Tomography of zero-energy end modes in topological superconducting wires
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Tomography of zero-energy end modes in topological superconducting wires
A. A. Aligia,
1, 2, 3
D. P´erez Daroca,
4, 3 and Liliana Arrachea
5, 3 Centro At´omico Bariloche, Comisi´on Nacional de Energ´ıa At´omica, 8400 Bariloche, Argentina Instituto Balseiro, Comisi´on Nacional de Energ´ıa At´omica, 8400 Bariloche, Argentina Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas, 1025 CABA, Argentina Gerencia de Investigaci´on y Aplicaciones, Comisi´on Nacional de Energ´ıa At´omica, 1650 San Mart´ın, Buenos Aires, Argentina International Center for Advanced Studies, Escuela de Ciencia y Tecnolog´ıa,Universidad Nacional de San Mart´ın, 25 de Mayo y Francia, 1650 Buenos Aires, Argentina (Dated: October 2, 2020)We characterize the Majorana zero modes in topological hybrid superconductor-semiconductor wires withspin-orbit coupling and magnetic field, in terms of generalized Bloch coordinates ϕ, θ, δ , and analyze theirtransformation under SU(2) rotations. We show that, when the spin-orbit coupling and the magnetic field areperpendicular, ϕ and δ are universal in an appropriate coordinate system. We use these geometric propertiesto explain the behavior of the Josephson current in junctions of two wires with di ff erent orientations of themagnetic field and / or the spin-orbit coupling. We show how to extract from there, the angle θ , hence providinga full description of the Majorana modes. Introduction . Topological superconductors host Majoranazero modes (MZMs) localized at the edges of the system[1, 2].A significant e ff ort is invested in the detection and manipula-tion of these states, because of their potential application forimplementing topological quantum computation[3, 4]. Quan-tum wires with spin-orbit coupling (SOC), proximity-induceds-wave superconductivity and a magnetic field having a com-ponent perpendicular to the direction of the SOC [1, 6], areone of the most prominent systems. Several experimentalworks investigated realizations of this platform for topolog-ical superconductivity in wires of InAs [7–12].The existence of MZMs leads to signatures in the behaviorof the Josephson current. In the ac case, the periodicity as afunction of the phase bias φ is 4 π , in contrast to the 2 π oneof the ordinary superconductors. This feature is common tothe non-time reversal invariant, [1, 13–36] and time-reversalinvariant [37–48] families.In the topological superconducting phase of the quantumwires proposed in Refs. [1, 6] the zero modes have a non-trivial spin texture [49–51]. A naive expectation suggestsMZMs polarized at both ends of the wire along a directiondetermined by the magnetic field and the SOC. The explicitcalculation of the spin density from the exact solution of themodel Hamiltonian shows that the zero modes also have mag-netization components perpendicular to the magnetic field andthe spin-orbit axis. Remarkably, for perpendicular magneticfield and SOC, the components of the spin polarization per-pendicular to the magnetic field are also perpendicular to theSOC and have opposite signs at the two ends of the wire[1, 49].In the present work we introduce a geometrical characteriza-tion of the MZMs in terms of their Bloch coordinates ϕ, θ as-sociated to the spin orientation and a phase δ . We denote thesethree parameters as generalized Bloch coordinates (GBC). Westudy the model introduced in Refs. [1, 6]. We show that,when the magnetic field and the SOC are perpendicular, thereexist an easy coordinate frame (ECF) where ϕ and δ are uni-versal and can be exactly calculated by means of symmetry arguments, up to a sign that can be obtained from the solu-tion in particular limits. We present a low-energy e ff ectiveHamiltonian to describe the MZMs in Josephson junctionsand show that the angle θ can be inferred from the behavior ofthe Josephson current in suitable designed junctions. Hence,the combination of geometric properties along with the infor-mation of the Josephson current enables a full tomography ofthe MZMs. Model for the wires.
We consider a lattice version of the modelfor topological superconducting wires introduced in Refs. [1,6], with arbitrary orientations of the magnetic field and SOC[52, 53]. The corresponding Hamiltonian is H w = H + H ∆ ,with H = X ℓ c † ℓ (cid:16) − t σ − i ~λ · ~σ (cid:17) c ℓ + + H.c. (1) − X ℓ c † ℓ (cid:16) ~ B · ~σ + µσ (cid:17) c ℓ , H ∆ = ∆ X ℓ c † ℓ ↑ c † ℓ ↓ + H.c. , where ℓ labels sites of a 1D lattice and c ℓ = ( c ℓ ↑ , c ℓ ↓ ) T . ~ B = B ~ n B and ~λ = λ~ n λ , with B , λ ≥ ~ n B and ~ n λ ,respectively. The components of the vector ~σ = (cid:16) σ x , σ y , σ z (cid:17) are the Pauli matrices and σ is the 2 × ~ n λ has a non-vanishing component perpendicular to the direction ~ n B . Theevaluation of topological invariants [54, 55], leads to the fol-lowing analytical expressions for the boundaries | | t | − r | < | µ | < | | t | + r | , B | ~ n λ · ~ n B | < | ∆ | < B , (2)with r = √ B − ∆ . Geometric characterization of the MZMs.
The MZMs of theHamiltonian of Eq. (1) can be written as η ν = γ ν † + γ ν , (3)where ν = L , R denotes the left and right end of the wires,respectively. We assume that the spin of γ † ν is oriented alongthe Bloch vector ~ n ν = (cos θ ν sin ϕ ν , cos θ ν cos ϕ ν , sin θ ν ). Theangles θ ν and ϕ ν in the Bloch sphere, as well as a phase δ ν —defined mod( π ) — are the GBC, which fully characterize theMZM through γ † ν = e i δ ν h cos( θ ν / c † ν ↑ + e i ϕ ν sin( θ ν / c † ν ↓ i . (4)Here, c † ν s are fermionic creation operators associated to thebasis of H w , acting at the ends of the wire (usually including afew sites). Importantly, not only the angles θ ν and ϕ ν , but also δ ν depend on the choice of the reference frame. The change inthe coordinates of ~ n ν under a rotation of the coordinate systemis a routine exercise. The corresponding change of δ ν leads toa function ξ L , R (cid:0) ~ n L , ~ n R (cid:1) , — see Eq. (S5) in the SM [56] —which is a vector potential that depends on ~ n ν but not on δ ν ,generated by a twist between the spin directions [57]. Thequantity δ L , R = δ L − δ R − ξ L , R (cid:0) ~ n L , ~ n R (cid:1) mod( π ) , (5)is invariant under rotations. Notice that the SU(2) invarianceof δ L , R is expected since it appears in the evaluation of ex-pectation values of observables, in particular, the Josephsoncurrent through the closing contact of a ring formed with thewire, which is threaded by a magnetic flux. In addition, thescalar product of the two unit vectors, ~ n L · ~ n R , is also an SU(2)-invariant, which defines the relative tilt of the Bloch vectors ofthe two MZMs. Symmetry-imposed properties of the MZMs of a wire.
We startby noticing that when ~ n λ · ~ y = ~ n B · ~ y =
0, the Hamiltonian isinvariant under inversion (defined by the transformation ℓ ↔ N + − ℓ , for a chain with N sites) and complex conjugation,implying δ R = − δ L , θ R = θ L = θ ϕ R = − ϕ L . (6)The Hamiltonian is also invariant under inversion and simul-taneous change in the sign of λ . For ~ n λ · ~ n B = ~ n B || ~ z , thelatter change of sign can be absorbed in a gauge transforma-tion ˜ c † ℓ ↑ = ic † ℓ ↑ , ˜ c † ℓ ↓ = − ic † ℓ ↓ . Therefore the MZM for ν = R , hasthe same form as the one for ν = L , replacing the operators c † ℓσ at the left end by the ˜ c † ℓσ at the right. Hence, the GBC at thetwo ends are related as δ R = δ L ± π , θ R = θ L = θ, ϕ R = ϕ L + π. (7)This means that the Bloch vectors of the MZMs have compo-nents perpendicular to ~ n B with opposite signs at the two edges,a conclusion that has been previously reached after the explicitcalculation of the wave function in particular frames [49, 51].We conclude that this property does not depend on the choiceof the coordinate frame, since the relative tilt of the spin ori-entations is invariant under rotations. Furthermore, combiningwith the condition of Eq. (6), we identify an ECF: ~ n B || ~ z and ~ n λ || ~ x . In that frame we have δ R = − δ L = ± π/ , ϕ R = − ϕ L = ± π/ , θ R = θ L = θ. (8)In order to conclude the full characterization of the MZMsof Eq. (1) in this frame, we still need to define the signs in Eq. (8) and find the relation between θ ν and the parameters ofthe Hamiltonian of Eq. (1). It is not simple to get analyticalexpressions in general. In what follows we present results forthe case of B ≫ λ, | ∆ | , which will lead to the exact values of ϕ ν and δ ν in all the parameter space. In the SM [56], we show thatthey coincide with the values for these parameters obtainedfrom the calculation of the continuum version of the modelin Ref. [1] in the limit of dominant SOC of the topologicalphase. Limit of dominant magnetic field.
This limit is intuitively re-lated to Kitaev’s model. Albeit, it is important to recall thatthe nature of the MZMs in the present case is quite di ff erent.In fact, they are not fully polarized in the direction of the mag-netic field, but the spins are tilted in opposite directions at thetwo edges, as concluded after Eq. (7).Our aim now is to explicitly calculate the GBC of the twoMZMs as functions of the Hamiltonian parameters λ, ∆ , B , µ when B dominates, in the ECF. To this end, it is useful torewrite the Hamiltonian H w of the wires in the basis that di-agonalizes H in Eq. (1). We introduce the unitary trans-formation in reciprocal space d k + = u k c k ↑ + v k c k ↓ , d k − = − v k c k ↑ + u k c k ↓ , being u k , v k / sgn( λ k ) = √ (1 ± B / r k ) /
2, with r k = q λ k + B , and λ k = λ sin k . This leads to H w = X k , s =+ , − (cid:16) ε k , s d † ks d ks + ∆ Tk d † ks d †− ks (cid:17) + X k ∆ Sk d † k + d †− k − + H.c.(9)being ε k , s = ξ k ∓ r k with ξ k = − t cos k − µ . The pairing in-teraction contains a triplet component with p-wave symmetry ∆ Tk = − λ k ∆ / r k — notice that λ k is an odd function of k — anda singlet one, ∆ Sk = B ∆ / r k .For B ≫ ∆ ≫ λ , the transformed model can be solved analyt-ically with the method of Alase et al. [2–5] (see SM [56], fordetails). For t , ∆ >
0, and µ = − B , ∆ , the results are δ L = − δ R = π , ϕ L = − ϕ R = − π , (10) θ ∼ ∆ B + p ( B − t ) + O ( λ B ) , ~ n B = ~ z , ~ n λ = ~ x . While Eq. (8) gives the values of δ ν and ϕ ν up to a sign, Eq.(10) gives their exact values. Although the calculation wasdone for dominant magnetic field, this result should be validfor continuity in all the topological phase with B , t , ∆ , λ > , µ <
0. The corresponding values for the opposite signs andorientations of these parameters can be deduced by means ofsymmetry arguments [62].It is important to highlight that the previous results and ap-propriate SU(2) rotations permit to obtain exactly δ ν and ϕ ν inany coordinate system for any value of the parameters of Eq.(1) with ~ n λ · ~ n B =
0, while θ needs an explicit calculation. Ourgoal now is to show that this angle can be inferred from thebehavior of the Josephson current in suitably designed junc-tions. Josephson current.
In order to calculate the Josephson currentwe consider two wires w1 and w2 with di ff erent phases φ , φ of the pairing potentials, related as φ − φ = φ and connectedby a tunneling term, as indicated in the sketches of Figs. 2and 3. Gauging out the dependence on φ in the operators ofthe wires, the Hamiltonian for the full system reads H ( φ ) = H w1 + H w2 + H c ( φ ), where H w1 , H w2 have the same structureas in Eq. (1). The connecting term reads H c ( φ ) = t c X σ = ↑ , ↓ (cid:16) e i φ/ c † ,σ c ,σ + H.c. (cid:17) , (11)with 1 and 2 denoting, respectively, the site at the right / leftend of w1 / w2. We can calculate the current numerically asdescribed in the SM [56]. In the topological phase, however, asimple description based on the coupling of the MZMs accu-rately explains the Andreev spectrum and the Josephson cur-rent. This is because, in a topological junction, Andreev statesare formed from the hybridization of the MZMs [19, 41, 45]. FIG. 1. (a) Reference frame with ~ z ′ along the Bloch vector of theMZM of w2. (b) Bloch vector of the MZMs of the two wires. (c)Reference frame of H wj , j = , In what follows we derive the low-energy e ff ective Hamilto-nian H e ff that describes the hybridization of the MZMs. Im-portantly, we consider di ff erent magnetic fields and SOC ori-entations in the two wires. H e ff takes a particularly simpleform if the quantization axis is chosen in the direction of theBloch vector of one of the MZMs next to the junction, whichwe choose to be ~ n . In the basis where ~ n ≡ ~ z ′ — see Fig. 1 (a)— the spin down operators of the sites nearest to the junctioncontribute only at high energies, while the low-energy compo-nent is precisely the contribution of the MZM. Concretely, wecan substitute the fermionic operators at the ends of the wiresby their projection on the MZMs, c ′ ↑ ≃ a e i δ ′ cos θ ′ ! η , c ′ ↑ ≃ a e i δ ′ η , (12)where θ ′ is the angle between ~ n and ~ n — see Fig. 1 (b) —and δ ′ , δ ′ are the corresponding phases. a i are real numbers, a i ≤ H e ff ( φ ) = t J ( θ ′ )2 sin (cid:18) φ + δ ′ − δ ′ (cid:19) i η η , (13) t J ( θ ′ ) = t c a a cos θ ′ ! , (14) which is solved by defining a fermion d = ( η + i η ) / i η η = d † d −
1. The ground-state energy is E e ff ( φ ) = − | t J ( θ ′ ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin φ ′ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (15)where φ ′ = φ + (cid:16) δ ′ − δ ′ (cid:17) . The Josephson current is J e ff ( φ ) = e ~ dE e ff ( φ ) d φ = − e | t J ( θ ′ ) | ~ cos φ ′ ! sgn ( sin φ ′ !) . (16)Performing the rotation sketched in Fig. 1 (see SM [56] fordetails), we can express this current in terms of the GBC ofthe MZMs of w1 and w2 next to the junction in the laboratoryframe through φ ′ = φ + (cid:0) δ − δ − ξ , (cid:1) , (17)where δ and δ are the corresponding phases and ξ , = arctan sin ( ϕ − ϕ )cos ( ϕ − ϕ ) + cot (cid:16) θ (cid:17) cot (cid:16) θ (cid:17) . (18)The di ff erent angles are indicated in Fig. 1. We would like tostress that all the quantities that determine the behavior of theJosephson current are SU(2)-invariant, as explicitly shown inRef. [56]. In particular, θ ′ does not depend on the referenceframe while δ − δ − ξ , is an invariant akin to Eq. (5) andfrom Eq. (17) we clearly see that this quantity plays the roleof a geometric phase that modifies the magnetic flux.The Josephson current of Eq. (16) has a jump at φ ′ = ff erent fermionparity. If parity is conserved, the typical 4 π -periodicity oftopological junctions is obtained. In the case of junctions ofwires with the same orientation of the magnetic field and SOC, δ − δ = ± π , as given by Eq. (7), and the jump occurs at φ = π .However, in junctions of wires having di ff erent orientations of ~ n B and ~ n λ , this may take place at other values of φ . In whatfollows, we analyze junctions of wires with di ff erent configu-rations of these vectors with the aim of using the behavior ofthe Josephson current to extract information of the MZMs. Junctions with tilt in the SOC.
We consider the same orien-tation of the magnetic field in both wires, but a tilt β λ in theorientation the SOC, i.e. ~ n λ, · ~ n λ, = cos β λ . This can berealized with a junction where the wires are placed on the su-perconducting substrate forming an angle β λ , as in the sketchof Fig. 2, where we also indicate the ECF for w2 ( ~ n B || ~ z and ~ n λ, || ~ x ). We focus on ∆ > µ <
0, in which case Eqs. (10)give δ and ϕ , while δ and ϕ can be be also derived fromthese Eqs. by performing a rotation of β λ around ~ z . This leadsto θ = θ = θ , ϕ − ϕ = π − β λ and δ = δ + ( π + β λ ) / φ ′ = φ − π − β λ + sin ( β λ )cos ( β λ ) − cot (cid:16) θ (cid:17) . (19)Therefore, from the position of the jump in the current as afunction of the flux it is possible to extract the angle θ be-tween the Bloch vector of the MZMs with respect to ~ n B . Thiscompletes the full description of the MZMs at both sides ofthe junction. In Fig. 2, we show results calculated with H e ff ,and by exact diagonalization of the full Hamiltonian H ( φ ) (seeRef [56] for technical details). Both calculations are in excel-lent agreement and also agree with results reported in the limitof weak SOC in the continuum model [63] and in the limit oflarge B [64]. FIG. 2. Josephson current as a function of the flux for several valuesof the angle β λ between the orientation of the SOC in the two wires,keeping ~ n B · ~ n λ = J e ff calculated using H e ff . Parameters are t = B = λ = ∆ = µ = − Junctions with tilt in the magnetic field.
We now focus onthe case where the SOC is equally oriented in the two wires, ~ n λ, = ~ n λ, = ~ x , while the orientation of the magnetic field ~ n B , is tilted by an angle β B with respect to ~ n B , || ~ z . We start with thecase ~ n λ, j · ~ n B , j = , j = ,
2, which can be realized in the twoconfigurations sketched in Fig. 3. As before, for ∆ > µ < δ and ϕ . On the other hand,the corresponding values of δ and ϕ can also be obtainedfrom these Eqs. by performing a rotation of angle β B around ~ x . These are δ = − π/ ϕ = ( π/ θ − β B )] and θ = θ − β B . Hence, the Josephson current is given by Eq.(16) with φ ′ = φ − π . Therefore, the shape of the function J ( φ )is the same for all values of β B , displaying a jump at φ = π .However, the magnitude of the current depends on the angles θ and β B according to Eq. (14), with θ ′ = θ − β B . Thisis illustrated in Fig. 3 and has a simple interpretation. For β B = ~ n and ~ n have the same z component, θ = θ , zero x component and opposite y components [see Eqs. (7) and (10)].Rotating ~ n B , around the x axis, ~ n is moved towards ~ n andboth vectors coincide when β B = θ . This angle correspondsto the maximum of t J ( θ ′ ), hence, the maximum of J ( φ ) atfixed φ . In addition, for fixed fermion parity, the Josephsoncurrent is 4 π -periodic in β B , in agreement with Ref. 20.When the magnetic field is tilted in such a way that there isa finite component along the direction of the SOC there is nosimple analytical expression relating the tilt in the magneticfield and the orientation of the Bloch vector of the MZMs andwe must rely on the full expressions given by Eqs. (16), (17) FIG. 3. (a) Configurations of wires corresponding to wires with atilt β B in the orientation of the magnetic field with ~ n B · ~ n λ =
0. (b)Amplitude of the Josephson current t J as a function of β B for ~ n B · ~ n λ =
0. (c) Josephson current as a function of φ for ~ n λ, = ~ n λ, , ~ n B , · ~ n λ, = ~ n B , · ~ n λ, = cos β λ B and all vectors in the same plane. Solid lines:numerical results. Symbols: J e ff calculated using H e ff . Parameters are t = B = λ = ∆ = µ = − and (S5). In Fig. 3 (c) we show the Josephson current J ( φ )for the case in which ~ n λ, = ~ n λ, , ~ n B , · ~ n λ, = ~ n B , istilted keeping it perpendicular to ~ n B , ∧ ~ n λ, , and forming anangle β λ B with ~ n λ, i . Without tilting J ( φ ) = J ( φ ) presents jumps at φ , π as in the caseof wires with SOC perpendicular to B but with a relative tilt,analyzed in Fig. 2. It is found again an excellent agreementbetween the description in terms of the e ff ective Hamiltonian H e ff ( φ ) and the numerical solution of the exact Hamiltonian(see Ref. [56] for details). For small β λ B , the topologicalphase is lost in w1 leaving its place to a non-topological phase— see of Eq. (2) — which is gapless for a wide parameterrange. There, H e ff ( φ ) is no longer useful and the numerical so-lution of H ( φ ) is necessary, which leads to a smooth behaviorof J ( φ ), typical of ordinary superconductors with small ampli-tude, albeit preserving some peculiar features of the topologi-cal phase, like a non-vanishing current for φ = Conclusions.
We have characterized the MZMs of topolog-ical superconducting wires with SOC and magnetic field interms of GBC ( ϕ, θ, δ ). We have analytically calculated ϕ, δ for the ECF where ~ n B ≡ ~ z and · ~ n λ ≡ ~ x . We have also de-rived the transformation of these quantities under changes ofthe reference frame. We used these results to derive exact ex-pressions for the Josephson current in wires having relativetilts in the orientations of the SOC and magnetic fields. Weshowed that for suitable configurations of the junctions, thedc Josephson current provides the necessary information tofully reconstruct the structure of the MZMs. These resultsmay be useful in the experimental implementation of quan-tum tomography of MZMs through the information providedby the dc Josephson current. The dc regime could be reached,for instance, by adiabatically switching on the magnetic fieldor by rotating it from the gapless non-topological phase ofnearly parallel spin-orbit coupling and magnetic field. This ispossible within the present experimental state of the art of thehybrid superconducting-semiconducting wires we have stud-ied [12]. Interestingly, this regime is free from the problemof the time-scales introduced by the poor equilibration of theMZMs which a ff ect readout processes of dynamical e ff ects[67–69]. Acknowledgments.
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CHANGE OF REFERENCE FRAMEGeneral case
The spin of the fermionic creation operators defined in Eq. (4) of the main text is expressed in a given reference frame O ,determined by the quantization axis of the Hamiltonian H w . Here, we analyze the transformation of the spin under a change ofbasis to a rotated frame O ′ . We remind the reader that under an active transformation, (a rotation of the physical system an angle α around de unit vector ~ v keeping the coordinates unchanged) a state | ψ i = ( ac †↑ + bc †↓ ) | i ), becomes R ~ v ( α ) | ψ i , where the SU(2)matrix R ~ v ( α ) is R ~ v ( α ) = cos (cid:18) α (cid:19) − i sin (cid:18) α (cid:19) ~ v · ~σ. (S1)If, instead, the physical system is fixed and the rotation is applied to the coordinate system O to transform it to O ′ , the state inthe new basis is | ψ ′ i = R − ~ v ( α ) | ψ i . Inverting the previous transformation we obtain for the creation operators c †↑ = (cid:18) cos (cid:18) α (cid:19) + e − i α v z (cid:19) ( c ′↑ ) † + sin (cid:18) α (cid:19) (cid:16) v y − iv x (cid:17) ( c ′↓ ) † c †↓ = − sin (cid:18) α (cid:19) (cid:16) v y + iv x (cid:17) ( c ′↑ ) † + (cid:18) cos (cid:18) α (cid:19) + v z e i α (cid:19) ( c ′↓ ) † , (S2)where v j is the component of ~ v in the direction j . Replacing this transformation in Eq. (4) of the main text we obtain theexpression of the creation component of the MZM in the rotated frame O ′ : γ † ν = e i δ ν h A ( c ′↑ ) † + B ( c ′↓ ) † i , A = cos (cid:18) θ ν (cid:19) (cid:18) cos (cid:18) α (cid:19) + e − i α v z (cid:19) − e i ϕ ν sin (cid:18) θ ν (cid:19) sin (cid:18) α (cid:19) (cid:16) v y + iv x (cid:17) , B = cos (cid:18) θ ν (cid:19) sin (cid:18) α (cid:19) (cid:16) v y − iv x (cid:17) + e i ϕ ν sin (cid:18) θ ν (cid:19) (cid:18) cos (cid:18) α (cid:19) + v z e i α (cid:19) . (S3)Expressing γ † ν in the same form as Eq. (4) of the main text we get γ † ν = e i δ ′ ν " cos θ ′ ν ! ( c ′ ν ↑ ) † + e i ϕ ′ ν sin θ ′ ν ! ( c ′ ν ↓ ) † , (S4)from where the parameters in the frame O ′ can be obtained. Writing A = | A | e i ξ ν , it is clear that δ ′ ν = ξ ν + δ ν being ξ ν = arctan Im A Re A ! = arctan − cos( θ ν ) sin( α ) v z − sin( ϕ ν ) sin( θ ν ) sin( α ) v y − cos( ϕ ν ) sin( θ ν ) sin( α ) v x cos( θ ν ) cos( α )(1 + v z ) − cos( ϕ ν ) sin( θ ν ) sin( α ) v y + sin( ϕ ν ) sin( θ ν ) sin( α ) v x . (S5)We see that in general, the phases δ ν transform in a non trivial way under rotations or a change in coordinates. Instead, asexpected, the directions ~ n ν (defined by θ ν and ϕ ν ) transform as ordinary vectors. Comparing Eqs. (S3) and (S4) we see that B / A = | B / A | e i ϕ ′ ν or ¯ AB = | ¯ AB | e i ϕ ′ ν ( ¯ A denotes the complex conjugate of A ), and | A | = cos( θ ′ ν / θ ν and ϕ ν are easilyobtained: θ ′ ν = | A | ) = | B / A | ) . (S6) ϕ ′ ν = arctan Im( ¯ AB )Re( ¯ AB ) ! , (S7) Derivation of Eq. (18)
In the main text, we evaluate the Josephson current through the connection between wires w1 and w2, with the parameters definedwith respect to a frame O ′ with ~ n || ~ z ′ , being ~ n the direction of the polarization of the MZM of the wire w2 that hybridizes withthe MZM of the wire w1 in the junction. The consequent expression for the Josephson current –see Eq. (16) of the maintext– depends on the Josephson phase φ , as well as on the phases δ ′ and δ ′ of the two hybridized MZMs, which depend onthe reference frame. Since we know the values of these phases, given the values of the parameters of the Hamiltonians for thewires only when the latter are written in the reference frame O where ~ n B || ~ z and ~ n λ || ~ x –see Eq. (10)– we need to implement atransformation between O ′ and O . The concrete transformation is sketched in Fig. 1 of the main text. In the formalism describedabove, this corresponds to a rotation R ~ v ( α ) that transforms O to O ′ such that R ~ v ( α ) ~ n = ~ z . We choose ~ v in the direction of ~ n ∧ ~ z ,so that it is perpendicular to both ~ n and ~ z , hence a positive rotation in the angle α = θ moves ~ n to ~ z . The components of theunit vector ~ v become v x = sin( ϕ ), v y = − cos( ϕ ), v z = . Replacing these values in Eq. (S5) for ν =
2, we see that the numerator vanishes, and therefore ξ = δ ′ = δ . Instead for ν = δ ′ = ξ + δ , with ξ = arctan sin ( ϕ − ϕ )cot (cid:16) θ (cid:17) cot (cid:16) θ (cid:17) + cos ( ϕ − ϕ ) . (S8)Combining the δ ′ − δ ′ , we get Eqs. (17) and (18) of the main text, with ξ , ≡ ξ , given above. SU(2) invariance of d In this section we prove the SU(2) invariance of the quantity d = δ − δ − ξ , , ξ , = arctan sin ( ϕ − ϕ )cot (cid:16) θ (cid:17) cot (cid:16) θ (cid:17) + cos ( ϕ − ϕ ) , (S9)mod( π ) for any two fermions of the form of Eq. (4) of the main text [same as Eq. (S4) without the superscript prime]. The factthat the quantity is defined mod( π ) means that the branch and discontinuities of the arctan are unimportant. The invariance of d is expected, since in the particular case discussed in Section , it enters the equation of the Josephson current through φ ′ [seeEqs. (17) of the main text] and the current is an observable. Here, we prove it explicitly for the general case.As is well known, any SU(2) rotation can be obtained by composing infinitesimal rotations around three mutually perpendicularaxis and the generators of these rotations ( i σ x , i σ y and i σ z in Section ) form a basis of the Lie algebra of the group. Twogenerators are enough for our purposes because the third one is the commutator of the other two times a factor. The invarianceof d under any rotation around z immediately verified since θ and θ , as well as δ − δ and ϕ − ϕ are unchanged underthis transformation. Therefore, it remains to prove that d is invariant under a rotation through an axis perpendicular to z . Wechoose the y axis in a reference frame with ϕ = π/ + ϕ with the x axisin the original reference frame).We use the results of Section for ~ v = ~ y , ϕ = α → ff erential d α of the angle of the rotation.In particular we replace cos( α/ ≃ α/ ≃ d α/
2. From Eq. (S5) we obtain the change in the phase under theinfinitesimal rotation, d δ ν = δ ′ ν − δ ν , d δ ν = d arctan( δ ν ) = − d α sin( ϕ ν ) sin( θ ν )cos( θ ν ) . (S10)Evaluating explicitly for ν = , d δ d α = −
12 tan (cid:18) θ (cid:19) sin( ϕ ) , d δ d α = . (S11)From Eqs. (S3) and (S6) we getcot θ ′ ν ! = cot (cid:18) θ ν (cid:19) − tan( θ ν ) cos( ϕ ν ) d α + cot( θ ν ) cos( ϕ ν ) d α = cot (cid:18) θ ν (cid:19) (cid:18) − cos( ϕ ν ) d α (cid:18) tan (cid:18) θ ν (cid:19) + cot (cid:18) θ ν (cid:19)(cid:19)(cid:19) (S12) d cot (cid:18) θ ν (cid:19) = − cot (cid:18) θ ν (cid:19) (cid:18) tan (cid:18) θ ν (cid:19) + cot (cid:18) θ ν (cid:19)(cid:19) cos( ϕ ν ) d α. (S13)Using that for any function r , dr = rdr we obtain d cot( θ / d α = − cos( ϕ )2 + cot (cid:18) θ (cid:19) ! (S14) d cot( θ / d α = − + cot (cid:18) θ (cid:19) ! (S15)The change of the angles d ϕ ν = ϕ ′ ν − ϕ ν are obtained using Eqs. (S3) and (S7)tan( ϕ ′ ν ) = sin( ϕ ν ) sin( θ ν ) cos( θ ν )cos( ϕ ν ) sin( θ ν ) cos( θ ν ) + (cos( θ ν ) − sin( θ ν ) ) d α (S16) d tan( ϕ i ) = − d α sin( ϕ i )2 cos( ϕ i ) (cid:18) cot (cid:18) θ i (cid:19) − tan (cid:18) θ i (cid:19)(cid:19) . (S17)Using d tan( r ) = (1 + tan( r ) ) dr d ϕ d α = − sin( ϕ ) (cid:18) cot (cid:18) θ (cid:19) − tan (cid:18) θ (cid:19)(cid:19) , d ϕ d α = . (S18)The remaining task to prove that dd / d α = ξ , = arctan( q ), where q = sin( ϕ )cos( ϕ ) + cot( θ ) cot( θ ) (S19)To simplify the algebra we use the notation c = cos ( ϕ ), s = sin ( ϕ ) and x i = cot ( θ i /
2) . With this notation the equations (S14),(S15), (S18) and (S19) become dx d α = − c (cid:16) + x (cid:17) , dx d α = − (cid:16) + x (cid:17) , d ϕ d α = − s x − x ! , q = sc + x x = sh . (S20)Di ff erentiating the last expression we get dqd α = c d ϕ d α h − s (cid:16) − s d ϕ d α + dx d α x + x dx d α (cid:17) h = d ϕ d α + cx x d ϕ d α − s (cid:16) dx d α x + x dx d α (cid:17) h (S21)and replacing Eqs. (S20) above, we obtain dqd α = s x + cx x + x x h (S22)On the other hand, from Eq. (S19) d ξ , d α = dqd α + q , with 1 + q = + s h = + cx x + x x h , (S23)and using Eq. (S22) we obtain d ξ , d α = s x = sin( ϕ )2 cot ( θ / . (S24)Finally, di ff erentiating Eq. (S9) and expressing it as dd d α = d δ d α − d δ d α + d ξ , d α , (S25)and substituting Eqs. (S11), (S19), and (S24) we get the desired result dd d α = . (S26) STRUCTURE OF THE MAJORANA STATES IN SOME LIMITING CASESSolution for dominant spin-orbit coupling with ~ n B ≡ ~ x and ~ n λ ≡ ~ z We apply the formalism of Section to the exact solution of the continuum version of the model of Eq. (1) of the main text,calculated in Ref. [S1]. A very simple expression was found for the left and right MZMs in the region of parameters where thespin-orbit coupling dominates, assuming ∆ > λ ≫ t , B > ∆ , µ ∼ µ ∼ − t in the lattice version). From there,we can easily examine the properties summarized in Eqs. (6) to (8) of the main text. The solution, as expressed in Ref. [S1]reads η L = (cid:16) ψ L , ↑ − i ψ L , ↓ + i ψ † L , ↓ + ψ † L , ↑ (cid:17) , η R = (cid:16) ψ R , ↑ + i ψ R , ↓ − i ψ † R , ↓ + ψ † R , ↑ (cid:17) , (S27)where the labels L , R in the field operators indicate that they are evaluated at spacial coordinates corresponding the the L , R ends,respectively. In order to make an explicit comparison to Eqs. (7) and (8), we need to perform a rotation of = π/ y -axis, corresponding to α = − π/ ~ v = (0 , ,
0) in Eq. (S2), and a change in the sign of λ which changes the sign of both δ and φ (see Ref. 53 of the main text). Under these transformations, the above operators transform to γ ′ L = e i π/ (cid:16) ψ ′ L , ↑ − i ψ ′ L , ↓ (cid:17) , γ ′ R = e − i π/ (cid:16) ψ ′ R , ↑ + i ψ ′ R , ↓ (cid:17) , (S28)in full agreement with Eqs. (7) and (8) of the main text. Solution for dominant magnetic field, B ≫ ∆ ≫ λ with ~ n B ≡ ~ z and ~ n λ ≡ ~ x In this Section, we obtain analytically the zero-energy modes at the ends of a finite long chain for 0 < λ ≪ ∆ ≪ t < B and µ ∼ − B . We start with the Hamiltonian Eq. (9) of the main text, which to linear order in λ/ B takes the form H = X k , s =+ , − ( − t cos k − µ ) d † ks d ks − B X k ( d † k + d k + − d † k − d k − ) + X k ∆ S d † k + d †− k − − ∆ T sin k X s =+ , − d † ks d †− ks + H.c. , (S29)with ∆ S = ∆ and ∆ T = λ ∆ / B . Transforming Fourier to Wannier functions localized at any site j , d † js = P k e i jk d † ks / √ N , theHamiltonian becomes H = − t X j , s =+ , − ( d † js d j + s + H.c.) − B X j ( d † j + d j + − d † j − d j − ) + X j ∆ S d † j + d † j − + i ∆ T X s =+ , − d † j + s d † js + H.c. . (S30)For later use we note that in the real-space basis, to linear order in λ/ B the transformation introduced in the main text to defineEq. (9) from Eq. (1) reads d † j + = c † j ↑ − i λ B (cid:16) c † j + ↓ − c † j − ↓ (cid:17) , d † j − = c † j ↓ + i λ B (cid:16) c † j + ↑ − c † j − ↑ (cid:17) . (S31)In order to eliminate the imaginary unit in the coe ffi cient i ∆ T of the triplet superconductivity in Eq. (S30) we define˜ d † j + = e i π/ d † j + , ˜ d † j − = e − i π/ d † j − (S32)and the triplet superconducting term takes the form ∆ T P j ( ˜ d † j + + ˜ d † j + − ˜ d † j + − ˜ d † j − + H.c.) . We obtain the solutions with zero energy of Eq. (S30) for a finite long chain of N sites using the method of Alase et al. [S2, S3]in the form used previously by some of us.[S4] As in the Nambu formalism, the operators are mapped to one particle states,using the following notation ˜ d js ↔ | js i , ˜ d † js ↔ | js i . (S33)The desired solutions are linear combinations of states of the form (not normalized) | zsi i = N X j = z j − | jsi i , s = ± , i = , , (S34)where z is a complex number with | z | < >
1) for the Majorana zero mode localized at the left (right) of the chain. Sinceboth modes are related by symmetry we focus here on the left mode only. The possible values of z are obtained from the bulkequation P B ( H − E ) | ψ i =
0, where in our case E = P B = P N − j = P si | jsi ih jsi | . In the basis | z , + , i , | z , + , i , | z , − , i , | z , − , i ,the matrix P B H takes the form P B H = − a − b ∆ S b a − ∆ S − ∆ S − a + B b ∆ S − b a − B , a = µ + B + t z + z ! , b = ∆ T z − z ! (S35)and its determinant is Det( P B H ) = ( a − b ) h ( a − B ) − b i − h a (2 B − a ) + b i ∆ S + ∆ S . (S36)To linear order in ∆ S / B , we can neglect ∆ S above and the four roots z k of Det( P B H ) = | z k | < ffi cients of the eigenvectors | e k i = P si β ksi | jsi i , for µ ′ = µ + B ≪ t are z = ic − µ ′ t + ∆ T ) , β + = β + = √ , β − = β − = , c = r t − ∆ T t + ∆ T , z = ¯ z = − ic − µ ′ t + ∆ T ) , β si = β si , z = B − µ ′ t + ∆ T ) − s B − µ ′ t + ∆ T ) ! − t − ∆ T t + ∆ T , β + = β + = , β − = β − = √ , z = B − µ ′ t − ∆ T ) − s B − µ ′ t − ∆ T ) ! − t + ∆ T t − ∆ T , β + = β + = , - β − = β − = √ . (S37)The zero mode state has the form | f i = P k α k | e k i , and the coe ffi cients are obtained from the boundary equation, which in ourcase takes the form P H | f i =
0, where P = P si | si ih si | . It is easy to see that the form of the matrix P H is similar to Eq.(S35) without the terms in 1 / z (due to the fact that there are no sites at the left of site 1), and z replaced by z k . Taking for thebasis state | b i , the four states | z , + , i , | z , + , i , | z , − , i , | z , − , i , h b | P H | f i = X k h − (cid:0) µ ′ + tz k (cid:1) β k + − ∆ T z k β k + + ∆ S β k − i α k = , X k h ∆ T z k β k + + (cid:0) µ ′ + tz k (cid:1) β k + − ∆ S β k − i α k = , X k h − ∆ S β k + + (cid:0) B − µ ′ − tz k (cid:1) β k − − ∆ T z k β k − i α k = , X k h ∆ S β k + + ∆ T z k β k − − (cid:0) B − µ ′ − tz k (cid:1) β k − i α k = . (S38)Using Eqs. (S37) and calling C = ∆ S ( α + α ) , C = B − µ ′ − tz − ∆ T z , C = B − µ ′ − tz + ∆ T z , (S39)the last two Eq. (S38) can be written as − C + C α − C α = , C − C α − C α = . (S40)The solution of this equation is α = α = C C , C = B − µ ′ + s(cid:18) B − µ ′ (cid:19) − t + ∆ T , (S41)where the expression of C has been obtained using Eqs. (S37) and (S39). From Eqs. (S37), (S39), and (S41) it is easy to seethat the contribution of α and α to the first two Eqs. (S38) is either of order ∆ S or zero. Therefore, it can be neglected to firstorder in ∆ S leading to X k = (cid:0) µ ′ + tz k + ∆ T z k (cid:1) α k = . (S42)Using the expressions for z k , the solution can be written in the form α = e i ω √ , α = e − i ω √ , ω = arctan " ( t + ∆ T ) µ ′ t + ∆ T ) c . (S43)Using | f i = P k α k | e k i , | e k i = P si β ksi | jsi i , Eqs. (S32), (S33), (S34), (S37), (S41), and (S43), we obtain the final expression ofthe Majorana zero mode at the left end of the chain (except for a normalization factor) η L = N X j = " Re( e i ω z j − ) (cid:16) e i π/ d † j + + e − i π/ d j + (cid:17) + ∆ S cos ω C z j − (cid:16) e − i π/ d † j − + e i π/ d j − (cid:17) . (S44)The amplitude of the mode is maximum at the first site and decreases exponentially for sites inside the chain with di ff erent decayrates for spin + and − .In order to make contact to Eqs. (7) and (8) , we need to express η L in terms of the operators c j ,σ of the original model. To thisend, we introduce the representation of Eqs. (S31) in to Eq. (S44) and focus on the limit λ →
0. The projection of Eq. (S44) onthe first site of the lattice reads η L = γ L + γ † L with γ † L ∼ e i π/ " c † , ↑ + ∆ S C e − i π/ c † , ↓ . (S45)We see that this solution has the structure of Eq. (4) with δ L = π/ , ϕ L = − π/ , tan( θ L / = ∆ S C + O ( λ B ) (S46)The results for δ L and ϕ L are valid for any value of the parameters in the topological phase with ∆ , t > µ <
0, with ~ n B ≡ ~ z , ~ n λ ≡ ~ x , and are in full agreement with the result of the continuum model discussed in Section . The value of θ L is however verysensitive to the values of the parameters of the Hamiltonian. As explained in the main text, our goal is to show that this anglecan be inferred from the behavior of the Josephson current in suitably designed junctions.In contrast to δ L and ϕ L (obtained for ~ n B || ~ z and ~ n λ || ~ x ), θ depends on the site. As a consequence for other directions of ~ n B and ~ n λ (orother systems of coordinates), δ L and ϕ L also depend on the site, since their transformation properties depend on θ . Neverthelessfor the calculation of the Josephson current we are only interested in the first and the last site of the chain. NUMERICAL CALCULATION OF THE JOSEPHSON CURRENT
The Hamiltonian of the system describing two wires and a Josephson junction is H ( φ ) = H w1 + H w2 + H c ( φ ) , H c = t c X σ = ↑ , ↓ (cid:16) e i φ/ c † R ,σ c L ,σ + H.c. (cid:17) , (S47)where H wi , i = ,
2, describe two topological superconducting wires, w1 at the left of w2, described by Eq. (1) of the main text,and with a di ff erence φ = φ − φ between the superconducting phases, with φ = π corresponding to one superconducting fluxquantum. The subscript 1 R (2 L ) indicates the last (first) site of w1 (w2). Denoting as N = P js c † , js c , js the operator of totalnumber of particles of w1, the current flowing through the junction from left to right is J ( φ ) = h e dN L dt i = h ie ~ [ N , H ] i = − et c ~ X σ Im h e i φ/ D c † R σ c L σ Ei . (S48)The above expectation value can be numerically calculated given the eigenmodes of the Hamiltonian which correspond toannihilation operators that satisfy [ Γ ν , H ] = λ ν Γ ν , with positive λ ν . The relevant part of these operators have the form Γ ν = X σ h A ν R σ c † R σ + A ν L σ c † L σ + B ν R σ c R σ + B ν L σ c L σ i + ..., (S49)where ... denotes the contribution of operators at site di ff erent from 1 R and 2 L . The coe ffi cients are known from the numericaldiagonalization. Inverting Eq. (S49) we have c † R σ = X ν (cid:16) A ν R σ Γ ν + B ν R σ Γ † ν (cid:17) , c L σ = X ν (cid:16) A ν L σ Γ † ν + B ν L σ Γ ν (cid:17) . (S50)Replacing in Eq. (S48) and taking into account that in the ground state the only non vanishing expectation values of a productof two Γ ν and / or Γ † ν ′ operators is D Γ ν Γ † ν E =
1, we obtain J ( φ ) = − et c ~ Im e i φ/ X νσ A ν L σ A ν R σ . (S51)An alternative expression can be derived from the numerical derivative with respect of the flux of the eigenvalues λ ν . Thissimplifies the diagonalization procedure at the cost of introducing numerical errors in the di ff erentiation.Noting that only H c depends on the flux, Eq. (S48) can be also related to the ground state energy E g as follows J ( φ ) = d h H i d φ = t c X σ D ie i φ/ c † R ,σ c L ,σ + H.c. E = e ~ dE g ( φ ) d φ . (S52)In turn, except for an additive constant, E g can be calculated as half the sum of all positive eigenvalues of the Hamiltonianmatrix S = P ν λ ν . The latter procedure can be justified by using symmetry arguments[S5] as follows. Considering the chargeconjugation operation C , acting as c † i , j σ ↔ c i , j σ plus complex conjugation. It is easy to see that CHC = − H − µ N , where N = N + N is the total number of particles. Taking the number of particles as fixed h N i , we can write this equation in the form˜ H ′ = CH ′ C = − H ′ , H ′ = H + µ h N i , which can be considered as change of representation of the same states. Since both ˜ H ′ and H ′ have the same many-body spectrum but inverted, the maximum energy of H ′ , which we denote as E ′ M and the ground state E ′ g are related by E ′ M = − E ′ g . On the other hand the state of maximum energy is obtained applying all the creation operators Γ † ν to the ground state. Therefore E ′ M − E ′ g = S = P ν λ ν , which leads to E g = − P ν λ ν − µ h N i . Hence, J ( φ ) = − e ~ d P ν λ ν d φ . (S53)We have verified that the results of Eq. (S51) and (S53) coincide within numerical precision. NUMERICAL CALCULATION OF δ ν , θ ν AND ϕ ν The Majorana modes that enter the e ff ective low-energy Hamiltonian H e ff for the Josephson current [see Eqs. (12) and (13) ofthe main text] have the form η ν = γ † ν + γ ν , γ † ν = a ν e i δ ν h cos( θ ν / c † e ↑ + e i ϕ ν sin( θ ν / c † e ↓ i + ..., (S54)where a ν is a real number that can be chosen positive, the subscript e refers to the site at the end of the chain (first or last) wherethe Majorana mode is localized and ... refers to the contribution of other sites which are not important for H e ff . The normalization η ν = a ν ≤ γ ν can be expressed as acombination of two Majorana operators η ν and ˜ η ν of the form, γ ν = ( η ν + i ˜ η ν ) / γ † ν = ( η ν − i ˜ η ν ) /
2, of which only η ν contributesat low energy, γ † ν ≃ η ν / ff ective mixing between the Majorana at the left ( L ) and right ( R ) end of the chain which byhermiticity should be proportional to i η L η R . Therefore, the one-particle eigenstates of lowest absolute value correspond to thefermions f = e i ζ ( η L + i η R ) / f † which diagonalize i η L η R . The phase ζ is unknown. Thus, for the end we are interested ( L or R ) we can write, including explicitly only the operators related with that end f = e i ζ ′ η ν + ... = Ac † e ↑ + Bc e ↑ + Cc † e ↓ + Dc e ↓ + ... (S55)where the coe ffi cients at the right side are determined by the numerical calculation. Comparing with Eq. (S54) we see that theparameters of η ν can be obtained from the following equations a ν = q(cid:0) | A | + | C | (cid:1) , δ ν =
12 arctan
Im[ A / B ]Re[ A / B ] ! , θ ν = | C || A | ! , ϕ ν = arctan Im[ C / A ]Re[ C / A ] ! . (S56) . . . . . . . a ν ∆ θ ν /π λ = 0 . λ = 1 . λ = 2 . .
2∆ = 1 .
0∆ = 2 . λ FIG. S1. Parameters θ ν (top panels) and a ν (bottom panels) as a function of ∆ ( λ ) for several values of λ ( ∆ ) and t = B = µ = − ~ n B ≡ ~ z and ~ n λ ≡ ~ x . The dependence of θ ν and a ν with the parameters, obtained numerically as described above is shown in Fig. S1. Both determinethe coe ffi cient t J of the Josephson current. The amplitude a tends to zero at the borders of the topological region. Curiously, ithas a maximum for intermediate values of λ . The angle θ tends to 0 or π (depending on the sign of ~ n B · ~ z ) when both λ and ∆ tend to zero as anticipated above.As explained in the main text, for perpendicular directions of the magnetic field and spin-orbit coupling, δ ν and ϕ ν can bedetermined from symmetry arguments and analytical calculations. In particular, for ~ n B || ~ z and ~ n λ || ~ x , δ L = − δ R = π , ϕ L = − ϕ R = − π . (S57)In Fig. S2 we show how these parameters change when the orientation of the spin-orbit coupling ~ n λ is rotated keeping it in the xy plane. We can see that the absolute values of δ ν and φ ν increase, keeping δ L = − δ R and ϕ L = − ϕ R , as anticipated in the maintext by symmetry arguments. [S1] Y. Oreg, G. Refael, and F. von Oppen, Helical Liquids and Majorana Bound States in Quantum Wires, Phys. Rev. Lett. − . − . . . − . . π/ . π . π . π π/ δ/π Left endRight end ϕ/π β λ Left endRight end
FIG. S2. Parameters δ ν (top panel) and φ ν (bottom panel) as a function of the angle between the magnetic field and spin-orbit coupling for t = B = ∆ = λ = µ = − ~ n λ ≡ ~ x , and ~ n B ≡ ~ z in the xz plane.[S2] A. Alase, E. Cobanera, G. Ortiz, and L. Viola, Exact Solution of Quadratic Fermionic Hamiltonians for Arbitrary Boundary Conditions,Phys. Rev. Lett. , 076804 (2016)[S3] A. Alase, E. Cobanera, G. Ortiz, and L. Viola, Generalization of Bloch’s theorem for arbitrary boundary conditions: Theory, Phys. Rev.B , 195133 (2017).[S4] A. A. Aligia and L. Arrachea, Entangled end states with fractionalized spin projection in a time-reversal-invariant topological supercon-ducting wire, Phys. Rev. B , 174507 (2018).[S5] A. A. Aligia and A. Camjayi, Exact analytical solution of a time-reversal-invariant topological superconducting wire, Phys. Rev. B100