DIII Topological Superconductivity with Emergent Time-Reversal Symmetry

Christopher Reeg, Constantin Schrade, Jelena Klinovaja, and Daniel LossDepartment of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland

(Dated: August 23, 2017)

We find a new class of topological superconductors which possess an emergent time-reversal sym-metry that is present only after projecting to an effective low-dimensional model. We show that atopological phase in symmetry class DIII can be realized in a noninteracting system coupled to ans-wave superconductor only if the physical time-reversal symmetry of the system is broken, and weprovide three general criteria that must be satisfied in order to have such a phase. We also providean explicit model which realizes the class DIII topological superconductor in 1D. We show that, justas in time-reversal invariant topological superconductors, the topological phase is characterized bya Kramers pair of Majorana fermions that are protected by the emergent time-reversal symmetry.

PACS numbers: 74.45.+c,71.10.Pm,73.21.Hb,74.78.Na

Introduction. Topological superconductors have beenintensively pursued in recent years [1–3] because the Ma-jorana fermions which are localized to their boundarieshave potential applications in the development of a topo-logical quantum computer [4, 5]. The most promisingproposals to date for engineering topological supercon-ductivity involve coupling a conventional superconductoreither to a nanowire with Rashba spin-orbit interactionthat is subjected to an external magnetic field [6–17] orto a ferromagnetic atomic chain [18–25].

Additionally, there have been several proposals to engi-neer topological superconductors in symmetry class DIII.Such systems possess both particle-hole symmetry andtime-reversal symmetry [26], with the presence of time-reversal symmetry ensuring that the Majorana fermionsexisting at the boundaries of class DIII topological su-perconductors come in Kramers pairs. In one dimen-sion (1D), where superconductivity is required to be in-duced by the proximity effect, it has been shown thata nontrivial topological phase in class DIII can be real-ized by proximity coupling a noninteracting multichannelRashba nanowire to an unconventional superconductor[27–30] or to two conventional superconductors forminga Josephson junction with a phase difference of π [31].Alternatively, an effective π-phase difference can be in-duced in a multichannel Rashba nanowire with repusliveelectron-electron interactions [32] or in a system of twotopological insulators coupled to a conventional super-conductor via a magnetic insulator [33]. It has also beenproposed to realize class DIII topological superconduc-tivity in a system of two Rashba nanowires [34–36] ortwo topological insulators [37] coupled to a single conven-tional superconductor, but repulsive interactions are alsonecessary to reach the topological phase in these setups,which require a strength of induced crossed Andreev (in-terwire) pairing exceeding that of the direct (intrawire)pairing [38–40]. While it would be beneficial to engineera DIII topological superconductor in a noninteracting 1Dsystem coupled to a single conventional superconductor,as such a setup could avoid relying on unconventional su-

perconductivity or interactions that are difficult to con-trol experimentally, it was recently shown that this is notpossible in a fully time-reversal invariant system [41].

In this paper, we show that such a 1D topological su-perconductor in class DIII can be realized when time-reversal symmetry is explicitly broken. While the fullHamiltonian (describing the 1D system, the supercon-ductor, and the tunnel coupling) possesses only particle-hole symmetry and is thus in symmetry class D, it is pos-sible to place the system in symmetry class DIII after in-tegrating out the superconductor [38, 42–48] and project-ing to an effective 1D model [49]. We establish three nec-essary criteria to realize a DIII topological phase. First,the 1D system must obey an “emergent” time-reversalsymmetry. That is, given the Hamiltonian density hk ofthe 1D system, there must exist a unitary matrix T1D

such that T †1DhkT1D = h∗−k and T 21D = −1. [While a spe-

cific example could be the physical time-reversal symme-try T1D = iσy, where σx,y,z is a Pauli matrix acting inspin space, we do not restrict ourselves to this case.] Sec-ond, the self-energy induced on the 1D system by the su-perconductor must preserve the emergent time-reversalsymmetry. Third, the anomalous (pairing) componentof the self-energy must have both positive and negativeeigenvalues.

After a general discussion, we provide a simple modelwhich realizes the DIII topological phase in 1D. We con-sider a system of two Rashba nanowires with oppositeZeeman splittings coupled to an s-wave superconductor.We show that such a system undergoes a topologicalphase transition under certain conditions. By explicitlysolving for the wave functions of the Majorana boundstates, we show that the topological phase is charac-terized by the presence of a Kramers pair of Majoranafermions that is protected by the emergent time-reversalsymmetry.

Minimum requirements for DIII topological phase.We consider a general 1D (noninteracting) system cou-pled to a conventional superconductor. We assume thatthe system is translationally invariant along the direction

arX

iv:1

708.

0675

5v1

[co

nd-m

at.m

es-h

all]

22

Aug

201

7

2

of the 1D system, allowing us to define a conserved mo-mentum k. The Hamiltonian of the 1D system is givenby

H1D =

∫dk

2πψ†khkψk, (1)

where ψk is a spinor annihilation operator acting on allinternal degrees of freedom of the 1D system (spin, sub-band, etc.) and hk is a Hermitian matrix. We assumethat there exists a unitary matrix T1D which acts asan effective time-reversal symmetry on the Hamiltonian,such that T †1DhkT1D = h∗−k and T 2

1D = −1. Introducing

the Nambu spinor Ψ†k = (ψ†k, ψT−kT1D), the Hamiltonian

Eq. (1) can be expressed as

H1D =1

2

∫dk

2πΨ†kH1D

k Ψk, (2)

where Hk = τzhk and τx,y,z is a Pauli matrix acting inNambu space.

The 1D system is coupled to a conventional supercon-ductor which is described by a BCS Hamiltonian,

Hsc =1

2

∫dk

2π

∫dr⊥ η

†k(r⊥)Hsc

k (r⊥)ηk(r⊥), (3)

where r⊥ denotes directions transverse to the 1D systemand η†k = (η†k↑, η

†k↓,−η−k↓, η−k↑) is a spinor creation op-

erator in Nambu ⊗ spin space. The Hamiltonian densityin this basis is given by Hsc

k (r⊥) = τz[(k2 −∇2

⊥)/2msc −µsc] + τx∆, where msc, µsc, and ∆ are the effective mass,chemical potential, and pairing potential of the supercon-ductor, respectively. The superconductor is time-reversalinvariant, T †scHsc

k Tsc = Hsc∗−k , where Tsc = iσy is the phys-

ical time-reversal operator, with σx,y,z the Pauli matrixacting in spin space.

We allow for a linear coupling between the 1D systemand the superconductor of the form

Ht =1

2

∫dk

2π

∫dr⊥[η†k(r⊥)Tk(r⊥)Ψk +H.c.]. (4)

The tunneling matrix in Nambu space, Tk(r⊥), can beexpressed generally as

Tk = T 0k + τzT

zk , (5)

where T 0k = [tk −T †sct∗−kT1D]/2, T zk = [tk + T †sct∗−kT1D]/2,

and tk is a tunneling matrix acting on all additional de-grees of freedom in the system [50]. Combining Eqs. (2)-(4), the full Hamiltonian can be expressed as

H =1

2

∫dk

2π

(Ψ†k η†k

)(H1Dk T †kTk Hsc

k

)(Ψk

ηk

), (6)

where we suppress explicit reference to r⊥ for brevity.Note that the time-reversal symmetry of the full Hamil-tonian is broken by having T 0

k 6= 0; i.e., T †HkT 6= H∗−k,

where Hk is the Hamiltonian density of Eq. (6) andT = diag(T1D, Tsc). For this reason, it was assumed thatT 0k = 0 in Ref. [41].We now project our system to an effective 1D model

by integrating out the superconductor [38, 42–48]. Thesuperconductor induces a self-energy on the 1D systemgiven by

Σk(ω) =

∫dr⊥

∫dr′⊥ T

†k (r⊥)Gsc

k,ω(r⊥, r′⊥)Tk(r′⊥), (7)

where Gsck,ω(r⊥, r

′⊥) is the Matsubara Green’s function

of the bare superconductor, defined such that [iω −Hsck (r⊥)]Gsc

k,ω(r⊥, r′⊥) = δ(r⊥− r′⊥). In the limit of weak

tunneling, where the relevant pairing energies in the 1Dsystem are ω � ∆, it is sufficient to evaluate the self-energy at ω = 0. In this case, the system is described byan effective 1D Hamiltonian given by Heff

k = H1Dk + Σk.

Since it was already assumed that H1Dk obeys an effective

time-reversal symmetry, the Hamiltonian Heffk is in class

DIII if the self-energy of Eq. (7) preserves this symmetry,

T †1DΣkT1D = Σ∗−k. (8)

Hence, T1D acts as an emergent time-reversal symme-try which exists only in the low-dimensional subspace.Assuming that the self-energy satisfies Eq. (8), we candecompose it into normal and anomalous parts as

Σk = τzΣNk + τxΣAk , (9)

where ΣNk = GNk (T 0†k T 0

k +T z†k T zk ) and ΣAk = GAk (T 0†k T 0

k −T z†k T zk ) [51]. In arriving at Eq. (9), we have utilized thefact that the superconducting Green’s function can besimilarly decomposed as Gsc

k = τzGNk + τxG

Ak , where GNk

and GAk are scalars.The anomalous self-energy, which represents the in-

duced pairing in the 1D system, can be expressed in aform

ΣAk = Σzk − Σ0k, (10)

where Σik = −GAk T i†k T ik. It was shown in Ref. [41] thatthe class DIII topological invariant can only take a non-trivial value if ΣAk has both positive and negative eigen-values. It was also shown that Σzk is always positivesemidefinite; hence, if the tunneling Hamiltonian is time-reversal invariant with Σ0

k = 0, it is not possible to realizea topological phase. By extension, it is straightforward toshow that Σ0

k must also be positive semidefinite. Conse-quently, the topological invariant must also take a trivialvalue if Σzk = 0 (in which case ΣAk is negative semidef-inite). However, if both terms in Eq. (10) are nonzero,ΣAk is not restricted to be either positive semidefinite ornegative semidefinite, and it is possible to have a topo-logically nontrivial phase.

We have thus established three minimal criteria to re-alize a class DIII topological phase in a noninteracting

3

1D system: the bare 1D system must obey an effectivetime-reversal symmetry, the self-energy induced by thesuperconductor must preserve this symmetry, and theanomalous self-energy must have both positive and neg-ative eigenvalues. The final requirement can be satisfiedonly if the full tunneling Hamiltonian [Eq. (6)] is nottime-reversal invariant. We will now provide a modelwhich satisfies all three criteria, showing indeed that theclass DIII topological phase can be realized.

Model. We consider the geometry shown in Fig. 1.Two Rashba nanowires, separated by a distance d (letus take one wire to be located at z = 0 and the otherat z = d), are coupled to an infinite 2D s-wave super-conducting plane. We take the two nanowires to haveopposite Zeeman splitting, which can be achieved by ap-plying an antiparallel external magnetic field to each wireor by applying a uniform external magnetic field to twowires with opposite g-factors.

The two nanowires are described by the Hamiltoniandensity

hk = ξk − αkσz −∆Zηzσx, (11)

where ξk = k2/2mw − µw (mw is the effective mass andµw the chemical potential of the nanowires), α is theRashba spin-orbit interaction constant (we choose ourspin quantization axis along the direction of the effectiveRashba field), and ∆Z = gµBB/2 is the Zeeman split-ting in an external magnetic field of strength B (g is thenanowire g-factor and µB is the Bohr magneton). ThePauli matrix ηx,y,z acts in left/right wire space. Cru-cially, we impose that the two nanowires are identical,with only a change in the sign of the Zeeman splitting.Although the Zeeman term in Eq. (11) explicitly breakstime-reversal, the Hamiltonian density obeys an effec-tive time-reversal symmetry T †1DhkT1D = h∗−k, whereT1D = iηxσy and T 2

1D = −1.The self-energy induced on the two nanowires by the

superconductor is given in Eq. (7). Assuming that µs �µw, we evaluate the Green’s function of the bulk 2D su-perconductor for momenta k � kFs (kFs =

√2msµs is

the Fermi momentum of the superconductor) to give

Gsc0,0(z, z′) = − 1

vFs[τx cos(kFs|z − z′|)

− τz sin(kFs|z − z′|)]e−|z−z′|/ξs ,

(12)

where ξs = vFs/∆ is the coherence length and vFs =kFs/ms the Fermi velocity of the superconductor (wehave also expanded in the limit µs � ∆). We as-sume local spin- and momentum-independent tunnel-ing of the form tk(z) =

[tδ(z) tδ(z − d)

], where t is a

(scalar) tunneling amplitude which has the same strengthin both nanowires. This gives T 0

k = tk(1 − ηx)/2 andT zk = tk(1 + ηx)/2. Evaluating the self-energy Eq. (7),we find

Σ = Γτzηx + τx(∆c + ∆dηx), (13)

SC

d

�Z ��Z

FIG. 1. Two nanowires, separated by a distance d, are cou-pled to an infinite superconducting plane. The two nanowiresare assumed to have opposite Zeeman splittings.

where we define a single-particle interwire tunnel cou-pling Γ = γ sin(kFsd)e−d/ξs , an induced direct (in-trawire) pairing potential ∆d = γ, and an inducedcrossed Andreev (interwire) pairing potential ∆c =γ cos(kFsd)e−d/ξs [38]. Here, γ = t2/vFs is a tunnel-ing energy scale. Note that the pairing potentials alwayssatisfy ∆d > ∆c.

Taking into account the self-energy, the effectiveHamiltonian describing the double nanowire system isgiven by

Heffk = τz(ξk − αkσz −∆Zηzσx + Γηx) + τx(∆c + ∆dηx).

(14)Because the self-energy preserves the effective time-reversal symmetry, T †1DΣT1D = Σ∗, we also have

T †1DHeffk T1D = Heff∗

−k . Additionally, the effective Hamil-

tonian possesses a particle-hole symmetry P†Heffk P =

−Heff∗−k , where P = τyηxσy is a unitary matrix satisfy-

ing P2 = 1. Finally, the effective Hamiltonian possessesa chiral symmetry {C,Heff

k } = 0, where C = T1DP = iτy.These three properties of the Hamiltonian place it in theDIII symmetry class. Additionally, the anomalous self-energy ΣA = ∆c + ∆dηx has both positive and negativeeigenvalues, ∆c ± ∆d. Because all of our previously es-tablished criteria are met in this setup, it is possible tohave a topological phase.

To determine whether such a topological phase existsin this setup, we search for a k = 0 gap-closing transitionby enforcing det(Heff

0 ) = 0. We find

det(Heff0 ) =

{∆4Z + 2∆2

Z(Γ2 −∆2d + ∆2

c − µ2w)

+ [(∆d −∆c)2 + (Γ− µw)2][(∆d + ∆c)

2 + (Γ + µw)2]}2,

(15)which yields a gap-closing at the critical Zeeman splitting(∆c

Z)2 = ∆2d−∆2

c+µ2w−Γ2±2i(µw∆c−Γ∆d). Therefore,

in order to have a physical transition, the chemical po-tential of the nanowires must be tuned to µw = Γ∆d/∆c.For simplicity, let us assume that the system is tuned insuch a way that the interwire tunnel coupling vanishes,Γ = 0 [which can be done by tuning sin(kFsd) = 0]. In

4

00

1

�c/�

d

�Z/�d

1

0.5

0.5 1.5

Trivial

Topological

FIG. 2. Phase diagram of our model. A phase boundary at∆2

Z = ∆2d − ∆2

c separates trivial and DIII topological super-conducting phases. The topological phase is characterized bythe presence of a Kramers pair of Majorana fermions that isprotected by the emergent time-reversal symmetry.

this case, the critical Zeeman splitting at which the gapcloses is given by

∆cZ =

√∆2d −∆2

c . (16)

The phase diagram of our model is displayed in Fig. 2.We find two distinct phases whose topological charac-terization can be inferred along the line ∆c = 0, corre-sponding to the case when the two wires are decoupled(d � ξs). For ∆Z < ∆d, both wires are in a topolog-ically trivial phase. For ∆Z > ∆d, both wires are in atopologically nontrivial phase, with each wire hosting itsown distinct pair of Majorana bound states. Because thenumber of Majorana bound states is a topological invari-ant that cannot be changed without closing the gap, weconclude that ∆2

Z < ∆2d −∆2

c corresponds to a topologi-cally trivial phase while ∆2

Z > ∆2d−∆2

c corresponds to atopologically nontrivial phase with two pairs of Majoranabound states.

To further establish the presence of a topologicallynontrivial phase, we explicitly solve for the wave func-tions of the Majorana bound states. We now take ournanowires to be semi-infinite (x > 0), and we assumethat the effective Hamiltonian of Eq. (14) remains validfor the semi-infinite case after replacing k → −i∂x [52].States in the nanowires obey a Bogoliubov-de Gennesequation given by Heff(x)φ(x) = Eφ(x). By construct-ing a general zero-energy solution and imposing a van-ishing boundary condition φ(0) = 0, we find two Ma-jorana bound state solutions in the topological phase(∆2

Z > ∆2d − ∆2

c) and no solutions in the trivial phase(∆2

Z < ∆2d −∆2

c). The full analytical expressions for thewave functions are given in the limit of strong spin-orbitinteraction (mwα

2 � ∆d,∆c,∆Z) in the SupplementalMaterial [51]. We find that the two Majorana wave func-tions in the topological phase are orthogonal,

φ†2(x)φ1(x) = 0, (17)

and related by the effective time-reversal symmetry,

φ1(x) = T1Dφ∗2(x),

φ2(x) = −T1Dφ∗1(x).

(18)

Hence, the two Majorana bound states in the systemform a Kramers pair that is protected by the emergenttime-reversal symmetry.Conclusions. We have shown that a topological su-

perconductor in symmetry class DIII can be realized in anoninteracting 1D system proximity coupled to a conven-tional superconductor. Crucially, the full Hamiltonian(incorporating the 1D system, the parent superconduc-tor, and the tunneling term) must not possess an effectivetime-reversal symmetry, with such a symmetry emerg-ing only after projection to an effective 1D model. Weprovide an explicit example realizing such a class DIIItopological superconductor, showing that the topologicalphase is characterized by a Kramers pair of Majoranabound states which is protected by the effective time-reversal symmetry of the system. We believe that ourgeneral criteria can be applied to realize class DIII topo-logical superconductivity in a multitude of additional sys-tems coupled to a bulk s-wave superconductor, for exam-ple in nanowires with helical magnetization of oppositehelicity, antiferromagnetically coupled spin chains, mag-netic topological insulators with opposite magnetization,or ferromagnetic atomic chains with opposite magnetiza-tion.Acknowledgments. This work was supported by the

Swiss National Science Foundation and the NCCR QSIT.Note. Upon completion of this manuscript, we be-

came aware of a very recent preprint, Ref. [53], whichdiscusses the realization of a class DIII topological phasein 2D antiferromagnetic quantum spin Hall insulators.

[1] J. Alicea, Rep. Prog. Phys. 75, 076501 (2012).[2] M. Leijnse and K. Flensberg, Semicond. Sci. Technol. 27,

124003 (2012).[3] C. W. J. Beenakker, Annu. Rev. Condens. Matter Phys.

4, 113 (2013).[4] A. Y. Kitaev, Physics-Uspekhi 44, 131 (2001).[5] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and

S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).[6] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev.

Lett. 105, 077001 (2010).[7] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett.

105, 177002 (2010).[8] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P.

A. M. Bakkers, and L. P. Kouwenhoven, Science 336,1003 (2012).

[9] M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson,P. Caroff, and H. Q. Xu, Nano Letters 12, 6414 (2012).

[10] A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, andH. Shtrikman, Nat. Phys. 8, 887 (2012).

[11] L. P. Rokhinson, X. Liu, and J. K. Furdyna, Nat. Phys.8, 795 (2012).

5

[12] H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen,M. T. Deng, P. Caroff, H. Q. Xu, and C. M. Marcus,Phys. Rev. B 87, 241401 (2013).

[13] A. D. K. Finck, D. J. Van Harlingen, P. K. Mohseni,K. Jung, and X. Li, Phys. Rev. Lett. 110, 126406 (2013).

[14] W. Chang, S. M. Albrecht, T. S. Jespersen, F. Kuem-meth, P. Krogstrup, J. Nygard, and C. M. Marcus, Nat.Nano. 10, 232 (2015).

[15] S. M. Albrecht, A. P. Higginbotham, M. Madsen,F. Kuemmeth, T. S. Jespersen, J. Nygard, P. Krogstrup,and C. M. Marcus, Nature 531, 206 (2016).

[16] M. T. Deng, S. Vaitiekenas, E. B. Hansen, J. Danon,M. Leijnse, K. Flensberg, J. Nygard, P. Krogstrup, andC. M. Marcus, Science 354, 1557 (2016).

[17] H. Zhang, O. Gul, S. Conesa-Boj, M. Nowak, M. Wim-mer, K. Zuo, V. Mourik, F. K. de Vries, J. vanVeen, M. W. A. de Moor, J. D. S. Bommer, D. J.van Woerkom, D. Car, S. R. Plissard, E. P. A. M.Bakkers, M. Quintero-Perez, M. C. Cassidy, S. Koelling,S. Goswami, K. Watanabe, T. Taniguchi, and L. P.Kouwenhoven, Nat. Commun. 8, 16025 (2017).

[18] S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, andA. Yazdani, Phys. Rev. B 88, 020407 (2013).

[19] F. Pientka, L. I. Glazman, and F. von Oppen, Phys.Rev. B 88, 155420 (2013).

[20] J. Klinovaja, P. Stano, A. Yazdani, and D. Loss, Phys.Rev. Lett. 111, 186805 (2013).

[21] M. M. Vazifeh and M. Franz, Phys. Rev. Lett. 111,206802 (2013).

[22] B. Braunecker and P. Simon, Phys. Rev. Lett. 111,147202 (2013).

[23] S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon,J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yaz-dani, Science 346, 602 (2014).

[24] Y. Peng, F. Pientka, L. I. Glazman, and F. von Oppen,Phys. Rev. Lett. 114, 106801 (2015).

[25] R. Pawlak, M. Kisiel, J. Klinovaja, T. Meier, S. Kawai,T. Glatzel, D. Loss, and E. Meyer, Npj Quantum Infor-mation 2, 16035 (2016).

[26] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W.Ludwig, Phys. Rev. B 78, 195125 (2008).

[27] C. L. M. Wong and K. T. Law, Phys. Rev. B 86, 184516(2012).

[28] S. Nakosai, J. C. Budich, Y. Tanaka, B. Trauzettel, andN. Nagaosa, Phys. Rev. Lett. 110, 117002 (2013).

[29] F. Zhang, C. L. Kane, and E. J. Mele, Phys. Rev. Lett.111, 056402 (2013).

[30] E. Dumitrescu, J. D. Sau, and S. Tewari, Phys. Rev. B90, 245438 (2014).

[31] A. Keselman, L. Fu, A. Stern, and E. Berg, Phys. Rev.Lett. 111, 116402 (2013).

[32] A. Haim, A. Keselman, E. Berg, and Y. Oreg, Phys.Rev. B 89, 220504 (2014).

[33] C. Schrade, A. A. Zyuzin, J. Klinovaja, and D. Loss,Phys. Rev. Lett. 115, 237001 (2015).

[34] J. Klinovaja and D. Loss, Phys. Rev. B 90, 045118(2014).

[35] E. Gaidamauskas, J. Paaske, and K. Flensberg, Phys.Rev. Lett. 112, 126402 (2014).

[36] C. Schrade, M. Thakurathi, C. Reeg, S. Hoffman, J. Kli-novaja, and D. Loss, Phys. Rev. B 96, 035306 (2017).

[37] J. Klinovaja, A. Yacoby, and D. Loss, Phys. Rev. B 90,155447 (2014).

[38] C. Reeg, J. Klinovaja, and D. Loss, Phys. Rev. B 96,081301 (2017).

[39] P. Recher and D. Loss, Phys. Rev. B 65, 165327 (2002).[40] C. Bena, S. Vishveshwara, L. Balents, and M. P. A.

Fisher, Phys. Rev. Lett. 89, 037901 (2002).[41] A. Haim, E. Berg, K. Flensberg, and Y. Oreg, Phys.

Rev. B 94, 161110 (2016).[42] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma,

Phys. Rev. B 82, 094522 (2010).[43] A. C. Potter and P. A. Lee, Phys. Rev. B 83, 184520

(2011).[44] N. B. Kopnin and A. S. Melnikov, Phys. Rev. B 84,

064524 (2011).[45] A. A. Zyuzin, D. Rainis, J. Klinovaja, and D. Loss, Phys.

Rev. Lett. 111, 056802 (2013).[46] B. van Heck, R. M. Lutchyn, and L. I. Glazman, Phys.

Rev. B 93, 235431 (2016).[47] C. Reeg and D. L. Maslov, Phys. Rev. B 95, 205439

(2017).[48] C. Reeg, D. Loss, and J. Klinovaja, arXiv:1707.08417.[49] This is analogous to the conventional Majorana proposal

of a single Rashba nanowire coupled to an s-wave super-conductor and subjected to an external magnetic field.In this setup, the field breaks time-reversal symmetryand places the full Hamiltonian in class D. However, af-ter projecting to 1D, it is possible to define an effectivetime-reversal symmetry (which squares to +1), thus plac-ing the 1D system in class BDI [54].

[50] Note that we do not consider the possibility that tk is atwo-body operator as in Ref. [33], as this corresponds toan interacting model.

[51] See Supplemental Material for a derivation of Eq. (9)and a detailed solution for the Majorana wave functionsof our model.

[52] Because the proximity effect was treated in the bulk ofthe nanowires, there could be additional boundary ef-fects that are not properly accounted for in the effectiveHamiltonian. However, the presence or absence of topo-logical Majorana modes is insensitive to such boundaryeffects.

[53] Y. Huang and C.-K. Chiu, arXiv:1708.05724.[54] S. Tewari and J. D. Sau, Phys. Rev. Lett. 109, 150408

(2012).

6

SUPPLEMENTAL MATERIAL

Form of Self-Energy

In this section, we prove that the self-energy can be decomposed into normal and anomalous components as inEq. (9) of the main text. The self-energy of the effective 1D model is given by

Σk(ω) =

∫dr⊥

∫dr′⊥ T

†k (r⊥)Gsc

k,ω(r⊥, r′⊥)Tk(r′⊥). (S1)

Substituting the tunneling term Tk = T 0k + T zk , and noting that the superconducting Green’s function (at ω = 0) can

be decomposed into normal and anomalous parts as Gsck = τzG

Nk + τxG

Ak , where GNk and GAk are scalars, we find a

self-energy given by (we suppress explicit reference to the transverse coordinates r⊥ and r′⊥ for brevity)

Σk = GNk [τz(T0†k T 0

k + T z†k T zk ) + τ0(T 0†k T zk + T z†k T 0

k )] +GAk [τx(T 0†k T 0

k − T z†k T zk )− iτy(T 0†k T zk − T z†k T 0

k )]. (S2)

We now require the self-energy to preserve the effective time-reversal symmetry of the 1D system by enforcingT †1DΣkT1D = Σ∗−k. Because the four terms of Eq. (S2) are linearly independent, each term must separately obeythis condition.

Let us first consider the term proportional to τ0; expanding out this term using the definitions T 0k = [tk −

T †sct∗−kT1D]/2 and T zk = [tk + T †sct∗−kT1D]/2 gives

T 0†k T zk + T z†k T 0

k =1

2(t†ktk − T

†1Dt

T−kt∗−kT1D). (S3)

Applying the time-reversal operator to Eq. (S3) and using the fact that T 21D = −1, we see that in order to preserve

the time-reversal symmetry of the 1D system, the self-energy must satisfy

T †1Dt†ktkT1D − tT−kt∗−k = tT−kt

∗−k − T †1Dt

†ktkT1D, (S4)

or, equivalently,

t†ktk = T †1DtT−kt∗−kT1D. (S5)

Hence, the term proportional to τ0 in Eq. (S2) must vanish.

Next, we consider the term proportional to τy in Eq. (S2),

T 0†k T zk − T z†k T 0

k =1

2(t†kT †sct∗−kT1D − T †1Dt

T−kTsctk). (S6)

Applying the time-reversal operator to Eq. (S6), we require

tT−kTsctkT1D − T †1Dt†kT †sct∗−k = tT−kT †sctkT1D − T †1Dt

†kTsct

∗−k. (S7)

However, because T †sc = −Tsc, this amounts to

t†kT †sct∗−kT1D = T †1DtT−kTsctk, (S8)

meaning that the product t†kT †sct∗−kT1D is necessarily Hermitian and that the term proportional to τy in Eq. (S2) mustalso vanish.

Using the relations defined in Eqs. (S5) and (S8), we see that the terms proportional to τz and τx in Eq. (S2) donot vanish in general. Hence, the self-energy can be expressed in the form

Σk = GNk τz(T0†k T 0

k + T z†k T zk ) +GAk τx(T 0†k T 0

k − T z†k T zk ), (S9)

as given in Eq. (9) of the main text.

7

Majorana Wave Functions

In this section, we explicitly solve for the Majorana wave functions in a semi-infinite geometry. States in thenanowire obey a Bogoliubov-de Gennes equation given by

Heff(x)φ(x) = Eφ(x), (S10)

where φ(x) is a spinor wave function and Heff(x) is given by

Heff(x) = τz(−∂2x/2mw + iα∂xσz −∆Zηzσx) + τx(∆c + ∆dηx). (S11)

Because the wires are semi-infinite, Majorana solutions are necessarily at zero energy. Setting E = 0, we rewriteEq. (S10) in the form

∂xφ(x) = Mφ(x), (S12)

where φ = (∂xφ, φ)T and

M =

(2mwiασz −2mw[∆Zηzσx + iτy(∆c + ∆dηx)]

I8×8 0

). (S13)

Any zero-energy solution to Eq. (S10) can be written in the form

φ(x) =∑n

cnχneiknx, (S14)

where ikn are the eigenvalues of M and χn are the corresponding eigenvectors.Eigenvalues of M must satisfy (for brevity, we temporarily set 2mw = 1)

0 = [k8 − 2α2k6 + k4(α4 + 2∆2d + 2∆2

c − 2∆2Z) + 8αk3∆d∆c + 2α2k2(∆2

d + ∆2c + ∆2

Z) + (∆2Z −∆2

d + ∆2c)

2]

× [k8 − 2α2k6 + k4(α4 + 2∆2d + 2∆2

c − 2∆2Z)− 8αk3∆d∆c + 2α2k2(∆2

d + ∆2c + ∆2

Z) + (∆2Z −∆2

d + ∆2c)

2].(S15)

While this equation cannot be solved analytically in general, we can solve it in the limit of strong spin-orbit interaction,α2 � ∆Z ,∆d,∆c. First, we expand in the vicinity of the exterior branches of the spectrum by setting k = ±α+ δkeand expanding for δke � α. We find

[(∆d + ∆c)2 + α2δk2

e ][(∆d −∆c)2 + α2δk2

e ] = 0. (S16)

Therefore, at both the positive and negative exterior branches, we find four allowed momenta with δke = ±i(∆d ±∆c)/α. Next, we expand in the vicinity of the interior branches of the spectrum by setting k = δki and againexpanding for δki � α; we find

[α4δk4i + 2α2(∆2

Z + ∆2d + ∆2

c)δk2i + (∆2

Z −∆2d + ∆2

c)2]2 = 0. (S17)

For the interior branches, we find four doubly degenerate momenta corresponding to δki = ± iα (√

∆2Z + ∆2

c ±∆d).For bound states, only momenta with positive imaginary part are allowed. Restoring the factors of 2mw, the allowed

momenta in the topological phase (∆2Z > ∆2

d −∆2c) are given by

k1 = 2mwα+ i(∆d + ∆c)/α,

k2 = 2mwα+ i(∆d −∆c)/α,

k3 = −2mwα+ i(∆d + ∆c)/α,

k4 = −2mwα+ i(∆d −∆c)/α,

k5 = k6 = i

(√∆2Z + ∆2

c + ∆d

)/α,

k7 = k8 = i

(√∆2Z + ∆2

c −∆d

)/α.

(S18)

8

Substituting Eqs. (S18) and (S14) into Eq. (S10) and similarly expanding in the limit of strong spin-orbit interaction,we find the corresponding eigenvectors

χ1 =1

2(−1, 0,−1, 0,−i, 0,−i, 0)T ,

χ2 =1

2(−i, 0, i, 0,−1, 0, 1, 0)T ,

χ3 =1

2(0, i, 0, i, 0, 1, 0, 1)T ,

χ4 =1

2(0,−i, 0, i, 0, 1, 0,−1)T , (S19)

χ5 =1

2√

∆2Z + β2

(β,−i∆Z , β, i∆Z ,−iβ,−∆Z ,−iβ,∆Z)T ,

χ6 =1

2√

∆2Z + β2

(∆Z ,−iβ,−∆Z ,−iβ, i∆Z , β,−i∆Z , β)T ,

χ7 =−1

2√

∆2Z + β2

(−β, i∆Z , β, i∆Z , iβ,∆Z ,−iβ,∆Z)T ,

χ8 =1

2√

∆2Z + β2

(i∆Z , β, i∆Z ,−β,−∆Z , iβ,−∆Z ,−iβ)T ,

where we define β =√

∆2Z + ∆2

c + ∆c.The coefficients c1−8 are determined by imposing the boundary condition φ(0) = 0. We find two solutions given by

φ1(x) = N1

(i∆Z√

∆2Z + β2

χ1eik1x − iβ√

∆2Z + β2

χ4eik4x + χ8e

ik8x

),

φ2(x) = N2

(− iβ√

∆2Z + ∆2

c

χ2eik2x +

∆Z√∆2Z + ∆2

c

χ3eik3x + χ7e

ik7x

),

(S20)

where N1(2) are normalization constants. The wave functions are normalized according to∫ ∞0

dxφ†i (x)φi(x) = 2. (S21)

Evaluating the integral, we find normalization constants

N1 =1√α

(1√

∆2Z + ∆2

c −∆d

+β2

(∆d −∆c)(∆2Z + β2)

+∆2Z

(∆d + ∆c)(∆2Z + β2)

)−1/2

,

N2 =1√α

(1√

∆2Z + ∆2

c −∆d

+β2

(∆d −∆c)(∆2Z + ∆2

c)+

∆2Z

(∆d + ∆c)(∆2Z + ∆2

c)

)−1/2

.

(S22)

Substituting Eq. (S22) into Eq. (S20), we see that the two Majorana wave functions are orthogonal,

φ†2(x)φ1(x) = 0, (S23)

and related by the emergent time-reversal symmetry (T1D = iτ0ηxσy),

φ1(x) = T1Dφ∗2(x),

φ2(x) = −T1Dφ∗1(x).

(S24)

Hence, the two Majorana bound states form a Kramers pair that is protected by the emergent time-reversal symmetry.Constructing a general zero-energy solution to Eq. (S10) in the trivial phase (∆2

Z < ∆2d −∆2

c) requires replacing

k7,8 → i(∆d −

√∆2Z + ∆2

c

)/α,

χ7 →1

2√

∆2Z + β2

(−β,−i∆Z , β,−i∆Z ,−iβ,∆Z , iβ,∆Z)T

χ8 →1

2√

∆2Z + β2

(i∆Z ,−β, i∆Z , β,∆Z , iβ,∆Z ,−iβ)T .

(S25)

9

After making these replacements in Eq. (S14), we find that the boundary condition φ(0) = 0 can only be satisfied bychoosing c1−8 = 0 and hence there are no Majorana bound states.