Origin of Topological Surface Superconductivity in FeSe_{0.45}Te_{0.55}
Eric Mascot, Sagen Cocklin, Martin Graham, Mahdi Mashkoori, Stephan Rachel, Dirk K. Morr
OOrigin of Topological Surface Superconductivity in FeSe . Te . Eric Mascot , Sagen Cocklin , Martin Graham , Mahdi Mashkoori , Stephan Rachel , and Dirk K. Morr University of Illinois at Chicago, Chicago, IL 60607, USA and School of Physics, University of Melbourne,Parkville, VIC 3010, Australia (Dated: February 11, 2021)
The engineering of Majorana zero modes intopological superconductors, a new paradigm forthe realization of topological quantum comput-ing and topology-based devices, has been ham-pered by the absence of materials with suffi-ciently large superconducting gaps. Recent ex-periments, however, have provided enthrallingevidence for the existence of topological surfacesuperconductivity in the iron-based superconduc-tor FeSe . Te . possessing a full s ± -wave gap ofa few meV. Here, we propose a mechanism forthe emergence of topological superconductivityon the surface of FeSe . Te . by demonstratingthat the interplay between the s ± -wave symme-try of the superconducting gap, recently observedsurface magnetism, and a Rashba spin-orbit inter-action gives rise to several topological supercon-ducting phases. Moreover, the proposed mecha-nism explains a series of experimentally observedhallmarks of topological superconductivity, suchas the emergence of Majorana zero modes in thecenter of vortex cores and at the end of line de-fects, as well as of chiral Majorana edge modesalong certain types of domain walls. We alsopropose that the spatial distribution of supercur-rents near a domain wall is a characteristic signa-ture measurable via a scanning superconductingquantum interference device that can distinguishbetween chiral Majorana edge modes and trivialin-gap states.Introduction The non-Abelian braiding statistics of Majorana zeromodes (MZMs) have opened a new route for the real-ization of topological quantum computing and topology-based devices [1]. Evidence for the existence ofthese modes has been observed in one- [2–8] and two-dimensional (2D) [9–11] topological superconductors,however, their unambiguous identification has been ex-perimentally hampered by the small superconductinggaps in these systems, which are often only of the or-der of a few hundred µeV . The recent report of topo-logical superconductivity in the iron-based superconduc-tor FeSe . Te . , as evidenced by the observation of asurface Dirac cone [12, 13], of MZMs in the vortex core[14–16] and at the end of line defects [17], and of chi-ral Majorana mode near domain walls [18], has therefore been greeted with much enthusiasm as this system pos-sesses a significantly larger superconducting gap of a fewmeV. The origin of these observations was ascribed toFeSe . Te . being a topological insulator whose surfaceDirac cone is gapped out by proximity induced supercon-ductivity, giving rise to topological surface superconduc-tivity. However, the observation of a single T c that simul-taneously gaps the Dirac cone and the (so-far assumed)topological trivial α -, β - and γ -bands [19], implies a cou-pling between them. This coupling, in turn, would de-stroy the topological character of the Dirac cone, raisingthe question as to the origin of the observed MZMs andof the underlying topological phase.In this article, we propose that the origin of theobserved Majorana modes lies in the emergence oftopological superconductivity in the α -, β - and γ -bands,arising from the interplay of a hard superconducting gapof s ± -wave symmetry, a Rashba spin-orbit interactionand surface magnetism. While a Rashba spin-orbit inter-action occurs naturally at the surface of FeSe . Te . due to the broken inversion symmetry, recent ARPESexperiments [19] have provided strong evidence for theonset of surface magnetism at T c . In particular, theyobserved that a gap opens up in the Dirac cone not onlyat the Fermi energy E F , as expected from proximityinduced superconductivity, but also at the Dirac point,which lies approximately 8 meV below E F . The latter isa direct consequence of a broken time-reversal symmetryon the surface of FeSe . Te . , with a non-vanishingmagnetic moment perpendicular to the surface. Wedemonstrate that this interplay does not only give riseto robust topological superconducting phases on thesurface of FeSe . Te . , characterized by a Z topo-logical invariant, the Chern number, but also explainsthe various observations of Majorana zero modes invortex cores [14, 15] and at the end of line defects [17],as well as the emergence of dispersive Majorana edgemodes along domain walls [18]. As such, the existenceof a Dirac cone within our scenario is secondary for theemergence of topological superconductivity and of theresulting MZMs, whose existence is primarily driven bythe topological nature of the α -, β - and γ -bands. Inaddition, we propose a novel experimental signature, thepresence or absence of supercurrents along domain walls,which can distinguish topological Majorana modes fromtrivial in-gap states and can be imaged via a scanning a r X i v : . [ c ond - m a t . s up r- c on ] F e b superconducting quantum interference device (SQUID)[20]. Results and Discussion
To investigate the emergence of topological supercon-ductivity on the surface of FeSe . Te . , we considera two-dimensional 5-orbital model [21] extracted from ananalysis of scanning tunneling spectroscopy (STS) exper-iments on the surface of clean FeSe . Te . [22]. Inaddition, we include in this model (a) the presence ofsurface magnetism, evidence for which was recently re-ported by ARPES experiments [19] through an exchangefield, and (b) a Rashba spin orbit (RSO) interaction thatarises from the breaking of the inversion symmetry onthe surface [for detail, see supplementary materials (SM)Sec. S1]. The resulting Hamiltonian in real space is givenby H = − X a,b =1 X r , r ,σ t ab r , r c † r ,a,σ c r ,b,σ − X a =1 X r ,,σ µ aa c † r ,a,σ c r ,a,σ + iα X a =1 X r , δ ,σ,σ c † r ,a,σ ( δ × σ ) zσσ c r + δ ,a,σ + J X a =1 X r ,σ,σ S r · c † r ,a,σ σ σσ c r ,a,σ + X a =1 X hh r , r ii ∆ aa rr c † r ,a, ↑ c † r ,a, ↓ + H . c . (1)Here a, b = 1 , ..., d xz -, d yz -, d x − y -, d xy -, and d z − r -orbitals,respectively, − t ab rr represents the electronic hopping am-plitude between orbital a at site r and orbital b at site r on a two-dimensional square lattice, µ aa is the on-siteenergy in orbital a , c † r ,a,σ ( c r ,a,σ ) creates (annihilates) anelectron with spin σ at site r in orbital a , and σ is thevector of spin Pauli matrices. The superconducting orderparameter ∆ aa rr represents intra-orbital pairing betweennext-nearest neighbor Fe sites r and r (in the 1 Fe unitcell), yielding a superconducting s ± -wave symmetry [22].Moreover, α denotes the Rashba spin-orbit interactionarising from the breaking of the inversion symmetry atthe surface [3] with δ being the vector connecting near-est neighbor sites. Due to the full superconducting gap,which suppresses Kondo screening, we consider the mag-netic moments to be static in nature, such that S r isa classical vector representing the direction of a surfaceatom’s spin located at r , and J is its exchange couplingwith the conduction electron spin. The experimentallyobserved opening of a gap at the Dirac point [19] im-plies that a considerable fraction of the ordered mag-netic moment is aligned perpendicular to the surface,such that for concreteness, we assume an out-of-planeferromagnetic alignment. Within this model, topologi-cal superconductivity is thus confined to the surface of FeSe . Te . . In the normal state, the above Hamilto-nian yields three Fermi surfaces in the 1Fe Brillouin zone,two closed around the Γ-point, arising from the α - and β -bands, and one closed around the X/Y -points, arisingfrom the γ -band [22]. Due to the orbital character of thethree Fermi surfaces, the superconducting order param-eter is only non-zero in the d xz -, d yz -, and d xy -orbitals[22]. The local density of states (LDOS) resulting formthe above Hamiltonian in the superconducting state re-produces all salient features of the differential conduc-tance, dI/dV , measured via STS (see SM Sec. S1 andFig. S1), and in particular shows the existence of severalsuperconducting gaps ranging from 1.6 meV to 2.4 meV.We note that due to the particle-hole symmetry of thesuperconducting state, and the broken time-reversal sym-metry arising from the presence of magnetic moments,FeSe . Te . belongs to the topological class D [23, 24].For a two-dimensional system, the topological invariantis therefore given by the Chern number, which can becomputed via [25] C = 12 πi Z BZ d k Tr( P k [ ∂ k x P k , ∂ k y P k ]) P k = X E n ( k ) < | Ψ n ( k ) ih Ψ n ( k ) | (2)where E n ( k ) and | Ψ n ( k ) i are the eigenenergies and theeigenvectors of the Hamiltonian in Eq.(1), with n beinga band index, and the trace is taken over Nambu andspin space. Topological Phase diagram
The existence of a hardsuperconducting gap, a Rashba spin-orbit interactionand an out-of-plane ferromagnetic order are in generalsufficient requirements for the emergence of topologicalsurface superconductivity [26–29]. To demonstrate thatthis emergence is a robust phenomenon in FeSe . Te . within the proposed model, we present in Fig. 1 thetopological phase diagram – in terms of the Chernnumber C – computed from Eq.(2) as a function of theeffective magnetic exchange strength JS , with S rep-resenting the ordered spin moment on the surface, andof a shift of the chemical potential, ∆ µ , from its valueextracted in Ref.[22]. We note that already for ratherweak magnetism, as reflected in a magnetic exchangecoupling JS of the order of a few meV, the systemundergoes transitions into topological superconductingphases. An increase in the RSO interaction does notshift the position of the topological phase transitions,but increases the topological gap. While the presence oftopological phases is robust against shifts in the chem-ical potential, varying ∆ µ induces transitions betweentopological phases characterized by different Chernnumbers. With increasing JS (keeping all other bandparameters fixed), the Fermi surfaces eventually crossthe nodal lines of the superconducting s ± -wave order FIG. 1.
Topological phase diagram . Topological phasediagram of FeSe . Te . in the (∆ µ, JS )-plane with α = 7meV. ∆ µ is a shift in the chemical potential in all orbitalsfrom the value used in Eq.(1). The solid, dashed and dot-ted black lines indicate gap closing at the Γ-, X/Y -, and M -points, respectively, which accompany the topological phasetransitions. The gray area denotes the gapless region. parameter, and the system becomes gapless (as denotedby the gray area in Fig. 1), and hence topologicallytrivial. We note, however, that this gapless region of thephase diagram can be shifted to larger values of JS byappropriately adjusting the band parameters in Eq.(1).The points in the (∆ µ, JS )-plane where the topologicalphase transitions occur are determined by the closingof the superconducting gap. Analytical expressions forthese phase transition lines can be obtained (see SMSec. S2) and are plotted as solid, dashed and dotted linesin Fig. 1) representing the closing of the gap at the Γ-, X/Y -, and M -points in the Brillouin zone, respectively.The X/Y -, and M -points possess a multiplicity of m = 2 and m = 1, leading to a change in the Chernnumber by ∆ C = +2 and ∆ C = −
1, respectively. Incontrast, while the Γ-point possesses a multiplicity of m = 1, the band in which the gap closing occurs istwo-fold degenerate, leading to a change in the Chernnumber by ∆ C = −
2. Having established the presenceof topological superconducting phases in FeSe . Te . ,we next turn to a discussion of their unique physicalphenomena that have been observed experimentally. MZM in a vortex core
The experimental observa-tion of MZMs localized in vortex cores [14, 15] representsa salient signature of the topological nature [30] of the su-perconducting surface in FeSe . Te . . As the detailedspatial, energy and spin-structure of these MZMs canprovide important insight into the microscopic origin ofthe topological phase, we next investigate the electronicstructure of MZMs near a vortex core in FeSe . Te . .To this end, we implement the magnetic field via the FIG. 2.
MZM in a vortex core. a
Spatial dependence ofthe superconducting order parameter in the d xz -orbital neara vortex core for a magnetic field of B = 2T. Linecut ofthe LDOS, N ( r , E ), through the center of the vortex corealong the x -axis for b the C = − JS, α, ∆ µ ) = (7 . , ,
0) meV (the MZM is indicated by ablack arrow) and c the topologically trivial phase ( C = 0)with JS = α = 0 (the dashed white line corresponds to E =0). d Energy dependence of the spin-resolved LDOS at thevortex center in the topological C = − Peierls substitution and compute the spatial dependenceof the superconducting order parameters in the d xz -, d yz -, and d xy -orbitals self-consistently (for details, seeSM Sec. S3). The resulting spatial structure of the su-perconducting order parameter in the d xz -orbital, whichvanishes at the center of the vortex, in the topological C = − a (the analogous plots forthe d yz -, and d xy -orbitals are shown in SM Sec. S3). Wefind that all three superconducting order parameters pos-sess the same spatial symmetry as the orbitals they arisefrom, i.e., a C -symmetry in the d xz -, and d yz -orbitals,and a C -symmetry in the d xy -orbital. To demonstratethe existence of a MZM localized at the vortex core, wepresent in Fig. 2 b a linecut of the energy-resolved LDOSthrough the vortex which captures all salient features ofthe experimental observations: (i) the presence of a zero-energy state centered at the vortex core, (ii) a spatialextent of the MZM of about 15 a , and (iii) a decrease inthe effective gap near the vortex core. In contrast, in thetopologically trivial phase, the LDOS linecut [see Fig. 2 c ]does not show a zero-energy state, but only topologicallytrivial Caroli-de Gennes-Matricorn (CdGM) states [31],similar to results obtained in other iron-based supercon-ductors [32]. We therefore can identify the zero-energystate in the center of the vortex core shown in Fig. 2 b as aMZM. We note, however, that though there is no MZM inthe vortex core in the trivial phase [see Fig. 2 c ], the low-est energy trivial CdGM states are located at E = ± . d ]reveals a strong spin-polarization of the MZM, which is adirect consequence of the surface magnetism, and hencea characteristic signature of the here proposed origin ofthe topological superconductivity in FeSe . Te . . Incontrast, in previously proposed scenarios [12, 33], wherethe topological surface state arises from an interplayof a bulk topological insulator and superconductivity[12], which does not break the time-reversal symmetry,no spin-polarization of the MZM is expected. We thusconclude that the scenario we propose for the emergenceof topological surface superconductivity in FeSe . Te . does not only capture the salient experimental obser-vations, but also possesses an essential signature in thestrong spin-polarization of the vortex core MZM. MZMs at the end of line defects
In addition tovortex cores, MZMs were also predicted to occur at theend of line defects that are embedded in topological p x + ip y -wave superconductors [34]. The recent obser-vation of MZMs at the end of line defects on the surfaceof FeSe . Te . [17] has therefore raised the question ofwhether these MZMs are a direct signature of the under-lying topological superconducting phase, and thus repre-sent a sufficient condition for its existence. To addressthis question, we represent the line defect for simplicityas a line of potential scatterers (though magnetic scat-terers could also be realized [35, 36]) described by theHamiltonian H def = U X a =1 X R ,σ c † R ,a,σ c R ,a,σ , (3)where U is the potential scattering strength, and thesum runs over all sites R of the line defect In Fig. 3 a , wepresent the energies of the three lowest energy states fora line defect of length L = 119 in the C = − U . One can clearly discern three regionsof U where the lowest energy state is essentially locatedat zero energy (keeping in mind that the LDOS exhibitsa full gap of 1.6 meV, see SM Sec. 1), suggesting theexistence of a MZM. Further evidence arises from aline-cut of the LDOS along the line defect with U = 71meV, located in the rightmost gray region in Fig. 3 a , forthe lowest and second lowest energy states with energies FIG. 3.
MZMs and line defects. a
Energies of the threelowest energy states, E , E , E as a function of scatteringstrength U for a line defect of length L = 119 sites in the C = − JS, α, ∆ µ ) = (7 . , , U where the line defect is in a topological phase are shownusing a gray background. b Linecut of the spatial LDOS alongthe line defect for the two lowest energy states at E , E and U = 71 meV. Plot of the spatial LDOS at energies c E and d E for U = 71 meV. Filled black circles in c indicate thepositions of the defects. E and E , respectively [see Fig. 3 b ]. The spectralweight of the lowest energy state at E is confined tothe end of the line defect – with essentially no spectralweight located inside the defect line – reflecting thelocalized nature of the MZM. In contrast, the state at E exhibits considerable spectral weight along the entirelength of the line defect. This result is also confirmedby a spatial plot of the LDOS at energies E and E ,shown in Figs. 3 c and d , respectively. These resultsprovide strong evidence for the existence of MZMs atthe end of line defects in the topological C = − FIG. 4.
Electronic structure near domain walls . Upper (lower) row shows results for a spin domain wall (a π -phasedomain wall). Electronic band structure as a function of momentum along the domain wall for a a spin, and b a π -phasedomain wall. Energy-resolved LDOS at c the spin, and d the π -phase domain wall along a linecut perpendicular to the domainwall. The position of the domain wall is indicated by a dashed gray line. Spatial plot of the zero-energy LDOS for e a spin,and f a π -phase domain wall. Spatial distribution of supercurrents near g the spin, and h the π -phase domain wall. For bothtypes of domain walls, the system is in the C = − JS, α, ∆ µ ) = (7 . , ,
0) meV.
The regions of U where the line defect itself is in atopological phase and thus exhibits MZMs, shown with agray background in Fig. 3 a , are bounded by topologicalphase transitions that are accompanied the closing ofthe superconducting gap. We note that while topologicalsuperconductivity on the surface of FeSe . Te . ischaracterized by a Z topological invariant, the line defectas a one-dimensional system possesses a topological Z classification, similar to the Kitaev chain [37]. Moreover,while we find similar results in the C = − . Te . is in the topologicaltrivial ( C = 0) phase (see SM Sec. 4). We thus concludethat the existence of MZMs at the end of line defectsis directly tied to and a sufficient condition for theexistence of an underlying topological superconductingphase on the surface of FeSe . Te . . However, the re-sults shown in Fig. 3 a also demonstrate that a failure toobserve MZMs at line defects does not necessarily implya trivial nature of the underlying superconducting phase. Chiral Majorana modes along domain walls
Thebulk-boundary correspondence dictates that Majoranaedge modes need to arise along domain walls thatseparate regions of different Chern numbers [37]. Indeed,the observation of a nearly constant LDOS at a domainwall in FeSe . Te . [18] was recently interpreted asa signature of a chiral Majorana mode. This raisesthe intriguing question not only as to which types ofphysical domain walls can give rise to the emergence ofMajorana modes, but also of how to distinguish themfrom trivial, in-gap modes. To address this question,we calculate the electronic structure near two differenttypes of domain walls: a spin domain wall at whichthe magnetic moment is inverted, i.e., S → − S , and a π -phase domain wall, where the superconducting orderparameter undergoes a π -phase shift, i.e., ∆ → − ∆,for all bands in FeSe . Te . . As the spin domainwall separates regions with different Chern numbers(since S → − S implies C → − C ), the bulk boundarycorrespondence requires the emergence of dispersiveMajorana edge modes that traverse the superconductinggap, as shown in Fig. 4 a where we present the system’selectronic band structure as a function of momentum k k along the domain wall (Majorana modes are shown asred lines). In addition, the system also exhibits trivialin-gap modes, which do not connect the upper andlower bands. In contrast, regions separated by a π -phasedomain wall exhibit the same Chern number, and theelectronic band structure therefore only exhibits trivialin-gap states, as shown in Fig. 4 b . In Figs. 4 c and d ,we present the LDOS as a function of energy along alinecut perpendicular to the domain wall. In both cases,we find that the LDOS near the domain wall exhibitsconsiderable spectral weight inside the superconductinggap, with the LDOS being nearly energy independent forthe spin domain wall, but exhibiting a pronounced peakat zero-energy for the π -phase domain wall. A linecut ofthe zero-energy LDOS, shown in Figs. 4 e and f , however,reveals that the zero-energy state is localized close tothe domain wall in both cases. Thus, the differences inthe LDOS between these two types of domain walls isquantitative rather than qualitative in nature, and STSmeasurements might therefore not be able to distinguishbetween topological Majorana edge modes, and trivialin-gap states. However, a qualitative difference betweenthese domain walls can be identified when consideringthe spatial structure of the induced supercurrents (seeSM Sec. S5). For a spin domain wall, the chirality of theinduced supercurrent, which is determined by the sign ofthe Chern number, changes between the two separatedregions, implying that the supercurrents associated witheach region flow in the same direction along the domainwall [28], as shown in Fig. 4 g , yielding a non-vanishingnet supercurrent. In contrast, for the π -phase domainwall, the chirality of the supercurrents in both regions isthe same, implying that they flow in opposite directionsalong the domain wall [see Fig. 4 h ], yielding a vanishingnet supercurrent. This qualitative difference, a non-zeronet supercurrent for a spin domain wall and a vanishingnet supercurrent for the π -phase domain wall, can beimaged using a SQUID, thus providing an unambiguousexperimental signature to distinguish the existence oftopological Majorana modes for a spin domain wall,from that of trivial in-gap states for a π -phase domainwall. Conclusions
We have proposed a microscopicmechanism for the emergence of topological surfacesuperconductivity in FeSe . Te . arising from theinterplay of surface magnetism, a Rashba spin-orbitinteractions, and a hard superconducting gap with s ± -wave symmetry. This mechanism explains not only theemergence of robust topological phases already for weaksurface magnetism, with effective magnetic exchangecouplings of a few meV, but also the experimentalobservations of (i) MZMs in vortex cores, (ii) MZMs atthe end of line defects, and (iii) chiral Majorana edgemodes at domain walls. In addition, we demonstrated that by measuring supercurrents along domain wallsusing a SQUID it is possible to distinguish topologicalMajorana modes from trivial in-gap states. Within ourscenario, the existence of a Dirac cone is secondary forthe emergence of topological superconductivity and theresulting Majorana modes, as they are primarily drivenby the topological nature of the α -, β - and γ -bands. Oneremaining interesting question for future work pertainsto the experimental signatures that might arise from thepotential coupling between the topological α -, β - and γ -bands and the Dirac cone. Materials and Methods
To compute the Chern number of FeSe . Te . ,yielding the topological phase diagram of Fig. 1, weemploy Eq.(2). To calculate the electronic structureof a vortex, we implement the magnetic field via thePeierls substitution and compute the spatial dependenceof the superconducting order parameters in the d xz -, d yz -, and d xy -orbitals self-consistently, as describedin SM Sec. S2. The LDOS is computed by rewritingthe Hamiltonian of Eq.(1) in terms of a Hamiltonianmatrix, ˆ H , in real and Nambu space, and calculat-ing the retarded Greens function matrix ˆ g r usingˆ g r ( ω ) = h ( ω + iδ )ˆ1 − ˆ H i − . The local, spin-resolveddensity of states, N ( r , ω, σ ) at site r is then obtained via N ( r , σ, ω ) = − Im [ˆ g r ( r , σ ; r , σ ; ω )] /π . The supercurrentis calculated using the Keldysh formalism, as describedin SM Sec. S5. Acknowledgments
The authors would like to thank A. Kreisel, C. Hess,and P.D. Johnson for stimulating discussions.
Funding
This work was supported by the U. S. Department ofEnergy, Office of Science, Basic Energy Sciences, underAward No. DE-FG02-05ER46225 (E.M., S.C., M.G.and D.K.M.) and through an ARC Future Fellowship(FT180100211) (S.R.).
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Topological Insulatorsand Topological Superconductors (Princeton UniversityPress, Princeton, 2013). rigin of Topological Surface Superconductivity in FeSe . Te . Supplementary Material
Eric Mascot , Sagen Cocklin , Martin Graham , Mahdi Mashkoori , Stephan Rachel, , and Dirk K. Morr University of Illinois at Chicago, Chicago, IL 60607, USA and School of Physics, University of Melbourne, Parkville, VIC 3010, Australia
Section S1: Band parameters and local density of states in FeSe . Te . The band parameters in the Hamiltonian of Eq.(1) in the main text were extracted from scanning tunnelingspectroscopy (STS) experiments on clean FeSe . Te . [1]. In order to reproduce a lineshape of the differentialconductance dI/dV measured in STS experiments in the topological C = − JS = 7 . α = 7 meV,and ∆ µ = 0, we used the following intra-orbital superconducting order parameter reflecting pairing between next-nearest neighbor sites in the 1 Fe unit cell: ∆ d xz rr = ∆ d yz rr = 1 . d xy rr = 0 .
76 meV, and ∆ d x − y rr = ∆ d z − r rr = 0.The resulting local density of states (LDOS), which is the sum of the LDOS for each orbital, is shown in Fig. S1.It reproduces all salient feature of the experimental data: a hard superconducting gap, with the most pronounced FIG. S1. LDOS of FeSe . Te . in the topological C = − coherence peaks located at E = 2 meV and 2.4 meV arising from the α - and β -bands, and an additional in-gap peakaround E ≈ . γ -band. These band parameters are used for all figures in the main text thatpresent results in the topological C = − c ofthe main text, we used J = α = ∆ µ = 0 and ∆ d xz rr = ∆ d yz rr = 0 .
55 meV, ∆ d xy rr = 0 . d x − y rr = ∆ d z − r rr = 0. Section S2: Analytical expression for the topological phase transition lines
An analytical expression for phase transition lines can be obtained by diagonalizing the Hamiltonian of Eq.(1) in themain text at time-reversal invariant high symmetry points in the 1Fe Brillouin zone. The closing of the superconductinggap at these points indicates a topological phase transition. One thus obtains the following expressions determiningthe relation between JS and ∆ µ where a topological phase transition occurs and the gap closes (i) at the Γ-point[i.e., at k = (0 , J = r(cid:16) E min k =(0 , − ∆ µ (cid:17) + 4∆ d xz,yz rr , (S1) a r X i v : . [ c ond - m a t . s up r- c on ] F e b with E min k =(0 , = 13 meV is the energy of the lowest energy band in the normal state at k = (0 , X/Y -points [i.e., at k = ( ± π, , (0 , ± π )] J = r(cid:16) E min k =(0 ,π ) − ∆ µ (cid:17) + 4∆ d xz,yz rr , (S2)with E min k =(0 ,π ) = −
11 meV is the energy of the lowest energy band in the normal state at k = (0 , ± π ) , ( ± π, M -points (i.e., at k = ( ± π, ± π )) J = r(cid:16) E min k =( π,π ) − ∆ µ (cid:17) + 4∆ d xy rr , (S3)with E min k =(0 ,π ) = − k = ( ± π, ± π ), yielding theso dotted black line in Fig. 1 of the main text. Section S3: MZM in a vortex core
To calculate the structure of a vortex in the presence of a magnetic field, and in particular the spatial dependence ofthe superconducting order parameters, we employ the Peierls substitution [2, 3], which replaces the electronic hoppingterms according to t ab r , r → t ab r , r e iθ r , r , (S4a) iα ( δ × σ ) zσσ → iα ( δ × σ ) zσσ e iθ r , r + δ . (S4b)The Peierls phase factor is given by θ r , r = πφ Z r r A ( s ) · ds , (S5)where φ = h/ e is the superconducting flux quantum and A is the magnetic vector potential. Here, we use thesymmetric gauge A ( r ) = Bˆz × r . The superconducting order parameters in the presence of the magnetic field are FIG. S2. Spatial plot of the superconducting order parameter in the a d yz -, and b d xy -orbital in the C = − JS, α, ∆ µ ) = (7 . , ,
0) meV. self-consistently computed using the gap equation,∆ aa r , r = V aa r , r h c r , a , ↓ c r , a , ↑ i . (S6)The pairing interaction, V aa r , r , is chosen such that it reproduces the values of the superconducting order parameters∆ d xz = ∆ d yz = 1 . d xy = 0 .
76 meV, and ∆ d x − y = ∆ d z − r = 0 meV for a clean system (i.e., in the absenceof a magnetic field), as discussed in Sec. S1. This yields the following values V d xz r , r = V d yz r , r = 62 . V d xy r , r = 54 . V d x − y r , r = V d z − r r , r = 0 (S7c)The expectation values on the right hand side of Eq.(S6) are calculated using the kernel polynomial method [4, 5]with 200 moments and the gap equation was solved iteratively using Anderson’s method [6, 7] until the largest errorwas below 8 . µ eV. In Fig. S2 we plot the resulting spatial structure of the superconducting order parameters in the FIG. S3. Linecut of the orbitally-resolved LDOS, N ( r , E ), through the center of the vortex core along the x -axis in the a d xz -, b d yz -, and c d xy -orbitals for the C = − JS, α, ∆ µ ) = (7 . , ,
0) meV. The MZM localized in thevortex core is clearly visible in all three orbitals. d yz - [Fig. S2 a ] and d xy -orbitals [Fig. S2 b ] [that in the d xz -orbital is shown in Fig. 2 a in the main text] in the C = − N a ( r , E ), through the center of the vortex core forthe d xz - [Fig. S3 a ], d yz - [Fig. S3 b ], and d xy -orbitals [Fig. S3 c ] in the C = − b of the main text, but also in theLDOS of each orbital. Section S4: Electronic structure of a line of defects
We showed in the main text, that when FeSe . Te . is in the C = − U . A similar result is also obtained in the C = − a where we present the three lowest energy states a function of scattering strength U for a linedefect with L = 119 sites. While MZMs also exist in the C = − U where MZMs exist areshown using a gray background), they do so over a smaller range of U than in the C = − . Te . is characterized by a Z topologicalinvariant, the Chern number, the line defect as a one-dimensional system possesses a topological Z classification,similar to the Kitaev chain. As a result, the line defect possesses a single set of MZMs (one at each end) when itis in the topologically non-trivial phase, independent of the Chern number of the underlying topological phase ofFeSe . Te . .In contrast, in the topologically trivial C = 0 phase, the line defect does not possess any MZMs, as follows froma plot of the three lowest energy states in Fig. S4 b as a function of U . These results support our conclusion that FIG. S4. Energies of the three lowest energy states, E , E , E as a function of scattering strength U for a line defect with L = 119 sites in a the topological C = − JS, α, ∆ µ ) = (12 , ,
5) meV, and ∆ d xz rr = ∆ d yz rr = 2 . d xy rr = 1 . b the topological trivial C = 0 phase with ( JS, α, ∆ µ ) = (0 , , d xz rr = ∆ d yz rr = 1 . d xy rr = 0 .
76 meV.The regions of U where MZMs exist are shown using a gray background in a . the existence of MZM at the end of line defects is a sufficient condition for the existence of an underlying topologicalsuperconducting phase on the surface of FeSe . Te . . Section 5: Persistent supercurrents along domain walls
The persistent supercurrent associated with the hopping of an electron from a site r to another site r + δ can becomputed via I r , r + δ = − e ¯ h X σ,σ X a,b =1 Z dω π Re h(cid:0) − t ab r , r + δ δ σσ + iα ( δ × σ ) zσσ (cid:1) g