C2B2−→(4)

B1B2−→

(7)≡(1)A2

MF 2: A2C2−→(2)

C1C2−→(5)

B2A2−→(6)

A1 (16)

respectively. The step(7) is not explicitly shown in Fig. 4,since it is equivalent to(1). Applying Eq. 9, the parametersλ

6

andλ simplify to

λ = −λ = sgn(ζA1A2)χ (17)

whereχ is the junction chirality.

Thus, using Eq. 17 and Eq. 9, we obtain the the result thatthe unitary time-evolution of the MFsγ1 and γ2 under ex-change can be described by the unique braid-matrix

U = eπ4χsgn(ζ12)γ1γ2 (18)

whereζ12 is the tunneling amplitude of the segment separatingγ1 andγ2. The quantitiesζ12 andχ for a specific network arecalculated in the appendix.

VII. CONCLUSION

Non-Abelian statistics for MFs at the ends of TS nanowiresegments can be realized by introducing time-varying gate-controllable tunnelings between MFs in a nanowire system toexchange the end MFs. Similar to the previous proposal forbraiding MFs in 1D wires, our system can also be embeddedin 3D leading to the possibility of non-Abelian statistics in3D. The isolated MFs being exchanged in the tunneling ge-ometry considered in this paper may be thought of as defectsin a dimer lattice i.e. sites that are unpaired by tunneling.Alternatively, this system may also be thought of as a dis-cretized implementation of the continuous nanowire networkproposal of Alicea et. al.26 However, the discrete implemen-tation discussed in this paper allows us to compute the braidmatrix explicitly in terms of MF overlaps. The non-Abelianbraid matrix for exchange around a triangular loop geometryis given by a product of the fusion channel of the MFsζ1,2and the junction-chiralityχ. The fusion channelζ1,2 is sim-ply the tunneling matrix-element between the MFs being ex-changed and the junction-chirality is the product of tunnelingterms around the triangular junctions. Thus the braid-matrixin the tunneling geometry considered in this paper is com-pletely determined in terms of microscopic tunneling param-eters by Eq. 18 making nanowire systems a well-controlledplatform to realize non-Abelian statistics.

Acknowledgments

We thank Parsa Bonderson, Anton Akhmerov and KirillShtengel for helpful discussions. We are grateful to the As-pen Center for Physics for hospitality during the 2010 sum-mer programLow Dimensional Topological Systems. D.J.C.is supported in part by the DARPA-QuEST program. S.T. ac-knowledges support from DARPA-MTO Grant No: FA 9550-10-1-0497. J.D.S. is supported by DARPA-QuEST, JQI-NSF-PFC, and LPS-NSA.

FIG. 5: (Color online)Schematic of orthogonal nanowire system ona superconductor (shown as light rectangle) that generatestunnelingof MFs (shown as light orange discs). The entire system is subject toan in-plane magnetic field to generate Zeeman coupling. Nanowiresegments in the TS phase are shown as dark blue rectangles withend MFs. Tunneling is generated between MF T and MF L by con-ventional tunneling across a nearly depleted nanowire in the NTSphase. The tunneling can be calculated using the Bardeen tunnelingformula33 as the matrix element of the current operator in the mid-dle of the wire (black dotted line). Similarly tunneling is generatedbetween MF L and MF R by lowering the topological gap so thatthe wave-functions have significant overlap at the middle ofthe wire(black dotted line).

Appendix A: Calculation of tunneling matrix elements for aspecific nanowire system

In the main text of the paper we saw how controlling thetunneling between MFs can be used to generate transport ofMajorana fermions from one point to another and eventuallygenerate exchanges and braids that are useful for TQC. Thesign of the resulting exchange was found to be determined bythe signs of various tunneling matrix elements. While the ex-istence of non-Abelian statistics is demonstrated in the paperin general, the signs of the tunneling matrix elements them-selves are depend on the microscopic details of the system.

In this appendix, we calculate the tunneling matrix ele-ments between the various MFs at a junction for a networkof orthogonal wires on a superconducting substrate as shownin Fig. 5. A Zeeman potential is applied at 45 degrees to thewires. The wires in the TS phase (shown in dark blue), whichsupport MFs (shown as orange discs) at their ends, are takento have a Rashba spin-orbit coupling generated from interac-tion with the superconducting substrate. The Bogoliubov deGennes (BdG) Hamiltonian for the wire alongx is given by

HBdG = (−η∂2x−µ(x))τz+Vzσ·B+ıα∂xσyτz+∆τx (A1)

and for the wire alongy is given by

HBdG = (−η∂2y − µ(y))τz + Vzσ · B − ıα∂yσxτz +∆τx.

(A2)The direction of the Zeeman field isB = (x+y)/

√2. Follow-

ing the spin-rotation and phase transformations in Ref. [20],for negative Rashba couplingα < 0, the Majorana wave-functions at the left and the right ends of thex wires have

7

the form

φL =

(

u(x)eıφ/2

ıσyu(x)e−ıφ/2

)

andφR = ıσx

(

u(−x)e−ıφ/2

ıσyu(−x)eıφ/2

)

(A3)

respectively,whereφ = sin−1 VZ

∆√2

and u(x) is a real 2-spinor. Note that in this geometry there is now an additionalcondition for the wire to be gapped i.e

√

∆2 + µ2 < VZ <

∆√2. This constraint impliesπ4 < φ < π

2 . The Majoranawave-functions for the Majorana fermions at the bottom andtop ends of the wires parallel to they axis have the form

φB = Q

(

u(y)e−ıφ/2

ıσyu(y)eıφ/2

)

andφT = ıQσx

(

u(y)eıφ/2

ıσyu(y)e−ıφ/2

)

(A4)

respectively, whereQ = e−ıπσz/4.Transport of MFs is generated by introducing tunneling into

the system of MFs shown in Fig. 5. The junction chiralityχ defined in the paper depends only on the tunneling from3 end MFs MF{L,R, T } with wave-functionsφL,R,T . Letus start by considering the MF overlap across the NTS seg-ments (shown as red dashed lines in Fig. 5) which is simplestto understand in the limit of low negative chemical potentialµ = −|µ| where|µ| ≫ VZ ,∆. Physically, this correspondsto a wire that is nearly depleted of electrons and only acts asa tunnel barrier. In such a case, the Majorana wave-functionin the NTS wire has the usual exponentially decaying formΨ(x) = Ψ(xI)e

−γ|x−xI| as in a barrier, whereγ ∼√

2m|µ|andxI is the position of the interface between the TS andNTS wire segments. The tunneling matrix elementsζij be-tween two MFs atxI,1 = −a/2 andxI,2 = a/2 across theNTS wire can be calculated from the matrix-elements of thecurrent operator and the wave-functions in the middle of thewire33 (x = 0 as shown by the dark dotted lines in Fig. 5)

ζ =1

2[Ψ†

1(0)τz∂xΨ2(x)|x=0 − ∂xΨ†1(x)|x=0τzΨ2(0)]

− iαΨ†1(0)σyτzΨ2(0) ∼ −γe−γaΨ†

2(−a

2)τzΨ1(

a

2)

= −ρΨ†2(−

a

2)τzΨ1(

a

2) (A5)

whereρ = γe−γa is the overall tunneling strength and wehave assumedλ ≫ α. In this limit, the overlap between a pairof Majorana wave-functionsΨ1 = (u1(x), ıσyu

∗1(x))

T andΨ2 = (u2(x), ıσyu

∗2(x))

T is given byM = 2ıρIm(u†1u2).

This is purely imaginary and manifestly anti-symmetric asexpected. Furthermore since the fundamental spinoru(x) interms of which each ofu1,2 are written is real, we can write

it asu = (cos θ, sin θ)T , where the parameterθ depends onVZ , µ, α etc. With the help of these relations it is easy to tab-ulate the Majorana tunneling matrix as an anti-symmetric ma-trix for the states in the order(L,R, T ) as

ζ = ıρ√2

0√2 cosφ sin 2θ sin 2θ

∗ 0 (sinφ+ cosφ cos 2θ)∗ ∗ 0

(A6)where the elements in the∗ have been left empty since theyare determined by the anti-symmetry constraint. The junc-tion chirality χ in the previous section, used only the Majo-rana modesL, T,R and is calculated using the expression,χ = ζRLζLT ζTR = ρ3 cos2 φ sin2 2θ[cos 2θ + tanφ], whichis always positive, sincetanφ > 1 for the Zeeman direc-tion B = (x + y)/

√2 and negative Rashba couplingα < 0.

Changing the Zeeman potential toB = (x − y)/√2 flips

the chirality. Changing the sign of the Rashba couplingαrequires us to changeφL → ı(u(x)eıφ/2,−ıσyu(x)e

−ıφ/2).Since all the other wave-functions are derived from symmetrytransformations applied toφL, the rest of the calculation goesthrough as is with the only difference thatu(x) changes toıu(x). Therefore the final result for the chirality of the junc-tion is independent of the Rashba coupling.

The signs acquired by MFs on exchange is dependentalso on the tunneling between the MFs MF L,R across aTS segment. The tunneling amplitude between Majoranafermions on the same topological segment is well-controlledand can be calculated in the limit of a long topologicalwire (wire lengthL > α/VZ). In this limit, the MFsonly overlap in the limit where the gate potential is tunedso that the wire is driven towards a phase transition bytuning µ near toµ =

√

V 2Z −∆2. The relevant slow-

est decaying spinor component then determines the tun-neling matrix elements and is given byu(x) = (VZ +

sgn(α)√

V 2Z − µ2,−µ)T exp(− x

|α|(√

V 2Z − µ2 − ∆)). The

tunneling matrix element is given byM ∼ −iαΨ†LσyτzΨR =

−iαRe[u†LσyuR]. Substituting u, we find the over-

lap to simplify to M ∝ −iα cosφe−2L/ξ where ξ =

|α|/(√

V 2Z −∆2 − µ). Thus the sign ofζL,R is determined

by the sign of the Rashba spin-orbit couplingα. Using theseresults together with Eq. A6, one can check the physically rea-sonable result that changing the sign of the Rashba couplingα, flips the sign of the MFs on interchange.

Thus, provided care is taken to ensure the conditions forthe braid discussed in this section, the sign of both clock-wiseexchanges is positive for positive Rashba couplingα and neg-ative other-wise. For a given Rasbha coupling the sign of thebraid can be altered by considering anti-clockwise braids.

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