Majorana splittings from critical currents in Josephson junctions

Majorana splittings from critical currents in Josephson junctions Jorge Cayao1 , Pablo San-José2 , Annica Black-Schaffer1 , Ramón Aguado2 , Elsa Prada3 1

arXiv:1707.05117v1 [cond-mat.mes-hall] 17 Jul 2017

2

Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC), Cantoblanco, 28049 Madrid, Spain 3 Departamento de F´ısica de la Materia Condensada, Condensed Matter Physics Center (IFIMAC) & Instituto Nicol´ as Cabrera, Universidad Aut´ onoma de Madrid, E-28049 Madrid, Spain (Dated: July 18, 2017)

The superconducting proximity effect in semiconductor nanowires with strong Rashba spin-orbit coupling has been shown to give rise to emergent Majorana bound states under external Zeeman fields. Here we theoretically investigate how this exotic physics manifests in Josephson junctions based on such nanowires. In particular, we focus on critical currents in the short junction limit (LN ξ, where LN is the junction length and ξ is the superconducting coherence length) and show that they contain important information about nontrivial topology and Majoranas. This includes signatures of the gap inversion at the topological transition and an oscillatory pattern that originates from Majorana overlaps. Interestingly, this pattern can be modified by tuning the transmission across the junction, thus providing an alternative detection scheme of Majoranas and their energy splittings beyond standard tunnel spectroscopy experiments, while offering further tunability by virtue of the Josephson effect.

I.

INTRODUCTION

The theoretical proposals1,2 for artificially creating topological superconductors3 out of semiconducting nanowires proximitized with conventional s-wave superconductors has spurred an unprecedented experimental effort towards realizing such exotic phase with mundane materials. The topological superconductor phase is expected to occur whenpan external Zeeman field exceeds a critical field Bc ≡ ∆2 + µ2 , defined in terms of the induced superconducting gap ∆ and the wire’s chemical potential µ. Since the electron density can be modified by external electrostatic gates and the external Zeeman field can be larger than the induced superconducting gap [owing to the strong spin-orbit coupling (SOC) in semiconducting nanowires such as InSb or InAs], the topological phase can in principle be tuned in situ. Such nontrivial phase is characterized by the emergence of non-Abelian Majorana zero modes (MZMs) localized at the nanowire ends. The emergence of MZMs is predicted to give rise to perfect Andreev reflection, and thus perfectly quantized conductance G = 2e2 /h at zero bias voltage, in normalsuperconductor junctions. During the last few years, this idea has led to a great number of important experiments in hybrid superconductor-semiconductor systems looking for such zero-bias anomaly (ZBA) in the differential conductance as the Zeeman field increases.4–9 However, and despite the initial excitement, the interpretation of these experiments in terms of Majoranas has been hotly debated owing to the difficulty in ruling out alternative explanations such as parity crossings of Andreev levels,9–11 Kondo physics,7,8,10,11 disorder,12–15 smooth confinement,16,17 etc. The nagging question of whether MZMs have been observed in nanowires and the so-called soft gap problem4–9 have resulted in a second generation of experiments, with improved devices, which show more con-

vincing signatures of MZMs.18–20 Despite these improvements, the topological origin of the ZBAs is still under debate since subtler effects, such as the sticking of Andreev levels owing to non-homogeneous chemical potentials,21 can mimic Majoranas. Therefore, it would be very useful to study other geometries and signatures22 beyond ZBAs in normal-superconductor junctions. A direct measurement of a 4π-periodic Josephson effect in nanowire-based superconductor-normalsuperconductor (SNS) junctions may provide one of such strong signature. Unfortunately, the 4π effect, which appears as a result of protected fermionic parity crossings as a function of the superconducting phase difference (φ) in the junction, is very hard to observe experimentally. First, the unavoidable presence of two additional MZMs at the wire ends, and their overlap with inner MZMs at the junction, renders the Josephson effect 2π-periodic.23 A second problem is stochastic quasiparticle poissoning which spoils fermionic parity conservation. The 4π-periodic Josephson effect may still be detected by out-of-equilibrium measurements, such as AC Josephson radiation or noise,23–25 as demonstrated recently in Josephson junctions fabricated with quantum spin Hall edges.26,27 Another route is offered by the multiple Andreev reflection (MAR) mechanism in a voltage-biased SNS junction28 which in current experiments29,30 could provide stronger evidence of MZMs. We here take an arguably easier route and focus on equilibrium supercurrents which, despite the overall 2πperiodicity in the ground state, still contain useful information about the nontrivial topology in the junction. Our main findings are summarized in Figs. 4 and 5. First, we find that the critical current Ic through a short junction (LN ξ, where LN is the normal region length and ξ is the superconducting coherence length) between two finite length superconductors (two proximitized wires of length LS ) and tunable normal transmission TN (see

2 Fig. 1), shows a distinct feature at the gap inversion for the critical Zeeman field B = Bc [Fig. 4(a)]. For B < Bc , the critical current does not depend on the superconducting region length LS , as expected for a trivial ballistic junction in the short junction regime.31 In contrast, the critical current in the topological regime increases with LS . This unique length-dependence in a short SNS junction is a direct consequence of Majorana overlap on each side of the junction. For long enough wires (LS 2ξ) and small transmission TN 1, we obtaine a re-entrant behaviour of Ic as a function of Zeeman field. This reentrance results from√an enhanced Majorana-mediated critical current Ic ∼ TN (for B > Bc ) relative to that of conventional Andreev levels Ic ∼ TN (for B < Bc ), as discussed in Refs. 32 and 33. The critical current for B > Bc , moreover, exhibits oscillations derived from the oscillatory splitting of overlapping Majoranas around zero energy. These features are robust against temperature and changes in TN . The latter, in particular, gives rise to a period doubling effect in the oscillations [Fig. 5] which can be unmistakably attributed to overlapping Majoranas: for small TN , the oscillatory pattern is that of two nearly decoupled finite-length topological wires with two overlapping Majoranas at the ends of each wire. As the transparency increases, this oscillatory behaviour evolves into a more complex pattern which, for fully transparent junctions (TN → 1) results in the hybridization of the four Majoranas, with two energy splittings oscillating out-of phase (Fig. 7). From the above we conclude that, despite the finite lengths involved in the geometry of the junction, the critical current for B > Bc contains distinct signatures of the bulk topological transition, derived from the specific four-Majorana hybridization pattern. Our findings thus constitute an interesting alternative beyond ZBAs in tunnelling measurements while offering further tunability by virtue of the Josephson effect. Furthermore, they hold relevance in view of recent proposals for topological quantum computation that crucially rely on controlling the Josephson coupling in nanowire-based Josephson junctions.34,35 The paper is organized as follows. In Sec. II we describe the model for SNS junctions based on nanowires with SOC. Then, in Sec. III we investigate the Zeeman field dependence of critical currents and discuss the emergence of oscillations as a result of finite MBSs overlaps. In Sec. IV we present our conclusions. II. SNS JUNCTIONS MADE OF SEMICONDUCTING NANOWIRES A.

Model

We consider a single channel semiconducting nanowire with Rashba SOC whose Hamiltonian is given by36 H0 =

αR p2x −µ− σy px + Bσx , 2m ~

(1)

FIG. 1. (Color online) Sketch of a SNS junction with S and N regions of finite length LS and LN . The transmission across the junction can be controlled by the parameters τL(R) ∈ [0, 1], which parametrise the coupling between neighbouring sites at the interfaces. (b) For B > Bc , the S regions become topological and the system hosts four Majorana states close to zero energy at φ = π: two outer γ1,4 and two inner γ2,3 ones. The Majorana modes are localized at the ends of the S regions and their wavefunctions exhibit an oscillatory exponential decay into the bulk of S. (c) Relation between normal transmission TN and coupling parameter τL(R) = τ for different values of the Zeeman field. Parameters: ∆ = 0 and µ = 0.5 meV, LN = 20 nm and αR = 20 meVnm.

where px = −i~∂x is the momentum operator, µ is the chemical potential that determines the filling of the nanowire, αR represents the strength of Rashba SOC mα2 (which results in a SO energy ESO = 2~2R ), B = gµB B/2 is the Zeeman energy resulting from the applied magnetic field B, g is the wire’s g-factor and µB the Bohr magneton. Notice that the Zeeman and the spin-orbit axes are perpendicular to each other. The Hamiltonian given in Eq. (1) is discretized into a tight-binding lattice following standard methods.37 In all our calculations we consider typical values for InSb nanowires4 that include the electron’s effective mass m = 0.015me , being me is the electron’s mass, and the spin-orbit strength αR = 20 meV nm, unless otherwise stated. In order to model the SNS junction, we now asume that the left and right regions of the nanowire are in good contact with two parent s-wave superconductors. Due to the proximity effect, this results in a nanowire containing left (SL ) and right (SR ) regions of length LS with induced superconducting pairing potential ∆SL(R) , leaving the central region, of length LN , in the normal state (N). This model is relevant in recent experiments where hard gaps are proximity-induced into nanowires.38 The full Hamiltonian describing the SNS junction is

3

LS µ. Notice that the tunnel regime remains almost unaltered even when the Zeeman field is large. B.

1

(2)

Phase-dependent Andreev spectrum

At finite magnetic field the S sectors develop two gaps ∆1(2) which correspond to low and high-momentum, respectively, and exhibit a different behaviour on the Zeeman field B. As the Zeeman field B is increased beyond B = Bc , each S segment undergoes a topological phase transition, determined by ∆1 , and crosses into a topological superconducting phase1,2 with ∆2 finite for strong SOC. In this phase, MBSs emerge localized at each end of SL(R) . Their wavefunction decay in an exponentially and oscillatory fashion within a Majorana localisation length ξM into the bulk of the superconducting

Energy ε/∆

given in Nambu space as hSNS ∆SNS HSNS = , ∆†SNS −h∗SNS

(a)

LS>>2ξM (b)

0

-1 0

1 2 0 1 2 Phase difference φ/π Phase difference φ/π

FIG. 2. (Color online) Low-energy spectrum as a function of the superconducting phase difference in a short SNS junction for LS < 2ξM (left) and LS 2ξM (right) at B = 2Bc . The two lowest levels, almost insensitive to phase, correspond to the two outer Majoranas. The next two levels below the topological gap (green dashed line), are strongly dispersive with phase, and come from the hybridization of the inner Majoranas which anticross the former at φ = π. Parameters: LN = 20 nm, αR = 20 meVnm, µ = 0.5 meV, ∆ = 0.25meV.

regions,39,40 see Fig. 1(b). As long as Majoranas do not overlap spatially, i.e. in long enough superconducting regions LS 2ξM , they remain zero-energy eigenstates. A finite overlap, however, enables the hybridization of any two given Majoranas into a fermionic state of finite energy. When the left and right topological regions in the Josephson junction become coupled by a finite TN , the two Majoranas in the junction also hybridise, although in a phase-dependent manner, which gives rise to a distinctive phase-dependent Andreev spectrum, as first discussed in Refs. 23, 25, 37, and 41. In Fig. 2 we present this low-energy spectrum for LS < 2ξM (a) and LS 2ξM (b). It consists of a discrete set of levels below the topological minigap (green dashed line), originated from the hybridisation of the four Majoranas at the junction, and a quasi-continuum above. The lowest two levels are (almost) insensitive to φ and come from the outer MBSs, γ1,4 . The other two energy levels below the minigap originate from the inner Majoranas, γ2,3 , strongly disperse with φ and anticross at φ = π with the outer ones. For LS < 2ξM , Fig. 1(a), a considerable splitting around φ = π appears, which arises due to the finite spatial overlap between the MBSs wave-functions within each S region.42 Such splitting can be reduced either by increasing the SOC strength and therefore reducing the Majorana localisation length,37 or by increasing the length of the superconducting regions LS . The φ = π splitting is considerably suppressed for LS 2ξM , as shown in Fig. 1(b). In this limit, the outer MBSs are true Majorana zero modes, while the inner ones approach

4

LS>2ξΜ

(a)

τ

Fig. 3 shows the phase-dependent supercurrents corresponding to the Andreev spectra of Fig. 2. The finite splitting at φ = π, observed for LS < 2ξM , gives rise to supercurrents with a sine-like bevahior as a function of φ, see Fig. 3(a). The opposite case of LS 2ξM that corresponds to an almost perfect crossing at φ = π results in a sawtooth profile, Fig. 3(b). The overall shape in both limits remains unaltered irrespective of the transmission across the normal region all the way from fully transparent junctions τ = 1 to the tunnel limit τ = 0.6. As expected, finite temperature is more detrimental and tends to wash out the sawtooth profile (dashed magenta curves).

(b) 1 0.9 0.8 0.7 0.6

0

-0.2 0

Finite temperature κΒΤ=0.02meV


FIG. 3. (Color online) Supercurrent in a short SNS junction as a function of φ at B = 2Bc for LS < 2ξM (a) and LS 2ξM (b). Different curves correspond to different values of τ. A sawtooth profile in (b) is developed when MBS wavefunctions within each superconducting section do not overlap, Fig. 1(b). This feature remains robust as τ is reduced, but not so with finite temperature (see dashed purple curve, with κB T = 0.02 meV). Parameters: LN = 20 nm, αR = 20 meVnm, µ = 0.5 meV, ∆ = 0.25meV and I0 = e∆/~.

zero energy at φ = π. As we argue in what follows, the critical current across the junction is very sensitive to the φ = π anticrossing structure and can, therefore, be used to extract useful information about the Majorana hybridizations in the system. C.

Current-phase characteristics

The supercurrent can be obtained from the free energy of the junction as I(φ) =

1 dF , Φ0 dϕ

(4)

~ where Φ0 = 2e is the reduced (superconducting) flux quantum. Taking care about the double counting inherent to the BdG description, the above expression can be rewritten as ε dε e X p p I(φ) = − tanh , (5) ~ >0 2κB T dϕ p

where κB is the Boltzman constant, T is the temperature and εp are the energy levels, obtained by exact diagonalization of the Hamiltonian in Eq. (2). Importantly, the sum must be performed over all degeneracies (spin in our case) explicitly.43 In what follows, we focus on zero temperature calculations, unless otherwise stated, and short junctions. The full phenomenology of current-phase relationships, including the long junction limit, will be discussed elsewhere.44

III.

CRITICAL CURRENTS

The critical current Ic is the maximum supercurrent that flows across the junction and we calculate it by maximizing the supercurrent I(φ) with respect to the superconducting phase difference φ, Ic = maxφ [I(φ)] .

(6)

In Fig. 4 we present Ic in short SNS junctions as the Zeeman field B increases from the trivial (B < Bc ) to the topological phase (B > Bc ). Different curves in Fig. 4(a) correspond to different values of the superconducting region length LS . The critical current is maximum at zero field and decreases as the Zeeman field B increases, tracing the reduction of the low-momentum gap ∆1 . Remarkably, the gap closes at B = Bc but the critical current remains finite and develops a robust kink-like feature. At B = Bc , Ic is roughly halved with respect to its B = 0 value, as a result of the system losing one channel (the nanowire bands change from spinful to spinless). As the Zeeman field is further increased, the critical current develop oscillations when LS . 2ξM that arise from the structure of the four-Majorana anticrossing around φ = π. The oscillations disappear as the length LS is increased, and the outer Majoranas decouple from the junction. As a side effect of this decoupling, the overall value of Ic for B > Bc increases as LS becomes longer, while for B < Bc the critical current remains almost unaltered. This remarkable length-dependence in a short junction when B > Bc is a direct signature of the nontrivial topology of the junction and provides a way to distinguish the effects described here from other trivial explanations such as π-junction behavior or oscillatory critical currents due to orbital effects.45 The dashed curve in Fig. 4(a) shows the asymptotic limit LS → ∞, using a simplified model for the subgap states in the transparent limit,28 I 0 ∆1 + ∆ 2 , B < Bc 2 ∆ I 0 ∆2 Ic ≈ , B > Bc , 2 ∆ Ic ≈

(7)

5

(a) LS [nm] Eq.(7) ∞ 6000 5000 4000 2500 2000 1500 1000

0.8 0.6 0.4

LS=2000nm (b) αR [meVnm] 100 40 20

LS= 2000nm

0.6 Critical current I/I0

Critical current Ic/I0

1

0.4

τ=1

0.2

0.2 0 0

1 1.5 2 0 1 2 0.5 Zeeman field B/Bc Zeeman field B/Bc

FIG. 4. (Color online) Critical current Ic in a short SNS junction as a function of the Zeeman field. In (a) different curves represent different values of the superconducting region length LS . As B increases towards Bc , Ic decreases, approximately following the evolution of the gap. At the gap closing B = Bc (red vertical dashed line), Ic exhibits a non-trivial feature but remains finite. For B > Bc , the critical current develops an oscillatory behaviour, which is related to the finite inner-outer Majorana splitting. As expected, these oscillations disappear as the length LS is increased. However, the magnitude of Ic for B > Bc increases with increasing LS while its value for B < Bc remains unchanged. This unconventional dependence is a direct signature of nontrivial topology of the junction. (b) Ic for fixed LS = 2000 nm and increasing SOC strength αR . Observe that at large fields and in the very strong SOC regime the critical current saturates to 1/2 of its B = 0 value. Parameters are the same as in Fig. 3.

with ∆1 = |B − Bc | and ∆2 ≈ q

2∆ESO , p 2 ) ESO (2ESO + B 2 + 4ESO

the low and high-momentum gaps, respectively. I0 = e∆/~ is the maximum critical current supported by an open channel. Notice that the approximate expression for Ic given by Eq. (7) closely matches the results of an exact calculation with LS = ∞ involving only the inner MZMs,28 see dash-dot curve in Fig. 4(a). As the Zeeman field increases, the ratio Ic /I0 decreases, which directly reflects the transition from spinful to spinless bands as the nanowire becomes topological.28 In the strong SO case (dashed curve in Fig. 4(b)), the outer gap is ∆2 ≈ ∆ and thus the ratio between the critical currents at zero and high Zeeman fields reaches 1/2, which is yet another strong evidence of a topological transition. Note, however, that the required SOC to reach this value ESO may be unrealistically large in most nanowire systems. In Fig. 5 we show the critical current as a function of B for different values of τ. As expected, the critical current in the trivial regime gets reduced as τ decreases,

0 0

τ=0.6 1 Zeeman field B/Bc

2

FIG. 5. (Color online) Ic for fixed LS = 2000 nm and increasing values of τ ∈ [0.6, 1], in steps of 0.01 (indicated by vertical orange arrow). In the trivial phase, B < Bc , the critical current is quickly suppressed as τ is decreased towards the tunnelling limit. In the topological phase the critical current is suppressed much more slowly with τ, which results in a reentrant Ic at Bc for small τ. For B > Bc , the period of the Ic oscillations is doubled as the transparency of the junction is reduced (magenta and blue dashed lines). Parameters are the same as in Fig. 3.

until it becomes negligible in the tunneling regime. The kink-like feature in Ic for a transparent τ ≈ 1 junction associated to the closing of the gap at B = Bc evolves into a step-like increase as τ is reduced. This re-entrant behaviour of the low-transparency Ic as B exceeds Bc can be understood from the fact that the critical √ current carried by the inner Majoranas scales as Ic ∼ TN (for B > Bc ). At small TN this current is much larger than the critical current associated to conventional Andreev levels Ic ∼ TN (for B < Bc ).32,33 Furthermore, the critical current oscillations for B > Bc shows a characteristic period doubling as τ is reduced, see Fig. 5. This period doubling is highlighted by the magenta and blue vertical dashed lines. The robustness of the effect against various asymmetries is shown in Fig. 6. This period doubling can be understood in terms of changes in the structure of the φ = π anticrossing of the four Majoranas in the system as τ is reduced. The low-energy spectrum at φ = π results from the Bdependent hybridization of the four MBSs, which gives rise to the oscillatory pattern as B > Bc increases. A fully transparent junction (τ = 1) exhibits the oscillatory behaviour of the four MBSs with crossings around zero energy as shown in Fig. 7(a). As the coupling τ is reduced, Fig. 7(b-c), two consecutive zero-energy crossings (magenta vertical dashed lines in panel a) tend to approach each other and finally fuse into one single crossing when the tunnel regime is reached (blue vertical dashed lines in Fig. 7(d)). In this tunnel limit, the periodicity is that of overlapping Majoranas in essentially decou-

τR=1

(b) τ=1

1

LS =0.5LS L

Energy ε/∆

0.6 (a) τ =1 0.4 L

R

0.2 0

τL=0.6

0.6 (c)

τ=0.6 (d)

τR=0.9

τ=1

0.4 τ =1 L

LS =0.75LS L

R

0.2 τ =0.6 τ=0.6 0 L 0 1 2 0 1 2 Zeeman field B/Bc Zeeman field B/Bc

FIG. 6. (Color online) Critical currents as a function of the Zeeman field in a short SNS junction. (a,c) Asymmetry in the couplings to the left and right superconducting leads, τL(R) , for fixed LS . The coupling to the right S region τR is fixed and the one for the left one varies from tunnel to the full transparent regime in steps of 0.1. This situation is expected to arise in experiments. Observe that the oscillations for B > Bc remain robust with a clear period doubling for decreasing τL . (b,d) Asymmetry in the length of the superconducting regions LS = 2000 nm for different values of the coupling parameter τL(R) = τ from tunnel to the full transparent regime in steps of 0.1. Parameters same as in Fig. 3.

Energy ε/∆

1 (a)

τ=1

(b)

τ=0.9

0

τ=0.9

(c)

τ=0.7

(d)

τ=0.6

0

-1 0


FIG. 8. (Color online) Low-energy Andreev spectrum as a function of the superconducting phase difference φ at B = 2Bc for different values of the coupling between N and S regions τ. Note that upon decreasing τ the outer (two lowest) MBSs acquire a finite energy at φ = 2πn, for n = 0, 1, 2, . . . , while developing a crossing at φ = π with their inner counterparts. This is the limit of two identical decoupled superconducting nanowires with overlapping Majoranas at each end. Parameters same as in Fig. 3 and LS = 2000 nm.

pled wires (namely, two independent pairs of Majoranas which do not couple through the normal region). This can be also seen in the phase-dependent Andreev spectra as shown in Fig. 8. Therefore, the Ic oscillations and the period doubling upon reducing the transparency of the junction, which can be tuned by a central gate in a realistic geometry, can be unmistakably attributed to overlapping Majoranas in the system.

0 IV.

-1 1 Energy ε/∆

(b) τ=1

(a)

-1 1 Energy ε/∆

Critical current Ic/I0 Critical current Ic/I0

6

(d)

(c) τ=0.7

τ=0.6

0

-1 0

1 2 0 Zeeman field B/Bc

1 Zeeman field B/Bc

2

FIG. 7. (Color online) Low-energy Andreev spectrum as a function of the Zeeman field in a short SNS junction at φ = π for different values of the coupling between N and S regions τ. Note that upon decreasing τ the low-energy zero-energy crossings for B > Bc merge in pairs, which results in a doubling of the period of oscillation in Ic . Parameters same as in Fig. 3 and LS = 2000 nm.

CONCLUSIONS

We have theoretically investigated how Majorana physics manifests in Josephson junctions based on Rashba nanowires proximitized with superconductors. We have shown that the expected sawtooth profile in the phase-dependent supercurrent for LS 2ξM , resulting from almost perfect crossings at φ = π, does not depend on the transparency of the normal section. On the contrary, the sawtooth shape is rapidly lost as the finite length of the superconducting sections becomes relevant, namely when LS ≤ 2ξM . In this limit, sizable energy splittings around φ = π resulting from Majorana overlaps, give rise to increasingly conventional currentphase characteristics. This makes it hard to distinguish trivial from non-trivial topology from current-phase measurements in short wires. Despite this, we demonstrate that the critical current still contains important information about nontrivial topology and Majoranas. This includes signatures of the gap inversion at the topological transition, a re-entrant effect,32,33 an oscillatory pat-

7 tern that originates from Majorana overlaps, a perioddoubling effect as a function of junction transparency, and an anomalous length dependence that can be also traced back to the presence of Majoranas in the junction. All these features can be modified by tuning the normal transmission across the junction, thus providing an alternative detection scheme of Majoranas and their energy splittings beyond tunnel spectroscopy.

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V.

ACKNOWLEDGEMENTS

We acknowledge financial support from the Spanish Ministry of Economy and Competitiveness through Grant No. FIS2015-65706-P (MINECO/FEDER) (P. S.-J), FIS2015-64654-P (R. A.), FIS2016-80434-P (AEI/FEDER, EU) (E. P.) and the Ramón y Cajal programme through grant No. RYC-2011-09345 (E. P). J.C. and A.B.S. acknowledge financial support from the Swedish Research Council (Vetenskapsr˚ adet), the G¨ oran Gustafsson Foundation, the Swedish Foundation for Strategic Research (SSF), and the Knut and Alice Wallenberg Foundation through the Wallenberg Academy Fellows program.

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Notice that H0 is written in the basis (ψ↑ , ψ↓ ). J. Cayao, E. Prada, P. San-José, and R. Aguado, Phys. Rev. B 91, 024514 (2015). The quality of the semiconductor-superconductor interface in nanowire devices has been improved dramatically during the last few years19,46–48 . This results in a very good proximity effect with induced superconducting gaps without residual subgap conductance. Moreover, very good transparencies, down to the single-channel limit, have been reported with almost perfect Andreev conductances G ∼ 4e2 /h.19,46 . S. Das Sarma, J. D. Sau, and T. D. Stanescu, Phys. Rev. B 86, 220506 (2012). E. Prada, P. San-José, and R. Aguado, Phys. Rev. B 86, 180503 (2012). E. B. Hansen, J. Danon, and K. Flensberg, Phys. Rev. B 93, 094501 (2016). A true crossing is reached when considering semi-infinite superconducting leads, LS → ∞. In such geometry the outer MBSs are not present in the description and the crossing at φ = π is protected by conservation of the total fermion parity.

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Note the missing factor 2 as compared to, e. g., Eq. (2.16) in Ref. [31]. There, spin degeneracy is assumed, hence the degeneracy factor 2 [this should not be confused with the factor 2 of Eq. (4) contained in Φ0 , which comes from the charge 2e of Cooper pairs]. In our case, spin degeneracy is broken by Zeeman, so the spin summation must be done explicitly, and no degeneracy factor 2 should be included. J. Cayao, E. Prada, P. San-José, and R. Aguado, To be published. K. Zuo, V. Mourik, D. B. Szombati, B. Nijholt, D. J. van Woerkom, A. Geresdi, J. Chen, V. P. Ostroukh, A. R. Akhmerov, S. R. Plissard, D. Car, E. P. A. M. Bakkers, D. I. Pikulin, L. P. Kouwenhoven, and S. M. Frolov, arXiv preprint arXiv:1706.03331 (2017). W. Chang, S. M. Albrecht, T. S. Jespersen, F. Kuemmeth, P. Krogstrup, J. N. rd, and C. M. Marcus, Nat. Nano. 10, 232 (2015). P. Krogstrup, N. L. B. Ziino, W. Chang, S. M. Albrecht, M. H. Madsen, E. Johnson, J. N. rd, C. M. Marcus, and T. S. Jespersen, Nat. Mat. 14, 400 (2015). D. J. van Woerkom, A. Proutski, B. van Heck, D. Bouman, J. I. V¨ ayrynen, L. I. Glazman, P. Krogstrup, J. Nyg˚ ard, L. P. Kouwenhoven, and A. Geresdi, Nat. Phys. advance online publication (2017).