Tunable topological states hosted by unconventional superconductors with adatoms
TTunable topological states hosted by unconventional superconductors with adatoms
Andreas Kreisel, Timo Hyart,
2, 3 and Bernd Rosenow Institut f¨ur Theoretische Physik, Universit¨at Leipzig, Br¨uderstrasse 16, 04103 Leipzig, Germany International Research Centre MagTop, Institute of Physics,Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland Department of Applied Physics, Aalto University, 00076 Aalto, Espoo, Finland (Dated: February 23,2021)Chains of magnetic atoms, placed on the surface of s-wave superconductors, have been estab-lished as a laboratory for the study of Majorana bound states. In such systems, the breaking oftime reversal due to magnetic moments gives rise to the formation of in-gap states, which hybridizeto form one-dimensional topological superconductors. However, in unconventional superconductorseven non-magnetic impurities induce in-gap states since scattering of Cooper pairs changes theirmomentum but not their phase. Here, we propose a realistic path for creating topological super-conductivity, which is based on an unconventional superconductor with a chain of non-magneticadatoms on its surface. The topological phase can be reached by tuning the magnitude and direc-tion of a Zeeman field, such that Majorana zero modes at its boundary can be generated, moved andfused. To demonstrate the feasibility of this platform, we develop a general mapping of films withadatom chains to one-dimensional lattice Hamiltonians. This allows us to study unconventionalsuperconductors such as Sr RuO exhibiting multiple bands and an anisotropic order parameter. INTRODUCTION
Combining topology and superconductivity has beenheralded as a new paradigm for the realization of ex-otic new particles – Majorana zero modes (MZMs) –whose non-Abelian braiding statistics would enable fault-tolerant quantum computations[1, 2]. Moreover, the ex-istence of MZMs is topologically protected, making theminert to disorder effects. To-date, two main approaches,based on the Kitaev chain[3], have been pursued in thequest for topological superconductors. In the first ap-proach s -wave superconductivity is proximity-inducedin nanowires with strong spin-orbit coupling[4–7], whilein the second approach the hybridization of impurity(Shiba) bound states gives rise to a topologically non-trivial superconducting phase[8–12]. Experimentally, thelatter has been realized by placing a chain of magnetic FIG. 1.
Setup –
Non-magnetic adatoms (purple balls) areplaced at a distance a to form a chain on the surface of ahelical triplet superconductor with order parameter ∆. TheCooper pairs exhibit equal spin (red arrows), and their orbitalangular momentum (black arrow) points opposite to the spindirection. An external Zeeman field h can be used to tunethe system into the topological phase supporting Majoranazero modes at the endpoints of the chain as sketched by theyellow plot of the magnitude | ψ | of the wavefunction. atoms on the surface of an s -wave superconductor[13–17]. In these platforms, evidences for MZMs at the endpoints of the system were found in transport measure-ments on nanowires[6, 7] and in scanning tunneling spec-troscopy on Shiba chains[14–18]. Proposals for moving,fusing and braiding the MZMs have been brought for-ward theoretically[19–22], but their implementations re-main open challenges experimentally.In this work, we propose a realistic path to realizeone-dimensional topological superconductivity by plac-ing non-magnetic atoms on the surface of an unconven-tional triplet superconductor, see Fig. 1. The key ad-vantage of our proposal is that it is possible to moveand fuse MZMs by controlling a magnetic field. Sincecandidate systems for the realization of triplet supercon-ductivity usually exhibit multiple bands, we go beyonda single-band description in order to model for exampleSr RuO , which has been subject to intense theoreticaland experimental investigations regarding the nature ofthe superconducting pairing[23–26]. Implementing ourproposal in Sr RuO or in other candidate triplet su-perconductors such as UPt [27, 28], UTe [29, 30] andLaNiGa [31] could establish a new MZM platform andimprove the understanding of the pairing symmetries inthese systems. RESULTSModel of bulk superconductor
Our starting point is a single-band Hamiltonian H = H BdG + H Z (1)where H BdG describes a bulk triplet superconductor ona two dimensional lattice, and H Z is the Zeeman term in a r X i v : . [ c ond - m a t . s up r- c on ] F e b FIG. 2.
Topological phase diagram – (a) Helical single-band superconductor: the topological phase ( Q = − topo ) can be reached for a wide range of impurity strengths V imp by tuning an external Zeeman field h x .The phase boundary is marked with a white dashed line. An energy gap ∆ topo ≈ ∆ min / ζ/a ≈ ∆ min / ∆ topo . Parameters are: ∆ /E F ≈ . a ≈ . ξ .(b) Multiband model for Sr RuO with a realistic superconducting order parameter (see Supplemental material) shows similartunability as the single-band model, with a maximum gap ∆ topo ≈ ∆ min /
3. (c) For the topological invariant Q = +1 thesystem is in the trivial phase (and we denote ∆ topo = 0), while for Q = − topo . an external field. In momentum space it has the matrixstructure H ( p ) = (cid:18) h ( p ) ∆( p )∆( p ) † − h ( − p ) T (cid:19) , (2)where h ( p ) = ξ ( p ) · σ + h · σσσ , ξ ( p ) is the energy-momentum dispersion, h the Zeeman field, and σ and σσσ are the unit matrix and Pauli matrices acting in spinspace. In the off-diagonal, ∆( p ) = i [ d ( p ) · σ ] σ y is thepairing term, whose minimum value we denote by ∆ min .Here, d ( p ) describes the vector order parameter of thetriplet superconductor. Here, we concentrate on the caseof a helical p -wave order parameter d h = − i ∆ t ( e x sin p y + e y sin p x ) , (3)and in the Supplemental information we discuss the gen-eralizations to a chiral p -wave order parameter and multi-band models using Sr RuO as an example. Chain of nonmagnetic impurities
The chain of atoms placed along the x -direction r n = na e x (with integer n ) is described by H imp = ˆ U (cid:88) n δ r , r n , (4)with the matrix ˆ U = V imp τ z σ mediating non-magneticimpurity scattering of strength V imp . Although the scat-tering of Bogoliubov quasiparticles preserves spin, thereare still Shiba in-gap states in this system because thescattering does not change the phase in order to matchthe p -wave momentum dependence of the order parame-ter. In the case of a chain, Shiba states localized in the vicinity of the impurity atoms hybridize and give rise toimpurity bands within the bulk gap ∆ min of the super-conductor. These bands can be accurately described byan effective Hamiltonian H eff ( k x ) = ˆ U − ˜ G − [ G I ( k x ) ˆ U − , (5)which depends on the momentum k x in a supercell Bril-louin zone with lattice constant a . We derive the matri-ces on the r.h.s. by linearizing the bulk Green functionwith respect to energy, and obtain G I ( k x ), which de-scribes the propagation of Bogoliubov quasiparticles be-tween the impurities, by Fourier transforming the bulkGreen function at the impurity sites. The matrix ˜ G − enters as a prefactor and contains the renormalization ofthe bandwidth (see Methods). Effective Hamiltonian
The mapping onto the effective Hamiltonian Eq. (5)can be understood as integrating out the quantum num-bers p y of momenta perpendicular to the chain. There-fore, in the absence of a Zeeman field the effective Hamil-tonian has the structure H ( k x ) = ξ eff ( k x ) τ z σ + ∆ eff ( k x ) τ x σ , (6)diagonal in spin space, and describing twofold degener-ate impurity bands inside the bulk superconducting gap.Here, ξ eff ( k x ) and ∆ eff ( k x ) are the effective dispersionand pairing for the impurity chain, containing further-neighbor coupling terms between the impurity states.The Hamiltonian H ( k x ) is time-reversal symmetric,so that the system can support a topological phase withan even number of MZMs at each end of the chain[32].We can realize unpaired MZMs by additionally breakingthe symmetry down to only particle-hole symmetry byuse of a Zeeman field such that the system, when tunedinto the topological phase, exhibits unpaired MZMs atthe end of the impurity chain A Zeeman field in z -direction can be described by H Zeff ,z = h eff ,z σ z τ z witha renormalized magnitude h eff ,z , which can be calculatedby including chemical potential shifts ξ ( p ) ± h z in thebulk dispersion of the spin up and down electrons, re-spectively. We find that h eff ,z (cid:28) h z because the energyof Shiba in-gap state depends only weakly on the chemi-cal potential (see Supplemental information). Thus, theeffective Hamiltonian is diagonal in spin space with eachblock ˜ H eff ± = ( ξ eff ( k x ) ± h eff ,z ) τ z + ∆ eff ( k x ) τ x exhibit-ing the same topological properties as the Kitaev chain.The effective field ± h eff ,z plays the role of the chemicalpotential, which can drive a topological phase transition.However, since the effective field h eff ,z is parametricallysmall, the topological phase can only be reached by finetuning the impurity strength V imp such that the impuritybands almost touch zero already without a Zeeman field.For the Zeeman field pointing in y -direction, the ad-ditional term to the effective Hamiltonian is H Zeff ,y = h eff ,y σ y τ . In this case, the two BdG bands are shiftedtrivially in energy with respect to each other, leavingtheir topological character unchanged, i.e. a field in y -direction cannot tune into the topological phase.Finally, for a field in x -direction, the Zeeman termreads H Zeff ,x = h eff ,x τ z σ x with weak renormalization ofthe effective Zeeman field from the bulk value h eff ,x ∼ h x .The effective Hamiltonian can now be rotated around the y axis in spin space σ x → σ z , such that in the new basisthe effective field h eff ,x plays again the role of a chemicalpotential. The difference to the case of a Zeeman fieldin z -direction is that the weakly renormalized h eff ,x candrive a topological phase transition much more efficientlythan the strongly reduced h eff ,z discussed above. Topological phase diagram
To quantitatively demonstrate the tunability of topo-logical superconductivity, we compute the topologicalphase diagram (see Figs. 2 and 3) as characterized bythe topological invariant Q = (cid:89) k x ∈ TRIM Q ( k x ) , Q ( k x ) = sign(Pf( H ( k x ) τ x ) . (7)It is given as the product of Pfaffians Q ( k x ) at the timereversal invariant momenta of the corresponding one-dimensional Brillouin zone. We supplement the fully nu-merical supercell calculation by computing the topolog-ical invariant also using Q eff ( k x ) = sign(Pf( H eff ( k x ) τ x ).The excellent agreement between Q eff and Q (see Supple-mental Material) indicates that H eff ( k x ) indeed faithfullydescribes the low-energy physics of the impurity chain. Inthe nontrivial case Q = − topo as the minimum of the eigenenergies of H ( k x ) in the Brillouin zone. To detect the non-Abelian propertiesof the MZMs, it is necessary that the coupling betweenMZMs is weak. Thus, the distance between neighboringMZMs needs to be much larger than ζ ≈ ¯ hv F, eff / ∆ topo ,where v F, eff is the Fermi velocity for the impurity band.An estimate of v F, eff yields ζ/a ≈ ∆ min / ∆ topo . In orderto avoid thermal excitations the temperature needs to besmaller than the topological gap, k B T < ∆ topo .In Fig. 2 we present the topological phase diagram forthe single-band model and for a multiband descriptionof Sr RuO , revealing that the topological phase can bereached in both cases by application of a Zeeman field h x in the direction along the impurity chain, rather in-dependently of the value of the impurity potential V imp .We summarize that our proposal might be feasible in ahelical p -wave superconductor where impurity adatomscan be controlled experimentally. The multiband helical p -wave order parameter considered in Fig. 2 is one of thepossible candidate pairing symmetries for Sr RuO [26].Controllable placing of adatoms might be facilitated bystep edges, and in the Supplemental material we showthat the topological phase can be reached by applicationof a Zeeman field also in this case. In Fig. 3 we illustratethat the direction of the Zeeman field can be used to tunea system into and out of the topological phase. Namely,tuning the azimuthal angle φ has a strong effect on thetopological gap ∆ topo . A similar analysis of the tunabil-ity of a system with chiral p -wave order parameter (seeSupplemental Material) reveals that rotating the Zeemanfield within the x - y -plane has no effect at all. Hence, thisdifference in behavior could be used to experimentallydiagnose and discriminate the helical p -wave from a chi-ral p -wave order parameter, an important question forexample in the Sr RuO system[23–26]. DISCUSSION
For the case of magnetic adatoms, the dependence ofadatom magnetic order on an external Zeeman field hasbeen suggested as a means to to create, braid, and fuseMZMs[21]. Here, we exploit the direct dependence ofthe topological phase diagram on the Zeeman field (seeFig. 3). So far, we have arbitrarily chosen that the impu-rity chain is oriented along the x -axis, and as a result aZeeman field in x -direction was most suitable to induce atopological phase. More generally however, the relevantparameter is the relative angle of the Zeeman field withthe impurity chain, and for a curved chain the relevantangle would be the angle between the field and the localtangential direction as defined in Fig. 3(a). Thus, MZMson curved impurity chains are located at all interfacesbetween trivial and nontrivial regions, i.e. at positionswhere the local tangent and the external field draw a crit-ical angle, see Fig. 4. Hence, MZMs can be moved alongthe impurity chain by either rotating the direction of theZeeman field or by changing the magnitude of the field,which modifies the critical angle. -1 -0.5 0 0.5 1 / / -1 -0.5 0 0.5 1 / / FIG. 3.
Field direction dependence of the topological gap – (a) The direction of the Zeeman field relative to thedirection of the impurity chain is parametrized by the azimuthal angle φ and the polar angle θ . Changing the direction of thefield allows to enter or leave the topological phase. (b) Topological phase diagram for a single-band system with parameters V imp /E F = − . | h | /E F = 0 . a ≈ . ξ ; the white dashed line is the boundary between topological and trivial phase. (c)Similar phase diagram for Sr RuO ( V imp /E F = − . | h | /E F = 0 . a ≈ . ξ ). In both cases, the topological gap ∆ topo ismaximal for the field along the impurity chain ( φ = 0, θ = π/ φ = π/
2, or towards θ = 0 , π , i.e. to a transverse directions relative to the chain. The isolated point at which thesystem remains gapless (white cross) corresponds to a in plane field perpendicular to the chain. FIG. 4.
Moving and fusion of Majorana zero modes –
A curved impurity chain might be tuned partially into thetopological state (red arcs) if the the angle between the Zeeman field and the local direction of the chain puts it into thetopological phase. MZMs (yellow dots) occur at the boundaries of trivial and nontrivial regions, which can be moved in twodifferent ways: (a) Rotating the field around the axis perpendicular to the plane moves the Majoranas in the same direction,while (b) changing the field magnitude or polar angle can move them in opposite directions. (c) In a wiggly impurity chainMZMs 1 and 2, 3 and 4 can be created pairwise from the vacuum by changing the field magnitude, and by tuning deeper intothe topological phase one eventually fuses the MZMs 2 and 3.
Increasing the Zeeman field along a wiggly impuritychain creates two pairs of Majoranas which can formallybe described by operators γ i , i = 1 , , ,
4, satisfying γ i = γ † i and anticommutation relations { γ i , γ j } = 2 δ ij (for the labeling of MZMs see Fig. 4(c)). Grouping theMZMs in pairs of two, we can define the left and rightnumber operators n l = (1+ iγ γ ) and n r = (1+ iγ γ )with eigenvalues 0 and 1, and define a Hilbert spacespanned by basis states | n l n r (cid:105) . Creating the MZMs fromthe vacuum, the initial state is given by Ψ = | (cid:105) inthis basis. Tuning deeper into the topological phase, theinner MZMs 2 and 3 will fuse such that the final statewill be a statistical mixture of 0 and 1 for the opera-tor n o = (1 + iγ γ ) (see Supplemental material). Inthe fusion process the projective measurement can beperformed by detecting the charge acquired by MZMs 2and 3 after they have hybridized [22]. The movementand projective measurements are the key ingredients formanipulation of MZMs, and they can be realized by con-trolling external magnetic field as discussed above. How- ever, we also need to preserve the quantum informationstored in the MZMs. In the Supplementary material wediscuss the corresponding requirements and propose tomanipulate the local magnetization by spintronic meansto obtain signatures of non-Abelian statistics of MZMs.In summary, we have shown that topological super-conductivity can be realized by placing nonmagneticadatoms on the surface of an unconventional supercon-ductor, and that the topological invariant can be con-trolled with the magnitude and direction of a Zeemanfield. Our considerations are based on a lattice Hamilto-nian which can describe materials exhibiting a complexstructure of the order parameter and multiple bands. Wehave identified the field direction which can most effi-ciently tune the system into the topological phase andwe have proposed a scheme to move and fuse MZMs. Anexperimental realization of this proposal could becomea scalable platform for topological quantum informationprocessing based on the non-Abelian statistics of MZMs. METHODSTight binding model
For the single-band model, we use the normal statedispersion ξ ( p ) = − t (cos p x + cos p y ) − µ on a squarelattice, where t is the nearest-neighbor hopping and µ ≈− . t is the chemical potential fixed such that the fillingis one quarter.The superconducting order parameter can be writtenin real space as∆ = (cid:88) ij (cid:88) αβ (cid:88) σσ (cid:48) ∆ ij,αβ,σσ (cid:48) c † i,α,σ c † j,β,σ (cid:48) + h.c.. (8)For the single-band system with a triplet order parame-ter, the coefficients read ∆ ij,αβ,σσ (cid:48) = ∆ d i,j · σσσiσ y withthe Pauli operators σσσ = [ σ x , σ y , σ z ] and a vector d i,j . Inthe main text, we consider the physical consequences ofthe helical p -wave order parameter d = − i ∆ t (sin p y e x +sin p x e y ) which in real-space leads to nearest-neighborpairings ∆ = δ i (cid:18) − (cid:19) with δ i = ∓ ∆ t / , ± = δ i (cid:18) i i (cid:19) with δ i = ∓ ∆ t / ± , RuO [33] is discussed inthe Supplemental material. Green function approach and effective Hamiltonian
Derivations of effective Hamiltonians for continuummodels have been worked out in detail for example forhelical Shiba chains in Ref. 9. Here, we generalize thisapproach to multiband lattice models and derive the ef-fective Hamiltonian for the impurity bands as cited inEq. (5) of the main text. Starting point is the eigenvalueequation of the Bogoliubov de Gennes Hamiltonian (in-cluding the impurity chain) ( H BdG + H Z + H imp )Ψ = E Ψwith the eigenstate Ψ and eigenenergy E . Next, we in-troduce the Green function operator of the bulk system G ( E ) = ( E − H BdG − H Z ) − (9)to obtain a nonlinear eigenproblemΨ = G ( E ) H imp Ψ . (10)Evaluating Eq. (10) only at the impurity sites r m = a m e x , and using that the impurity Hamiltonian [Eq.(4)] is diagonal, we obtainΨ( r m ) = (cid:88) r n G ( E, r m − r n ) ˆ U Ψ( r n ) . Because of the periodicity with respect to translations bythe impurity chain lattice vector a e x it is useful to trans-form this equation to momentum space (with respect to the supercell) Ψ( k x ) = (cid:80) m Ψ( r m ) e − ik x a m such that theeigenvalue equation can be rewritten asΨ( k x ) = (cid:88) n G ( E, r n ) e − ik x a n ˆ U Ψ( k x ) . (11)The real space Green function can be obtained via itsFourier representation G ( E, r ) = 1Ω BZ (cid:90) BZ d p G ( E, p ) e i p · r ,G ( E, p ) = [ E − H BdG ( p ) − H Z ] − , (12)where the integral is over the bulk Brillouin zone withmomentum space area Ω BZ . By linearizing the bulkGreen function at E = 0, we obtain G ( E, p ) = G (0 , p ) − E ˜ G ( p ) , (13)where G (0 , p ) = − [ H BdG ( p ) + H Z ] − and ˜ G ( p ) =[ H BdG ( p ) + H Z ] − exist for fully gapped systems. Fur-thermore, we keep the linear correction ∝ E only in theonsite term of the Green function to obtain G ( E, r n ) = G (0 , r n ) − Eδ r n , ˜ G, (14)where ˜ G = 1Ω BZ (cid:90) BZ d p [ H BdG ( p ) + H Z ] − . (15)We now insert Eqs. (12), (13) and (14) into Eq. (11)and introduce the Fourier transforms with respect to thesupercell as G I ( k x ) = (cid:88) n G (0 , r n ) e − ia nk x (16)to obtain Ψ( k x ) = [ G I ( k x ) ˆ U − E ˜ G ˆ U ]Ψ( k x ) . (17)Rearranging the terms and multiplication with inversematrices brings the eigenvalue equation in the form H eff ( k x )Ψ( k x ) = E Ψ( k x ) , (18)where the effective Hamiltonian is H eff ( k x ) = ˆ U − ˜ G − (cid:2) G I ( k x ) ˆ U − (cid:3) . (19)This Hamiltonian becomes exact at E = 0 where thetopological phase transition occurs and therefore can beused to determine the phase diagram exactly.This approach is general and can be applied to all lat-tice Hamiltonians. In this work we have applied it tothe single-band p -wave superconductors and multibandmodel for Sr RuO [33], but similar theoretical investiga-tions can be performed also for other candidate materialsfor multiband triplet superconductors [27–31]. Topological invariant
A finite magnetic field breaks time-reversal symmetrysuch that the remaining symmetry of the Hamiltonian,Eq. (1) is particle-hole symmetry, described by a particle-hole operator P anticommuting with the Hamiltonian, { H, P } = 0. The superconducting pairing has the prop-erty ∆ T = − ∆ and the normal state block is Hermitean, h † = h . Therefore, P = τ x K is the desired anticommut-ing operator, where τ x is the Pauli matrix in particle-holespace and K the complex conjugation. The parity oper-ator ˆ P = ( − ˆ N , with ˆ N being the particle number op-erator, commutes with the Hamiltonian [ H, ˆ P ] = 0, thusthere is a common system of eigenstates. Since the parityoperator has the eigenvalues ±
1, the ground state of thesystem is either of odd or even parity. The parity can becalculated by Eq. (7), where we formally have factorizedout a prefactor of ( − n because Pf( Hiτ x ) = Pf( Hτ x )for matrices of size 4 n with n an integer number. Inthe fully numerical approach, we set up the Hamilto-nian for the superconductor subject to the Zeeman fieldand including the impurity potential, and use a supercellmethod to obtain H ( k x ) of an (infinite) impurity chainalong the x direction. For the calculation of the invari-ant using Eq. (7) the supercell Hamiltonian needs to beconstructed for the two time-reversal invariant momenta, k x = 0 , π/a , corresponding to periodic or antiperiodicboundary conditions. Finally, the Pfaffian is calculatedusing an efficient numerical algorithm[34].The effective Hamiltonian, Eq. (5) inherits the symme-tries from the bulk Hamiltonian in Eq. (2), i.e. it satisfiesthe particle-hole symmetry τ x H ∗ eff ( − k x ) τ x = − H eff ( k x )which can be read of from Eq. (5) by using that also theother matrices in the expression obey the same symme-try, e.g. τ x ˆ U ∗ τ x = − ˆ U , τ x ˜ G ∗ τ x = ˜ G , τ x G ∗ I ( − k x ) τ x = − G I ( k x ), which can be derived from the original prop-erty τ x [ H BdG ( p ) + H Z ] ∗ τ x = − [ H BdG ( p ) + H Z ] of thebulk Hamiltonian. Therefore, the effective Hamiltoniancan be used to calculate the topological invariant usingEq. (7), while the numerical effort is greatly reduced be-cause of the small size of the corresponding matrices. Topological gap
For the calculation of the topological gap, i.e. theminimal positive eigenvalue of the supercell Hamiltonianas a function of k x , we calculate the eigenvalues fora grid of a few k x points between 0 and π/a , selectthe k x with the smallest positive eigenvalue, and thenuse an iterative procedure to find the smallest positiveeigenvalue by a bisection bracketing algorithm to obtain∆ topo . This procedure is is implemented for boththe supercell and the effective Hamiltonian approachto investigate the reliability of the approximation inderiving Eq. (5), see Supplemental Material. Findingvery good agreement, we show in the main text onlyresults stemming from the effective Hamiltonian sincethe calculation of eigenvalues is orders of magnitudefaster once the expansion coefficients, Eqs. (15) and(16), for H eff ( k x ) have been calculated. DATA AVAILABILITY
The data that has been used to generate the plotswithin this paper and other findings of this study areavailable from the corresponding author upon reasonablerequest.
CODE AVAILABILITY
The code to numerically calculate the spectra andtopological invariants discussed in this paper is availablefrom the corresponding author upon reasonable request.
ACKNOWLEDGEMENTS
The research was partially supported by the Founda-tion for Polish Science through the IRA Programme co-financed by EU within SG OP.
ADDITIONAL INFORMATION
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Supplementary Note S1. Tight-binding modelsSupplementary Note S2. Effective Hamiltonian for the single-band model without Zeeman fieldSupplementary Note S3. Effective Hamiltonian in the presence of the Zeeman fieldSupplementary Note S4. Topological phase transition without Zeeman fieldSupplementary Note S5. Numerical results for the energy of the impurity bound stateSupplementary Note S6. Numerical results for impurity bands in the presence of Zeeman fieldSupplementary Note S7. Phase diagramsSupplementary Note S8. Curved chains: Manipulation of phase boundariesSupplementary Note S9. Fusion of Majorana zero modesSupplementary Note S10. Majorana spintronics: Fusion and Exchange of Majorana zero modesFigs. S1 to S18
Supplementary Note S1. TIGHT-BINDING MODELS
1. Single-band model
The normal state dispersion of the single band model on a square lattice is given by ξ ( p ) = − t (cos p x + cos p y ) − µ, (S1)where t is the nearest neighbor hopping which we use as energy unit, i.e. t = 1, and µ ≈ − . t is the chemicalpotential fixed such that the filling is one quarter yielding a Fermi surface as shown in Fig. S1(a). In the following weuse t = 1. The superconducting order parameter can be written as ∆( p ) = i [ d ( p ) · ~σ ] σ y , where d ( p ) can be classifiedaccording to the basis functions of the irreducible representations of the symmetry group. (If the instabilities related tothe different representations of the symmetry group have accidentally similar critical temperatures, a second transition − − x / π k y / π / | v F | (t) D O S ( /t ) -5 0 5 (t) D O S ( /t ) FIG. S1. Single band model. (a) Fermi surface of the single band model at filling n = 0 .
25 together with the (inverse) Fermivelocities. The color scale is included to emphasize that the Fermi velocity stays approximately constant along the Fermi line.(b) The density of states in the superconducting state. (c) A blowup of the low energy density of states exhibiting a full gapwith an anisotropy as expected from the lattice model. -0.5 0 0.5k x /-0.500.5 k y / d xy =d xz d xy d yz =d xy d yz d xz =d yz d xz -4 -2 0 (t) D O S ( /t ) d xz d yz d xy total -0.04 -0.02 0 0.02 0.04 (t) D O S ( /t ) FIG. S2. Three-band model for Sr RuO . (a) Fermi surface including orbital weights. (b) Density of states in the normal state.(c) Density of states in the superconducting state showing a V-shaped feature from the strongly anisotropic order parameter. to a mixed state can occur upon lowering of the temperature, but this possibility is neglected here.) For the squarelattice, the possible triplet order parameters are d , ( p ) = ∆( e x sin p y ± e y sin p x ) , d , ( p ) = ∆( e x sin p x ± e y sin p y ) , (S2) d A ( p ) = ∆ A e z sin p x , d B ( p ) = ∆ B e z sin p y . (S3)The basis functions of the one-dimensional representations d m ( p ) ( m = 1 , , ,
4) are distinguished by the differentsigns in the mirror transformations d m,x ( − p x , p y ) = ± d m,x ( p x , p y ) and d m,x ( p y , p x ) = ± d m,y ( p x , p y ). In the case ofthe two-dimensional irreducible representation the basis functions d A ( p ) and d B ( p ) are parallel to e z .All basis functions d m ( p ) ( m = 1 , , ,
4) give rise to helical p -wave superconductors and from the view point ofour analysis they are essentially equivalent. Therefore, we consider d h ( p ) = − i ∆ t ( e x sin p y + e y sin p x ) (S4)as a representative example of helical p -wave superconductors. If the highest critical temperature corresponds to thetwo-dimensional irreducible representation the order parameter that is realized (by minimization of the free energy)is a complex linear combination of d A ( p ) and d B ( p ), which describes a chiral p -wave superconductor d c ( p ) = ∆ t (sin p x + i sin p y ) e z . (S5)We choose ∆ t = 0 . n = 0 .
25, the magnitude of the order parameter on the Fermi surface | ∆( p ) | is almost constant, so isthe (inverse) Fermi velocity, Fig. S1(a). The density of states is fully suppressed within the energy interval | ω | . . t and yields coherence peaks at the gap maxima, see Fig. S1(b,c).
2. Multiband model for Sr RuO The model for Sr RuO is based on a tight binding parametrization proposed earlier with hoppings on asquare lattice giving rise to a Hamiltonian H ( p ) = H ( p ) + H SO with the spin independent part H ( p ) = P ab,s t ab ( p ) c † a,s ( p ) c b,s ( p ) where the sum runs over the Ru- d orbitals ( a, b ) = { d xz , d yz , d xy } and the spin s = ± t ( p ) = − t cos p x − t ⊥ cos p y − µ , t ( p ) = − t ⊥ cos p x − t cos p y − µ , t ( p ) = − t [cos p x +cos p y ] − t cos p x cos p y − µ , t ( p ) = t ( p ) = − t sin p x sin p y with( t, t ⊥ , t , t , t ) = (1 . , . , . , . , .
01) and ( µ, µ ) = (1 . , .
1) and the on-site spin-orbit coupling H SO = 2 η P i L i · S i where the sum runs over all sites of the lattice and η = 0 .
1. In momentum space, this yields constant coupling oflongitudinal type and spin flip type H SO = P ab,ss ˜ t ss ab c † a,s ( p ) c b,s ( p ) with the nonzero elements ˜ t ss = isη , ˜ t ss = − isη ,˜ t s − s = − iη , ˜ t s − s = isη , ˜ t s − s = − sη , ˜ t s − s = − sη . The Fermi surface of this model together with the density of statesis shown in Fig. S2.For the calculation in real space with system size of ( N x , N y ) lattice points in x and y direction, the hoppingelements and contributions from the spin-orbit coupling are set up in sparse matrices to be used to calculate theeigenvalues, or the topological invariant, see below.The order parameter for Sr RuO is considered to be of helical- p wave type with the same parametrization ofthe higher harmonics of the pairing components as in Ref. 1 which we state here again for convenience. The orderparameters in orbital space are given by the following ansatz d ah ( p ) = − i ∆ t X j =1 , , (cid:0) ∆ ax,j g x,j ( p ) e y + ∆ ay,j g y,j ( p ) e x (cid:1) (S6) g x, ( p ) = sin( p x ) (S7) g x, ( p ) = sin( p x ) cos( p y ) (S8) g x, ( p ) = sin(3 p x ) (S9)where a = xz, yz, xy is the orbital index, g y,j ( p x , p y ) = g x,j ( p y , p x ) and ∆ zyy,j = ∆ zxx,j ; ∆ zxy,j = ∆ zyx,j = 0; ∆ xyx,j = ∆ xyy,j ∀ j with the following parameters: (∆ zxx, , ∆ zxx, , ∆ zxx, ) = (0 , . , .
0) and (∆ xyx, , ∆ xyx, , ∆ xyx, ) = (0 . , . , − . t = 0 . Supplementary Note S2. EFFECTIVE HAMILTONIAN FOR THE SINGLE-BAND MODEL WITHOUTZEEMAN FIELD
In the next sections, we study the effective Hamiltonian for the impurity chains along x direction, with the impuritypositions are parametrized as r n = ( na,
0) ( n ∈ Z ), in chiral and helical p -wave superconductors H eff ( k x ) = ˆ U − ˜ G − (cid:2) G I ( k x ) ˆ U − (cid:3) , (S10)where ˆ U = V imp τ z σ , ˜ G = 1Ω BZ Z BZ d p [ H BdG ( p ) + H Z ] − , (S11)and G I ( k x ) = X n G (0 , r n ) e − iank x , G ( E, r ) = 1Ω BZ Z BZ d p G ( E, p ) e i p · r , G ( E, p ) = [ E − H BdG ( p ) − H Z ] − . (S12)We start by studying the Hamiltonian in absence of a Zeeman field h = . In this case, we obtain for both chiraland helical p -wave superconductors[ H BdG ( p )] = [ ξ ( p ) + | ∆( p ) | ] τ σ , | ∆( p ) | = | ∆ t | (sin p x + sin p y ) (S13)so that ˜ G = A τ σ , A = 1(2 π ) Z BZ d p ξ ( p ) + | ∆( p ) | , (S14)and H eff ( k x ) = − A V imp τ z σ + 1 A τ z σ G I ( k x ) τ z σ . (S15)
1. Single impurity
In the case of single impurity the effective Hamiltonian is H eff = − A V imp τ z σ + 1 A τ z σ G ( E = 0 , r = 0) τ z σ . (S16)Now G ( E = 0 , r = 0) = − π ) Z BZ d p ξ ( p ) + | ∆( p ) | H BdG ( p ) . (S17)By noticing that ∆( − p ) = − ∆( p ), we see that the momentum integral of ∆( p ) / ( ξ ( p ) + | ∆( p ) | ) vanishes, andtherefore G ( E = 0 , r = 0) = −B τ z σ , B = 1(2 π ) Z BZ d p ξ ( p ) ξ ( p ) + | ∆( p ) | . (S18)Since only the square of the order parameter enters, the single impurity Hamiltonian for both chiral and helical p -wavesuperconductor is H eff = − (cid:15) τ z σ , (cid:15) = 1 + B V imp A V imp . (S19)
2. Impurity chain in a chiral p -wave superconductor In the case of chiral p -wave superconductor without field we can write the BdG Hamiltonian as H BdG ( p ) = ξ ( p ) τ z σ + ∆ t sin p x τ x σ x − ∆ t sin p y τ y σ x (S20)and the bulk Green function at zero energy is G ( E = 0 , p ) = − ξ ( p ) + | ∆( p ) | H BdG ( p ) . (S21)Therefore, G ( E = 0 , r n ) = − τ z σ π ) Z d p ξ ( p ) ξ ( p ) + | ∆( p ) | e ip x na − τ x σ x π ) Z d p ∆ t sin p x ξ ( p ) + | ∆( p ) | e ip x na . (S22)Thus, we obtain H eff ( k x ) = ξ eff ( k x ) τ z σ + ∆ eff ( k x ) τ x σ x , (S23)with ξ eff ( k x ) = − (cid:15) + X n =0 h n e ik x n , h n = − A π ) Z d p ξ ( p ) ξ ( p ) + | ∆( p ) | e − ip x na , (S24)and ∆ eff ( k x ) = X n =0 ∆ n e ik x n , ∆ n = 1 A π ) Z d p ∆ t sin p x ξ ( p ) + | ∆( p ) | e − ip x na . (S25)By reordering the Nambu basis from ψ † k = ( c † k ↑ , c † k ↓ , c − k ↑ , c − k ↓ ) to ˜ ψ † k = ( c † k ↑ , c − k ↓ , c † k ↓ , c − k ↑ ), we can turn theHamiltonian into a block-diagonal form˜ H eff ( k x ) = (cid:18) ˜ H ( k x ) 00 ˜ H ( k x ) (cid:19) = ˜ σ ˜ H ( k x ) , ˜ H ( k x ) = (cid:18) ξ eff ( k x ) ∆ eff ( k x )∆ eff ( k x ) − ξ eff ( k x ) (cid:19) = ξ eff ( k x )˜ τ z + ∆ eff ( k x )˜ τ x . (S26)Here ˜ τ i and ˜ σ i are Pauli matrices in the new basis which still correspond to particle-hole and spin degrees of freedom.With the help of these matrices we can write ˜ H eff ( k x ) = ξ eff ( k x )˜ σ ˜ τ z + ∆ eff ( k x )˜ σ ˜ τ x .Although the Hamiltonian is block-diagonal, these blocks are not independent degrees of freedom because theparticle-hole symmetry connects these. Let us remind the reader that particle-hole symmetry is in the original basis τ x σ H T eff ( − k x ) τ x σ = − H eff ( k x ) (S27)and it now reads in the new basis as ˜ σ x ˜ τ x ˜ H T eff ( − k x )˜ σ x ˜ τ x = − ˜ H eff ( k x ) . (S28)The fact that the particle-hole symmetry is ˜ τ x ˜ σ x means that if there is a positive energy solution in the first blockthere is a negative energy solution in the second block.
3. Impurity chain in a helical p -wave superconductor In the case of helical p -wave superconductor without Zeeman field we can write the BdG Hamiltonian as H BdG ( p ) = ξ ( p ) τ z σ + ∆ t sin p x τ x σ − ∆ t sin p y τ y σ z (S29)and G ( E = 0 , r n ) = − τ z σ π ) Z d p ξ ( p ) ξ ( p ) + | ∆( p ) | e ip x na − τ x σ π ) Z d p ∆ t sin p x ξ ( p ) + | ∆( p ) | e ip x na . (S30)Thus, we obtain the effective Hamiltonian as stated in the main text H eff ( k x ) = ξ eff ( k x ) τ z σ + ∆ eff ( k x ) τ x σ . (S31)Because of the common factor σ , this Hamiltonian is already formally block-diagonal without basis transformation.The important difference to the case for the chiral order parameter is that here the particle-hole symmetry stilloperators inside each block since it still has the form σ τ x . Supplementary Note S3. EFFECTIVE HAMILTONIAN IN THE PRESENCE OF THE ZEEMAN FIELD
In this section we include the Zeeman term H Z = h · σσσ in the Hamiltonian and study how the magnitude anddirection of the Zeeman field h = ( h x , h y , h z ) influences the topological properties of the system. We assume thatthe | h | (cid:28) ∆ t , so that the Zeeman field does not influence the superconducting order parameter . The effect of theZeeman field can be analyzed numerically using the effective Hamiltonian (S10). Here we try to give an analyticallytransparent expressions for the effect of Zeeman field.The first simplification of the effective Hamiltonian H eff ( k x ) is obtained by noticing that τ z σ ˜ G − exists as acommon factor in H eff ( k x ) and therefore its exact structure could be important for the topology only if the Zeemanfield would cause a gap closing in the bulk Hamiltonian H BdG ( p ). We will consider only weak Zeeman fields which donot cause gap closings in the bulk. Therefore, without modifying the topology of the effective Hamiltonian H eff ( k x )for the impurity chain, we can evaluate ˜ G in the absence of the Zeeman field.Although we have not managed to evaluate G I ( k x ) analytically in the case of the general direction of the Zeemanfield, we have obtained approximate expressions for the effective Hamiltonian. We express these results by utilizingthe matrix obtained for one block of the Hamiltonian in the previous section˜ H ( t, µ, ∆ t , V imp , a, k x ) = (cid:18) ξ eff ( t, µ, ∆ t , V imp , a, k x ) ∆ eff ( t, µ, ∆ t , V imp , a, k x )∆ eff ( t, µ, ∆ t , V imp , a, k x ) − ξ eff ( t, µ, ∆ t , V imp , a, k x ) (cid:19) . (S32)The dependence of the effective dispersion ξ eff ( t, µ, ∆ t , V imp , a, k x ) and pairing ∆ eff ( t, µ, ∆ t , V imp , a, k x ) on the param-eters t, µ, ∆ t , V imp , a and k x is determined by the equations described in the previous section. Notice that althoughthe Hamiltonian is not necessarily exactly block-diagonal in the presence of the Zeeman field, we find that it canalways be expressed approximately in a block-diagonal form in the suitable basis. In the following, we directly expressthe results in the basis where the Hamiltonian is approximately block-diagonal. A. Chiral p -wave superconductor
1. Zeeman field along z -direction In the case of chiral p -wave superconductor and Zeeman field in z -direction h = (0 , , h z ) the effective model forthe impurity chain is˜ H eff ( k x ) = (cid:18) ˜ H ( t, µ, ∆ t , V imp , a, k x ) + h z τ
00 ˜ H ( t, µ, ∆ t , V imp , a, k x ) − h z τ (cid:19) . (S33)Here, the Zeeman field trivially shifts one block upwards in energy and the other block downwards in energy. Thus,it cannot cause a transition to a topologically nontrivial state. However, the Zeeman field causes a transition fromgapped phase into a gapless phase.
2. Zeeman field along x - or y -direction In the case of chiral p -wave superconductor and Zeeman field in x - or y -direction h = ( h x , ,
0) or h = (0 , h y ,
0) theeffective model for the impurity chain is˜ H eff ( k x ) = (cid:18) ˜ H ( t, µ + h x ( y ) , ∆ t , V imp , a, k x ) 00 ˜ H ( t, µ − h x ( y ) , ∆ t , V imp , a, k x ) (cid:19) . (S34)The Zeeman term enters the effective Hamiltonian indirectly via renormalization of the chemical potential µ → µ ± h x ( y ) in the two blocks. This can cause topological phase transitions but typically, the impurity bound state energy, andthus the impurity bands are only weakly dependent on the chemical potential via the change of the (normal state)density of states at the Fermi level (see Fig. S7). Thus, one needs relatively strong Zeeman field to cause a transitionand the topological gap stays small. B. Helical p -wave superconductor
1. Zeeman field along z -direction In the case of helical p -wave superconductor and Zeeman field in z -direction h = (0 , , h z ) the effective model forthe impurity chain is ˜ H eff ( k x ) = (cid:18) ˜ H ( t, µ + h z , ∆ t , V imp , a, k x ) 00 ˜ H ( t, µ − h z , ∆ t , V imp , a, k x ) (cid:19) . (S35)Therefore the topological phase diagram is exactly the same as in the case of chiral p -wave superconductor withZeeman field in x - or y -direction.
2. Zeeman field along x -direction In the case of helical p -wave superconductor and Zeeman field in x -direction h = ( h x , ,
0) the effective model forthe impurity chain is˜ H eff ( k x ) = (cid:18) ˜ H ( t, µ, ∆ t , V imp , a, k x ) + h eff ,x τ z
00 ˜ H ( t, µ, ∆ t , V imp , a, k x ) − h eff ,x τ z (cid:19) . (S36)Here we have made approximations during the derivation of the effective Hamiltonian, so that the expression onlyserves as a good approximation for calculation of the topological phase diagram. The Zeeman field in this case is veryeffective in causing topological phase transitions. It acts almost directly to the impurity states. The magnitude ofthe effective Zeeman field h eff ,z is renormalized from the bare value of Zeeman field h eff ,z ≈ h x /
3. Zeeman field along y -direction In the case of helical p -wave superconductor and Zeeman field in y -direction h = (0 , h y ,
0) the effective model forthe impurity chain is˜ H eff ( k x ) = (cid:18) ˜ H ( t, µ, ∆ t , V imp , a, k x ) + h eff ,y τ
00 ˜ H ( t, µ, ∆ t , V imp , a, k x ) − h eff ,y τ (cid:19) . (S37)Here we have also made some approximations. After these approximations it seems that the effect Zeeman field inthis case is similar as in the case of chiral p -wave superconductor with Zeeman field along z -direction. Therefore, itjust shifts the blocks in different directions in energy and cannot induce a topological transition. However, it causesa transition from gapped phase into a gapless phase. The magnitude of the effective Zeeman field h eff ,y is againrenormalized from the bare value of Zeeman field h y . −6 −5 −4 −3−1−0.500.51 V imp e , Q Q D Q DIII e (0.1t) FIG. S3. Magnitude of the lowest eigenvalue e together with the class D and class DIII invariants showing that the class DIIIinvariant changes when the eigenvalue hits zero. The calculation is done for impurity spacing a = 15 using a supercell of size N x = 15 sites along the chain and N y = 35 sites perpendicular to the chain. -5 0 5 d/a -5-4-3-2-10 l og | G n | -100 -50 0 50 100 d/a -6-4-20 l og | G n | -100 0 100 d/a -15-10-50 l og | G n | FIG. S4. Norm of the expansion coefficients of the quasiparticle propagator showing exponential decay with distance for thedifferent models: (a) Single-band model with chiral p -wave order parameter. (b) Single-band model with helical p -wave orderparameter, which is strongly anisotropic so that the gap is suppressed by a factor of 10 along p y . (c) Multiband model forSr RuO with helical p -wave order parameter, which is strongly anisotropic so that there is a gap minima along the 45 degreedirection in ( p x , p y )-plane. Supplementary Note S4. TOPOLOGICAL PHASE TRANSITION WITHOUT ZEEMAN FIELD
In the case of helical p -wave superconductivity without the Zeeman field the system satisfies a time-reversal sym-metry. Thus, it belongs to class DIII in the Altland Zirnbauer classification scheme allowing for a possibility of atopological phase transition as a function of the impurity strength. In the topologically nontrivial phase, there existstwo degenerate Majorana zero modes at each end of the impurity chain. In the case of chiral p -wave superconductivitywithout Zeeman field the system supports a spin-rotation symmetry that allows to block-diagonalize the Hamiltonian.Thus also in this case the system can support a topologically nontrivial phase with two degenerate Majorana zeromodes appearing at each end of the chain. (To be more precise the effective Hamiltonian in both cases also supportsa chiral symmetry allowing infinite number of topologically distinct phases to appear, but the Majorana end modesalways appear in pairs in the absence of the Zeeman field.)Because the Majorana end modes appear in pairs they are not so useful for topological quantum computing.However, we can check the numerical implementation by calculating the corresponding DIII invariant as described inRef. 3 which is based on the calculation of the product of Pfaffians at time reversal invariant momenta. We thereforetune the impurity band through the chemical potential by varying the impurity potential V imp from below V ∗ imp toabove that value. Indeed, the topological invariant as calculated numerically from the Pfaffian changes sign when theenergy of the bound state e hits zero. When looking at the energy bands as function of k x one can also observe thatsuch an impurity band is pushed through zero in this case. Due to time-reversal symmetry, the eigenvalues still comein pairs, therefore the class D invariant (as described in the main text) stays at +1 as expected, see Fig. S3. − − − − µ V i m p N x =N y =15N x =N y =25 − − − − − µ V i m p chiral p-wavehelical p-wave − − − − − − − helical p-wave d helical p-wave d V i m p µ chiral p-wave order parameter N x =N y =25 N x =N y =20 FIG. S5. Results for the impurity potential V ∗ imp that yields a bound state at zero energy as a function of the chemical potential.The order parameter is set to ∆ t = 0 .
5. (a) Already at a system size of N x = N y = 25, no finite size effects are visible, while N x = N y = 15 show some oscillations from discrete energy levels. (b) The bound state energy is exactly identical for chiral- p wave order parameter, Eq. (S5), and helical p-wave order parameter, Eq. (S4), as expected from the analytical result, Eq.(S19). (c) The same is true when comparing the results for the different basis functions, Eq. (S2). Supplementary Note S5. NUMERICAL RESULTS FOR THE ENERGY OF THE IMPURITY BOUNDSTATE
The effective Hamiltonian Eq. (S10) describes the topological phase transitions (energy gap closings) exactly, pro-vided that the Brillouin zone integrals are evaluated with sufficient accuracy and all the longer range hoppings andpairing amplitudes, given for example in Eqs. (S24) and (S25), are included. These longer range terms are expectedto decay exponentially with distance because the bulk Hamiltonian is fully gapped. To verify this, we have examinedthe norm of G (0 , r n ) as function of distance d = | r n | for the various models. The exponential decay is demonstratedin Fig. S4 for all models considered indicating that the errors are exponentially small if the expansion is truncated atfinite r n .We have also numerically studied the tight-binding models using finite size supercells with the impurity in thecenter. This approach leads to finite size effects in the energy of the order of the bandwith divided by the numberof quantum states. To estimate the required system sizes, we perform a check of finite size effects by varying thesystem size and calculating the impurity potential V ∗ imp where the bound state robustly crosses the zero energy dueto a change of a topological invariant as described in detail in Ref. 4 (see Fig. S5). Plotting this quantity as functionof the chemical potential µ one can easily estimate the effects of the energy spacing as small oscillations. These canbe seen in Fig. S5(a) for a system size of 15x15 elementary cells (single band model with isotropic order parameter),while for 25x25 elementary cells these effects are not present any more. According to Eq. (S19) the bound state energyis the same for chiral and helical p -wave order parameters because only the absolute magnitude of the gap enters thecalculation, and this is verified also numerically in Fig. S5(b-c). Note further that the results of V ∗ imp ( µ ) are veryflat (except close to the van Hove singularity at µ = 0). As explained above, this property makes it very difficult tocontrol the topological phase transition by the use of a Zeeman field in z direction in the case of helical p -wave orderparameter and an in-plane field in the case of chiral p -wave order parameter.To compare our analytical result [Eq. (S19)] with the numerical implementation, we calculate the bound stateenergy at a fixed filling of n = 0 .
25 as a function of the impurity potential V imp and show the result in Fig. S6.The zero-energy crossing is captured exactly, while there are small deviations at non-zero energies arising from theexpansion of the Green function in powers of the energy. Finite size effects of the numerical implementation are clearlyseen when plotting the impurity bound state energy as function of chemical potential (see Fig. S7). Note again thatthe bound state energy depends only weakly on the chemical potential. Supplementary Note S6. NUMERICAL RESULTS FOR IMPURITY BANDS IN THE PRESENCE OFZEEMAN FIELD
In Fig. S8 we show the impurity bands for chiral p -wave superconductor in the presence of the Zeeman field. IfZeeman field is applied in-plane it only weakly breaks the degeneracy of the impurity bands [see Fig. S8(a),(b)] asexpected from Eq. (S34). Therefore, a strong field is required to induce a topological phase transition. If Zeemanfield is applied along z -direction the degeneracy of the impurity bands is broken strongly so that one band is shiftedup and the other down in energy [Fig. S8(c)] such that the bands crosss yielding a gapless system as expected from −100 −50 0 50 100−0.2−0.100.10.2 V imp e −100 −50 0 50 100−0.2−0.100.10.2 V imp e −100 −50 0 50 100−0.2−0.100.10.2 V imp e −8 −6 −4 −2−0.2−0.100.10.2 V imp e −8 −6 −4 −2−0.2−0.100.10.2 V imp e −8 −6 −4 −2−0.2−0.100.10.2 V imp e FIG. S6. Bound state energy as a function of the impurity potential. It crosses the zero-energy at V imp = V ∗ imp . Bottom rowshows a zoom-in on the the crossing. Size of the real space lattice from left to right: N x = N y = 30 , ,
60. Red curves areevaluated using the effective Hamiltonian (S19). −2 −1.5 −1 −0.5−0.0500.05 µ e −2 −1.5 −1 −0.5−0.0500.05 µ e −2 −1.5 −1 −0.5−0.0500.05 µ e FIG. S7. Bound state energy as a function of the chemical potential at fixed impurity potential V imp = − . N x = N y = 30 , , Eq. (S33).In Fig. S9 we show the impurity bands for helical p -wave superconductor in the presence of the Zeeman field. IfZeeman field is applied in x -direction the degeneracy of the impurity bands is strongly broken, but apart from thetopological phase transition point the system remains gapped [Fig. S9(a)] as expected from Eq. (S36). If Zeemanfield is applied along the y -direction the bands are just shifted in energy and the system becomes gapless [Fig. S9(b)]as expected from Eq. (S37). Finally, the field in z direction affects the bands only very weakly [Fig. S9(c)] so that astrong field is required to induce a topological phase transition as expected from Eq. (S35).0 − x | E k | − x | E k | − x | E k | FIG. S8. Impurity bands for the chiral p -wave superconductor with V imp = − . a = N x = 15, N y = 35 and Zeeman field | h | = 0 .
02 applied in (a) x , (b) y and (c) z direction. − x | E k | − x | E k | − x | E k | FIG. S9. Impurity bands for the helical p -wave superconductor with V imp = − . a = N x = 15, N y = 35 and Zeeman field | h | = 0 .
05 applied in (a) x , (b) y and (c) z direction. Supplementary Note S7. PHASE DIAGRAMS
With the discussion of the effects of the external Zeeman field in various directions, we can also understand thegeneral properties of the phase diagram obtained by calculating the topological invariant Q = Y k x ∈ TRIM Q ( k x ) . (S38)We have studied the topological phase diagram both by using the full tight-binding Hamiltonian and the effectiveHamiltonian (Figs. S10, S11, S12 and S13).Figs. S10 and S11 show Q for the chiral p -wave superconductor as a function | h | and V imp as obtained from theeffective Hamiltonian and the full tight-binding Hamiltonian, respectively. The results are in excellent agreementwith each other. The in-plane fields can cause a topological phase-transition to a topologically nontrivial phase with Q = − z -direction can make Q = − p -wave superconductor. The Zeeman field in x -direction is very effective in causing a transition to a topologically nontrivial phase with Q = − y -direction can make Q = − z -direction can cause a transition to a topologically nontrivial phase with Q = − p -wave superconductors. Thus, it can be used as a diagnostic tool to determine theorder parameter symmetry of triplet superconductors.In Fig. S16 we show that the topologically nontrivial phase can be reached also by placing the impurities at thestep edge appearing on the surface of the system.1 h x -6-5-4-3-2 V i m p h y -6-5-4-3-2 V i m p h z -6-5-4-3-2 V i m p FIG. S10. Phase diagram (white Q = +1, black Q = −
1) for the chiral p -superconductor as obtained from the effectiveHamiltonian with a = 15 and ∆ t = 0 .
2. Note that for the field in z direction, the system remains gapless, i.e. the zero energystates are not localized at the ends of the impurity chain, but are bulk states instead. h x -6-5-4-3-2 V i m p h y -6-5-4-3-2 V i m p h z -6-5-4-3-2 V i m p FIG. S11. Phase diagram for the chiral p -wave superconductor as obtained from the tight-binding model with a = N x = 15, N y = 35 and ∆ t = 0 . Supplementary Note S8. CURVED CHAINS: MANIPULATION OF PHASE BOUNDARIES
To illustrate the tuneability of the phase boundaries, we consider a Y-junction geometry of two curved chains withdifferent lattice constants (see Fig. S17), where Majorana zero modes appear at the interfaces of topologically trivial(black) and nontrivial (red) regimes which can be shifted by controlling the magnetic field direction. The local phasediagrams on the symmetrically placed example points P and P are shown in Fig. S17(c) and (d), respectively, todemonstrate that for a field with azimuthal angle φ = 0 and polar angle of θ = 0 . π , point P is in the topologialphase and P on the chain with the different lattice constant is in the trivial phase. Note that the phase diagrams h x -6-5-4-3-2 V i m p h y -6-5-4-3-2 V i m p h z -6-5-4-3-2 V i m p FIG. S12. Phase diagram for the helical p -wave superconductor obtained from the effective Hamiltonian with a = 15 and∆ t = 0 . V i m p V i m p V i m p h x h y h z FIG. S13. Phase diagram for the helical p -wave superconductor as obtained from the tight-binding model with a = N x = 15, N y = 35 and ∆ t = 0 . FIG. S14. (a-c) The topological energy gaps for the helical p -wave superconductor. (d-f) The same for the chiral p -wave orderparameter. The parameters are a = N x = 15, N y = 35 and ∆ t = 0 . (e,f) are in principle shifted by π/ h might help a bit; we have not attempted tooptimize this procedure, but instead propose to use local magnets to overcome this difficulty.3 -1 -0.5 0 0.5 1 / / -1 -0.5 0 0.5 1 / / FIG. S15. Phase diagram as function of the direction of the field for (a) helical p -wave superconductor and (b) chiral p -wavesuperconductor. The parameters are a = N x = 15, N y = 35, ∆ t = 0 . | h | = 0 . V imp = 3 . p -wave order parameterand the topological phase can be reached by application of a Zeeman field, but in this case a component perpendicular to thechain is needed. Here, h ≈ | h | (0 . , . , t /E F ≈ . a ≈ . ξ ,while we have used the fully numerical approach with N y = 75. Supplementary Note S9. FUSION OF MAJORANA ZERO MODES
As discussed in the main text, on curved impurity chains, two pairs of Majorana zero modes (MZMs) are generatedupon increasing the Zeeman field along the chain. The corresponding zero modes can be described by the anticom-muting operators γ i , i = 1 , , , { γ i , γ j } = 2 δ ij . These are related to ordinary fermonic operators c and d with { c, c † } = { d, d † } = 1 with c = 12 ( γ + iγ ) , d = 12 ( γ + iγ ) . (S39)The occupation operators of these fermions are given by n c = c † c = 1 + iγ γ , n d = d † d = 1 + iγ γ . (S40)and the many-particle ground states can be defined as ( c | i = 0, d | i = 0) | i , c † | i = | i , d † | i = | i , c † d † | i = | i . (S41)The back transformation reads γ = c + c † , γ = i ( c † − c ) , γ = d + d † , γ = i ( d † − d ) . (S42)In order to describe the state after the fusion, we introduce another set of fermionic operators as e = 12 ( γ + iγ ) , f = 12 ( γ + iγ ) . (S43)The degenerate many-particle ground states can be written using these fermion operators as ( e | i e,f = 0, f | i ef = 0) | i e,f , e † | i e,f = | i e,f , f † | i e,f = | i e,f , e † f † | i e,f = | i e,f . (S44)4 / // / FIG. S17. Nontrival movement of phase boundaries. A Y junction can be realized by joining two curved impurity chains withdifferent lattice constants (full line and dashed line). MZMs, labelled 1-4 are at the boundaries between topological regions(red) and trivial regions (black). (a-f) Changing the direction of the Zeeman field moves the boundaries when the direction ofthe magnetic field follows a closed loop in the ( φ, θ ) plane shown in (e) and (f). (e),(f) Topological phase diagrams at points P and P (open black circles in panel (a)). The differences in the two phase diagrams are due to the different tangential directionsof the chains at these points and the different lattice constants. The magnitude of the topological gap (not shown) remainssmall along the trajectory, thus MZMs will almost certainly overlap. In the calculation of the topological phase diagrams wehave assumed lattice constants (e) a ≈ ξ and (f) a ≈ . ξ . Other parameters are identical to the ones in the main text,∆ /E F ≈ . In order to find the basis transformation between the states , we e † = 12 ( c + c † − d + d † ) , f † = i − c + c † − d − d † ) . (S45)We observe that the unitary transformation between the two bases does not change the total parity. Therefore, usingthe equations (S45) we find that the basis transfomation between states (S41) and (S44) is given by | i e,f = 1 √ | i + | i ) , | i e,f = 1 √ | i + | i ) , | i e,f = i √ | i − | i ) , | i e,f = i √ | i − | i ) . (S46)From these expressions it is clear that when the MZMs 2 and 3 are fused corresponding to projective measurementof f † f , the measurement outcomes 0 and 1 have equal probabilities. After the measurement the system has equalprobabilities to be in the two different eigenstates of e † e .As discussed in the main text the MZMs can be moved by varying the direction and the magnitude of the magneticfield. Moreover, the parity of the Majoranas can be measured as soon as the MZMs 2 and 3 hybridize and acquire a5charge . These are the key ingredients for performing a fusion and braiding experiments. The curved chain geometrycan also be generalized so that braiding and more complicated manipulations of MZMs can be performed alongsimilar lines as proposed in Ref. 6. However, to preserve the quantum information stored in the MZMs the followingconditions have to be satisfied : (i) The time scale of operations t has to be much shorter than the tunnelingtime t tunneling ∝ e L/ζ M and thermal excitation time t thermal ∝ e E gap /k B T , where L is the distance between spatiallyseparated Majoranas (the ones which are not fused intentionally), ζ M is the localization length of the Majoranas, E gap is the minimum excitation energy of the quasiparticles (other than the MZMs) during the braiding cycle, k B isthe Boltzmann constant and T is the temperature. (ii) The time-scale t has be much shorter than the quasiparticlepoisoning time t poisoning (which is typically determined by non-equilibrium quasiparticles). (iii) The time-scale t should be long compared to ~ /E gap to avoid dynamical excitations of the quasiparticles.The curved chain geometry leads to slowly varying parameters along the chain so that in addition to the MZMthere exists low-energy Andreev bound states at the domain wall between nontrivial and trivial regions. This meansthat E gap is much smaller than the topological gap ∆ topo which can be achieved in a linear chain. Thus, also thelocalization length ζ M in the curved chain is much longer than the corresponding length scale ζ of the linear chaindiscussed in the main text. Therefore, t tunneling and t thermal are much shorter in curved chain geometry than in thelinear chain, so that it is challenging to satisfy the requirements ~ /E gap (cid:28) t (cid:28) t tunneling , t thermal . We also point outit is difficult to vary the external magnetic field fast so that it is also difficult to satisfy the requirement t (cid:28) t poisoning .In the next section we discuss how it is possible to overcome these problems by using small magnets where themagnetization directions are controlled fast using spintronic techniques . Supplementary Note S10. MAJORANA SPINTRONICS: FUSION AND EXCHANGE OF MAJORANAZERO MODES
In order to probe the non-Abelian statistics of the MZMs, we propose a tri-junction geometry (see Fig. S18),where Majorana zero modes appear at the interfaces of topologically nontrivial (red) and trivial (black) regimes.The topology of each segment i = 1 , , M iA and M iB on the different sides of the chain. The magnets should be placed within thesuperconducting coherence length from the impurity sites (in the case of Sr RuO this is approximately ∼
70 nm)and they should be close enough to the surface of the superconductor to cause a magnetic exchange field due to themagnetic proximity effect. If the magnetizations M iA and M iB point in the same direction the segment i realizesapproximately an impurity chain in the presence of a homogeneous Zeeman field. On the other hand, if M iA and M iB point in opposite directions, their effect on the impurity bound states cancel each other so that as a goodapproximation we obtain the Hamiltonian in the absence of Zeeman field. Thus, by designing the magnets so thatthe magnetizations M iA and M iB have an easy-axis anisotropy along the direction of the segment i , the resultsobtained in the previous sections demonstrate that we can choose the magnitudes of the exchange fields so that theparallel (antiparallel) magnetizations M iA and M iB lead to topologically nontrivial (trivial) phase with large energygap E gap and short localization length of MZMs ζ M . Thus, the MZMs can be robustly manipulated by switching themagnetization directions fast using the spintronic techniques .To perform an exchange of MZMs we utilize the anyon teleportation scheme , where the exchange of MZMsis obtained via a sequence of projective measurements shown in Fig. S18. According to the universal non-Abelianbraiding statistics of the MZMs the exchange of MZMs γ and γ is described by applying the unitary operator U = e − π γ γ = 1 √ (cid:0) γ γ (cid:1) (S47)on the state of the system . The teleportation scheme is based on the decomposition Π Π Π Π = r
18 Π ⊗ √ (cid:0) γ γ (cid:1) , (S48)where each operator Π kl = 12 (1 + iγ k γ l ) (S49)describes a projective measurement of the parity P kl = iγ k γ l of MZMs γ k and γ l with the outcome of the measurementbeing +1. Therefore, based on Eqs. (S47) and (S48), it is clear that the exchange of MZMs γ and γ can be performedas follows: (i) Initialize the parity P = 1 by fusing MZMs γ and γ , performing a projective measurement of iγ γ and continuing only if the outcome of the measurement is +1. (ii) Perform similarly a measurement of the parity6 Starting point: all segments topologically nontrivial Magnets have easy axis anisotropy. Parallel (antiparallel) magnetizations on the di ff erent sides of the segment lead to nontrivial (trivial) phase. Turn M so that the corresponding segment topologically trivial f u s i o n P =1 Go back to original configuration of magnets if even parity measuredTurn M so that the corresponding segment topologically trivial f u s i o n P =1 Go back to initial configuration of magnets if even parity measured Turn M so that the corresponding segment topologically trivial f u s i o n P =1 Go back to initial configuration of magnets if even parity measured Turn M so that the corresponding segment topologically trivial f u s i o n P =1 Go back to original configuration of magnets if even parity measured M M M M M M FIG. S18. Exchange of MZMs by utilizing the anyon teleportation scheme. The unitary operation describing the non-AbelianMajorana braiding statistics [Eq. (S47)] can be decomposed into four successive projective measurements of parity operators[Eq. (S48)]. 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