Topological Larkin-Ovchinnikov Phase and Majorana Zero Mode Chain in Bilayer Superconducting Topological Insulator ﬁlms

ARTICLE

https://doi.org/10.1038/s42005-019-0126-8 OPEN Topological Larkin-Ovchinnikov phase and Majorana zero mode chain in bilayer superconducting topological insulator ﬁlms

Lun-Hui Hu1,2,3, Chao-Xing Liu3 & Fu-Chun Zhang1,4,5 1234567890():,;

Topological superconductors possess a bulk superconducting gap and boundary gapless excitations, known as “Majorana fermion”. Search for new systems with topological superconductivity is of fundamental and application importance due to the potential application of Majorana fermions in topological quantum computation. Here we show that the Larkin- Ovchinnikov superconducting phase with a finite momentum pairing can emerge in a model of bilayer superconducting topological insulator films, in which superconductivity appears for both the top and bottom surface states, and can be topologically non-trivial. This “topological Larkin-Ovchinnikov phase” is induced by an in-plane magnetic field and possesses a Majorana mode chain along the edge perpendicular to the in-plane magnetic field direction Z due to its non-trivial 2 topological nature. Our theoretical model can be naturally realized in superconductor/topological insulator sandwich structure or in Fe(Te, Se) film, a topological material with superconductivity, and thus provides a route to explore unconventional superconductivity in existing systems.

1 Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, 100190 Beijing, China. 2 Department of Physics, Zhejiang University, Hangzhou, 310027 Zhejiang, China. 3 Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA. 4 CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Science, 100190 Beijing, China. 5 Collaborative Innovation Center of Advanced Microstructures, Nanjing University, 210093 Nanjing, China. Correspondence and requests for materials should be addressed to C.-X.L. (email: [email protected]) or to F.-C.Z. (email: [email protected])

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agnetism and superconductivity are two fundamental experimental signature for probing such MZM chain through Mstates of matter in condensed matter physics, and the scanning tunneling microscope (STM) is proposed to distinguish interplay between them continues to bring us intriguing from other topological SCing phase (e.g., tFF phase). The possible phenomena. Unlike the conventional Cooper pairs with zero material realization of tLO phase in the SC/TI sandwich structure momentum in the Bardeen–Cooper–Schrieffer (BCS) theory, or in the newly discovered Fe(Te, Se) films with topological magnetism can induce a superconducting (SCing) state with a surface states is also discussed. finite momentum pairing. The pairing function of such a state can either carry a single finite momentum Q, known as Fulde- Results 1 fi = … Ferrell (FF) state , or multiple nite momenta Qi (i 1,2, ), Bilayer SCing TI film. Our model system consists of a TI film known as Larkin–Ovchinnikov (LO) state2. The FFLO state has fi – under an in-plane magnetic eld, in which both top and bottom been theoretically proposed3 5 and experimentally tested in a surface states (bilayer) are in proximity to the conventional s- variety of systems, including heavy fermion superconductors wave SC pairing, as shown in Fig. 1a. The low energy physics of (SCs)6,7, organic SCs8, ultrathin crystalline Al films9, and even in this system is described by the Hamiltonian cold atom systems10. Another recent development is to realize H¼H þH ð Þ topological SCs by integrating magnetism, spin–orbit coupling 0 pair 1 and superconductivity into one hybrid system11–16, in which gapless excitations exist at the boundary or in the vortex core, “ ” “ ” H dubbed Majorana fermions or Majorana zero mode (MZM) . Here, 0 describes the surface states at the top and bottom MZM possess exotic non-Abelian statistics and thus can serve as surfaces under an in-plane magnetic field along the x direction the building block for topological quantum computation17,18. and is given by24,25 Since both finite momentum pairing and topological super- Z y conductivity require magnetism and superconductivity, it is H ¼ dr~c ðrÞmτ þ vτ ð^p σ À ^p σ ÞþB σ ~cðrÞð2Þ natural to ask if these two SCing phenomena can coexist and if 0 x z y x x y x x there is any interplay between them. In particular, one may ask (1) if topological SC phases can exist for FF or LO state with finite T where ~c ¼ ^c ;";^c ;#;^c ;";^c ;# are the electron annihilation momentum pairing; (2) how to find an experimentally feasible t t b b ð^ ; ^ Þ system for a robust realization of such state; and (3) what types of operators, px py are in-plane momentum operators, v is the boundary modes can emerge in such system. Recent theoretical Fermi velocity, and σ and τ are Pauli matrices, representing spin developments have revealed the possibility of two-dimensional and pseudo-spin (top(t) and bottom(b) surfaces),ÀÁ respectively. fi ¼ þ ^2 (2D) topological FF (tFF) phase with a non-zero Chern number For the Hamiltonian (2), the rst term m m0 m1p in the bulk and chiral Majorana modes at the boundary in cold describes the tunneling between the two surface states (called 19–22 atom systems . More recently, it was proposed that half- inter-layer tunneling below), the second term is the Dirac vortex in certain type of topologically trivial LO phase is able to Hamiltonian for two surface states, and the third term gives the 23 host MZMs . Zeeman coupling between electron spin and in-plane magnetic In this work, we propose a topologically non-trivial LO (tLO) fields. The sketch of the Fermi surface is shown in Fig. 1b. For state in a model system of bilayer SCing topological insulator (TI) μ simplicity, we have absorbed the parameters h into v and g B into thin films under an in-plane magnetic field. Based on the self- fi ¼ þ 2 Bx. In the calculation, we de ne mF m0 m1pF, which gives consistent gap equation and Ginzburg–Landau free energy, we μ2 ¼ 2 þ 2 2 fi rise to mF v pF in the absence of magnetic eld. In this construct the phase diagram of superconductivity in this model work, our calculation are based on realistic parameters for the system and demonstrate the existence of LO phase in a wide H model Hamiltonian 0, which are described in Supplementary parameter range. The topological nature of this LO phase is – Z Note 4. To include the SC pairing, we consider electron electron revealed by the calculation of edge modes and the non-trivial 2 attractive interaction term26 topological invariant in the D class. In the tLO phase, a chain of Z “ ” X numerous MZMs, dubbed MZM chain , is predicted along the H ¼À ^ ð Þ^ ð ÞÀ ^2ð Þ fi pair dr 2Vnt r nb r U nτ r ð3Þ edge perpendicular to the in-plane magnetic eld. The unique τ

abcU py

U = – SC (A1g) 2 V 2 ) m / Bx V (1 – 2 Q/2 U = SC (A2u) (0,0)

SC Bottom layer V

/ –Q/2 TI Metal SC (A ) Top 1u SC

Fig. 1 Sketch of the bilayer system and phase diagram. a Illustration of the superconductor/topological insulator/superconductor (SC/TI/SC)

heterostructure under an in-plane magnetic field Bx (big green arrow) possessing Majorana zero mode (MZM) chain (marked by the green spheres) at the boundary. b Illustration of the two Fermi surfaces with opposite spin textures (green arrows) for bilayer TI thin films with m0 = m1 = 0 in Eq. (2) under an in-plane magnetic field Bx. c Phase diagram of superconducting TI film in the absence of magnetic field, as a function of intra-layer interaction U and inter- ^ ^ þ ^ ^ layer interaction V. It consists of one metallic phase and three superconducting phases: the intra-layer A1g pairing ct"ct# cb"cb#, the inter-layer A1u pairing ^ ^ À ^ ^ ^ ^ À ^ ^ ct"cb# cb"ct#, and the intra-layer A2u pairing ct"ct# cb"cb#

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P ^ ð Þ¼ ^ ð Þ where nτ r σ¼";# nτ;σ r is the electron density operator on a τ = 1.65 layer t, b. 0.75 The phase diagram of SCing phases can be constructed by – L minimizing Ginzburg Landau (GL) free energy obtained from 0.60 III IV the microscopic Hamiltonian (1)27. The quadratic GL free energy 1.10

reads, c0 0.45 T /

X X x

1 1 Ã 1 Ã B L ¼ Δ ðqÞΔ ðqÞÀ Δ ðqÞΔ ðqÞχ ðqÞ ð Þ 2 i i i j ij 4 0.30 II 0.55 2 q;i Vi 2 i;j 0.15 I where the the superconductivity susceptibility is X 0.00 χ ð Þ¼À1 ½½γyG ðω; þ Þ½γ G ðω; À Þ 0.00 ij q β Tr i e p q j h p ð5Þ 0.75 0.80 0.85 0.90 0.95 1.00 ω;p T/Tc0 G ðω; Þ G ðω; À Þ ’ where e p and h p are the single-particle Green s b γ 0.40 functions of electrons and holes, and represents the super- T/Tc0 = 0.8 γ ∈ τ τ τ τ τ conducting pairing matrix: {sy, xsy, ysx, zsy, y, ysz}. The m = 20 linearized gap function is derived from ∂L =∂Δ ¼ 0 and can be 0.32 F 2 i mF = 10 used to construct the phase diagram. More technical details of m = 6.7 0.24 F constructing the phase diagram and the realistic parameters of the c0 m = 5

T F 24,28–32 / 1.0 microscopic model are discussed in Supplementary Note 3. C m = 4 B 0.16 F At zero magnetic ﬁeld B = 0, the Hamiltonian (1) has D group c0 x 3d T / 0.5

symmetry, which possesses three one- dimensional (1D) vQ 0.08 irreducible representations: A1g, A1u, and A2u; and one 2D 0.0 irreducible representation: Eu. Consequently, the momentum- 0.00 0.25 0.50 26 B /T independent SCing pairing forms can be classiﬁed . According 0.00 x c0 to the linearized gap equation (See Supplementary Note 2), the 0 4 8 12 16 20

stable SCing phases are shown as a function of interaction mF (meV) parameters U and V in Fig. 1c. It is found that intra-layer A1g ^ ^ þ ^ ^ pairing ct"ct# cb"cb# is favored for a strong attractive U while ^ ^ À ^ ^ inter-layer A1u pairing ct"cb# cb"ct# (intra-layer A2u pairing ^c "^c # À ^c "^c #) can appear for a strong attractive (repulsive) V. t t b b − More precisely, the phase boundary between A and A is, Fig. 2 Bx T phase diagram. a Phase diagram of Hamiltonian (1) for the set- 1g 1u ﬁ 2m2 up in Fig. 1a, in parameter space of in-plane magnetic eld Bx and U ¼ À F = − 1 μ2 ; and the boundary between A1g and A2u is U V. V temperature T, in unit of Tc,0 (the superconducting transition temperature However, in the realistic materials, we expect the intra-layer at Bx = 0). It comprises conventional Bardeen–Cooper–Schrieffer theory interaction U dominates over the inter-layer interaction V and (BCS) phase (I), Fulde-Ferrell(FF) phase (II), Larkin–Ovchinnikov (LO)

thus gives rise to superconductivity described by A1g pairing in phase (III), and normal metal (IV). Parameters used here are m0 = −13.5 2 the phase diagram. meV, m1 = 25 eV ⋅ Å , v = 2.67 eV ⋅ Å, μ = 100 meV, which are probe for Next, we focus on the case with a strong attractive U term below NbSe2/Bi2Se3/NbSe2, and UN0 = 0.27, and ωD = 20 meV, so that Tc,0 = = – and set V 0, which is most common, and discuss the T Bx phase 1.134 ωDexp(−1/UN0) = 0.5587 meV. b The critical field Bc/Tc,0 (transition fi diagram. With magnetic eld Bx, two intra-layer superconducting between BCS and LO phase) as function of mF at fixed T/Tc,0 = 0.8. And Bc pairings, Δ s and Δ τ s ,canbemixedwitheachother. is reduced if the inter-layer tunneling is reduced. The inset figure shows A1g y A2u z y In addition, an in-plane magnetic field is possible to induce the vQ/Tc,0 as function of Bx/Tc,0 for different values of mF. The non-zero superconducting pairing with the finite momentum q = (0, Q). momentum Q develops when the phase transition between BCS and FF or Therefore, we consider the following three typespffiffiffiffiffiffiffiffiffiffiffiffiffi of pairing forms: LO occurs. Here, we only consider intra-layer superconducting pairing and Δ ¼ Δ À 2σ þ τ σ ignore inter-layer interaction, namely V = 0 (1) BCS state with pairing functionpffiffiffiffiffiffiffiffiffiffiffiffiffiBCS 0 1 b y b z y, Δ ¼ Δ ð Þ À 2σ þ τ σ (2) FF state FF pffiffiffiffiffiffiffiffiffiffiffiffiffi0 exp iQy 1 b y b z y and (3) LO Δ ¼ Δ À À 2 ð Þσ þ ð Þτ σ energetically favored at a large Bx (phase III in Fig. 2a), while the FF state LO 0 i 1 b sin Qy y b cos Qy z y,inwhich Δ and b are parameters to be determined. Based on these three state only exists in a small region (phase II in Fig. 2a) between the 0 BCS and LO states. This small region for FF state may disappear for pairingforms,wecanminimizetheGLfreeenergywithadditional ¼ þ 2 Δ asmallermF m0 m1pF where pF is the Fermi momentum (see forth order terms numerically with respect to the parameters 0 and = b (see Supplementary Note 3 for details of the numerical procedure) Supplementary Note 3 for the analytical calculation in the mF 0 − limit). The critical Bc depends on the coupling strength mF,as and construct the T Bx phase diagram, shown in Fig. 2a. Four phases, including (I) BCS state, (II) FF state, (III) LO state and (IV) shown by different color lines in the inset of Fig. 2b. Figure 2b normal metallic state, are identified in the phase diagram of Fig. 2a. depicts the dependence of Bc on mF, from which one can see that Bc For a small B , the BCS state is energetically favored. As B can be greatly decreased when mF is reduced toward zero. x x To understand the occurrence of the LO state, we may increases, the self-consistently calculated momentum Q of the fi pairing becomes non-zero at a critical magnetic field B ,asshownin rst consider the energy spectrum of the single- c particle Hamiltonian H in Eq. (2), which is given by the inset of Fig. 2b, leading a first-order phase transition to a finite rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 2 momentum pairing state. For B far above the critical value B ,we 2 2 2 2 2 x c E ; ðpÞ¼ ± v p þ m þ v p þ sB , with s = ±1. In find that the momentum Q of the pairing can be approximated by 0 s x F y x fi → 2Bx/v.Twopossible nite momentum pairings, FF and LO states, the decoupling limit (mF 0), the Fermi surfaces of two are considered. Our calculation suggests that the LO state is surface states are shifted with ±Q/2 in the opposite directions

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ab 0.6 0.6 0.4 0.4 0.2 0.2 0 0 Δ Δ / 0.0 / 0.0 E E –0.2 –0.2 –0.4 –0.4 –0.6 –0.6

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ky/Q ky/Q

cd75 0.15 75 0.15 0.13 0.13 50 50 0.11 0.11 25 0.09 25 0.09 L L a a

/ 0 0.07 / 0 0.08 x x 0.06 0.06 –25 –25 0.04 0.04 –50 –50 0.02 0.02 –75 0.00 –75 0.00 –75 –50 –25 0 25 50 75 –15 –10 –5 0 5 10 15

y/aL y/aL = Fig. 3 Majorana zero modepffiffiffiffiffiffiffiffiffiffiffiffi chain. Low eigenenergy spectrum (a, b) and probability distribution of Majorana zero mode (MZM) state (c, d)atky Q/2 in Δ ½À À 2 ð Þσ þ ð Þτ σ = fi the LO phase with 0 i 1 b sin Qy y b cos Qy z y where b 0.77. We nd that the variation of b almost doses not affect the results. Open boundary along x-axis and periodic boundary along y-axis are used. In a, c, Bx = 2.5 Tesla so that Q = 2π/157 and the unit cell size is Nx = Ny = 157; MZM chain appears with a large minigap ΔE ~ 0.2Δ0.Inb, d, Bx = 10 Tesla so that Q = 2π/39 and the unit cell size is Nx = 157, Ny = 39; Majorana bands disperse due to the hybridization between the intra-edge MZMs, however, two pair of MZMs at ky = Q/2 are protected. Parameters used here are the same as that 2 in Fig. 2: m0 = −13.5 meV, m1 = 25 eV ⋅ Å , v = 2.67 eV ⋅ Å, μ = 100 meV, g = 23. And other parameters are aL = 16 Å and Δ0 = 1 meV pffiffiffiffiffiffiffiffiffiffiffiffiffi with Q = 2B /v due to the Zeeman term, as illustrated in Fig. 1b. H ¼ Δ ð Þ¼Δ ½À À 2 ð Þσ þ ð Þτ σ x eh LO y 0 i 1 b sin Qy y b cos Qy z y . Spin textures of the surface states are also depicted on the Fermi Δ + π = Δ H ð þ π= Þ¼ Since LO(y 2 /Q) LO(y). We have BdG y 2 Q surfaces, from which one can see that zero momentum pairing H ð Þ H BdG y . Below, the realistic parameters for 0 are chosen the can only occur for electrons with the same spin and opposite Δ same as those in the caption of Fig. 2, while the parameters in LO layers (inter-layer equal spin triplet pairing), while intra-layer Δ ( 0, Q and b) are obtained from the self-consistent calculations at spin-singlet pairing is only possible for a finite momentum. In the = = → fi Bx 0.75Tc,0 and T 0.8Tc,0. limit mF 0, the nite momentum pairing is favored and thus To investigate the possible gapless modes at the boundary of a two FF phases with opposite momenta for each surface state finite system, we implement the tight-binding (TB) regularization appear, similar to the case of bilayer transition metal dichalco- of the continue Hamiltonian H (including both H and Δ ) 33 fi BdG 0 LO genides (TMDs) system .A nite mF can induce Josephson ! 1 ð Þ 2 ! 2 ð À ð ÞÞ by setting px a sin pxaL and px a2 1 cos pxaL , where coupling between two FF phases at the opposite surfaces and thus L L = drive the whole system into the LO phase. We notice that such aL is the effective lattice constant. With aL 16 Å, the single- H coupling also tends to induce the inter-layer pairing (BCS type particle energy dispersion of 0 reproduces well that of the with zero momentum) between two surfaces in order to lower the continuous model near the chemical potential, as depicted in free energy L. As a result, there is a competition between Supplementary Fig. 3 of Supplementary Note 4, thus validating H Josephson coupling due to finite m , which favors BCS pairing, the TB regularization. We numerically diagonalize BdG in a slab F fi fi and the momentum shift due to in-plane magnetic fields, which con guration which is in nite in the y direction and has open favors FF or LO state. This analysis explains the dependence of boundary condition along the x direction. We can choose a super- the critical magnetic field B on m [See Fig. 2b]. We notice the cell with Nx × Ny lattice sites (See Supplementary Note 4 for c F ¼ 2π details), in which Nx labels the width of the slab while N similarity between our phase diagram (Fig. 2a) and that of 2D y QaL Rashba SCs34,35, presumably due to the same spin textures gives the wave length of the LO phase. The energy dispersions of (Fig. 1b) in these two systems. the slab are shown in Fig. 3a for Nx = Ny = 157 (Bx = 2.5 Tesla) = = = fl and Fig. 3b for Nx 157,Ny 39 (Bx 10 Tesla). Strikingly, at Majorana zero mode chain. We now study on topological bands (referred as Majorana flat bands below) appear with the −6 property of the LO phase (phase III in Fig. 2a). We consider the energy close to zero (E0 ~±10 ) for a large Ny (corresponding to below Bogoliubov-de Gennes (BdG) Hamiltonian, a small Bx) in Fig. 3a. The probability distribution for the state in fl = ! the Majorana at bands at ky Q/2 is shown in Fig. 3c, from H H ee eh which localized MZMs are well separated in one super-cell. We H ¼ ð6Þ BdG Hy H find two MZMs located approximately at y = π/2Q,3π/2Q in one eh hh super-cell at each edge of the slab. In addition to the Majorana flat bands at zero energy, there are also Andreev bound states H ¼Hð ; À ∂ Þ H ¼ÀHÃðÀ ; À ∂ Þ where ee 0 kx i y and hh 0 kx i y with within the SCing gap, which are well separated from MZMs with H Δ Δ 0 given by Eq. (2), and the pairing Hamiltonian an energy gap E ~ 0.2 0. At large Bx or small Ny, adjacent

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ð Þτ σ Z MZMs are hybridized with each other and the Majorana bands b cos Qy z y , and thus we can directly evaluate 2 topological become dispersive, as shown in Fig. 3b, d. We notice that these invariant M. Direct calculation and theoretical analysis described = two Majorana bands cross with each other at ky Q/2. This in Supplementary Note 6 suggest that M¼À1 always exists in fi crossing is robust against varying magnetic eld Bx and implies our model once the chemical potential is large enough, thus the possible non-trivial topological nature of these two Majorana demonstrating the robust realization of the tLO phase in the fi Z bands, which will be discussed in details below. bilayer SCing TI lms. The non-trivial 2 topological invariant suggests the existence of Majorana modes at the boundary of the Topological LO state. To understand the topological nature of system. As for the two decoupled BdG Hamiltonian (subspace) the system, we will develop a general theoretical framework for Heven Hodd ¼ Q Z BdG and BdG at ky 2, there is a non-trivial 2 topological topological LO state. We focus on the general BdG Hamiltonian invariant in each subspace. Therefore, two degenerate MZMs H fi (6), here ee needs not to be speci ed. Due to the periodicity of ~ Δ = Δ + π H protected by the new particle-hole symmetry C can exist at one the gapP function (y) (y 2 /Q)in ee, we can expand ¼ Q Δð Þ¼ Δ inQy edge for the momentum ky 2, which is consistent with y n ne with the wave-vector Q and an integer n. Δ numerical results in Fig. 3. Finally, we emphasize that tLO state Only one n is non-zero in the FF state while multiple non-zero Z Δ H with 2 topological invariant belongs to a different topological n exist in the LO state. Consequently, BdG can also be expanded in the momentum space as class from tFF state with non-zero Chern number discussed in ðÞH ¼ δ H ð ; þ Þ ðÞH ¼Àδ HÃðÀ ; À literature19,20 and thus possesses a different type of boundary ee nm nm 0 kx nQ ky , hh nm nm 0 kx nQ Þ ðÞH ¼ Δ ji mode, the MZM chain. ky and eh nm nþm (n, m are integers) on the basis en and In the decoupling limit with m → 0, the LO phase in our ji F hn with the wave-vector nQ. Here, the momentum ky is within system can be viewed as two decoupled FF states, and each FF the reduced Brillouin zone [0,Q]. The Hamiltonian (6) possesses state can be adiabatically connected to the Fu-Kane model36 by C¼ K particle-hole symmetry tx where the Pauli matrix tx acts on tuning magnetic field to zero. It is well-understood that although K the particle-hole space and is the complex conjugate. Based Fu-Kane model can host Majorana zero mode inside the vortex on the Hamiltonian (6), we can extract the topological property of core, it does not possess chiral Majorana edge state due to the FF state, as shown in Supplementary Note 5, which is consistent time reversal symmetry. Since there are two FF states in our with the recent results on topological FF state in cold atom sys- system, we expect two Majorana zero modes inside the vortex 19–22 tems . core and thus the statistics is no longer non-Abelian when Δ Next, we focus on the LO state with non-zero ±1, for which exchanging two vortices. Therefore, our system possesses neither the Hamiltonian (6) can be split into two decoupled blocks. All chiral (or helical) Majorana edge states nor non-Abelian statistics H ð ; þ Þ the electron part 0 kx nQ ky with even (odd) n is only through exchanging vortex core in the decoupling limit. This also ÀHÃðÀ ; À Þ coupled to the hole part 0 kx nQ ky with odd (even) n. makes the tLO phase found here different from the tFF phase Heven We call these two blocks as even and odd block, denoted as BdG with chiral Majorana edge modes discussed in literature. Hodd and BdG, respectively. Here, the even block is written on the ji Discussion basis e2n and h2nþ1 , while the odd block is written on the basis ji ji C In this work, we develop a general theory of tLO phase with Z e2nÀ1 and h2n . The global particle-hole symmetry relates 2 À fi these two blocks, CHevenC 1 ¼ÀHodd , and thus there is in classi cation and propose its realization in a theoretical model of BdG BdG bilayer SCing TI films. The tLO phase and the corresponding general no particle-hole symmetry within one block. However, at = MZM chain also open a new route in the study of MZMs for the momentum ky Q/2, a new particle-hole symmetry operator C~ fi C~ji¼ C~ ¼ ji topological quantum computation. The 1D MZM chain also can be de ned as e2n h2nþ1 and h2nþ1 e2n for the provides a natural platform to study the interacting Majorana Heven C~ji¼ ji C~ji¼ ji 37,38 even block BdG and e2nÀ1 h2n and h2n e2nÀ1 for chains . Below we will discuss the possible material realization Hodd CH~ evenð ; = ÞCÀ1 ¼ the odd block BdG, and we have BdG kx Q 2 of our model and the possible unique experimental signatures to ÀHevenðÀ ; = Þ CH~ odd ð ; = ÞC~À1 ¼ÀHodd ðÀ ; distinguish the tLO phase from other topological SCing phase. BdG kx Q 2 and BdG kx Q 2 BdG kx Q=2Þ, as shown in Supplementary Note 5. The existence of this Our proposed model can be realized in SC/TI/SC ~ heterostructure30,31,39, e.g., NbSe /Bi Te /NbSe heterostructure. new particle-hole symmetry C suggests that the Hamiltonian 2 2 3 2 With Δ = 1 meV, μ = 100 meV, ℏv = 2.67 eV ⋅ Å, the g-factor Heven and Hodd at k ¼ Q can be viewed as a 1D SC chain in the 0 BdG BdG y 2 g = 23, and m = 20.1 meV, we can estimate the critical field at Z 17 F D class along the x direction. A 2 topological invariant can be tricritical point to be about 0.15 Tesla according to gμ B /k T ≈ fi B c B c,0 de ned as, 0.35 from Fig. 2b. The distance between the two MZMs is esti- = πℏ μ ≈ μ M¼sgnðÞð Pf½AkðÞ¼ 0 Pf½AkðÞ¼ πðÞor 1 7Þ mated as ly v/4g BBc 1.2 m, which is four times larger x x ξ =Δ ¼ : μ than the localization length of MZMs hv 0 0 27 m

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a direction, demonstrating the existence of MZM chain, while a full x SCing gap is found for the parallel magnetic ﬁeld. This calculation Bx suggests that a simple scanning tunneling microscopy measure- y ment of the edge LDOS with parallel and perpendicular magnetic (0,0) ﬁelds can give rise to unambiguous experimental signature to

By identify the tLO phase and the MZM chain.

Methods b In the Supplementary Note 1 and Supplementary Note 2, we review the 2.5 Ginzburg–Landau theory and use the linearized gap equation to classify the pos- By sible SCing pairings of SCing TI thin ﬁlms in the absence of magnetic ﬁeld. Then, we show the details for the self-consistent calculation for the T − B phase diagram 2.0 Bx x by minimizing the Ginzburg–Landau free energy in Supplementary Note 3, show how to construct the tight-binding model in Supplementary Note 4, analyze the 1.5 calculation of topological invariant in Supplementary Note 6, and discuss how to calculate LDOS in Supplementary Note 7.

LDOS 1.0 Data availability The data and codes that support the ﬁndings of this study are available from the cor- 0.5 responding author upon reasonable request.

0.0 Received: 19 September 2018 Accepted: 28 January 2019 –2 –1 0 12 Δ E/ 0

Fig. 4 Experimental signature. The calculated local density of states (LDOS) NðEÞ at a fixed site (black point illustrated in a). In b, the red line is for the parallel magnetic field direction (By); while the blue line is for the References perpendicular magnetic field direction (B ). Parameters used here are the 1. Fulde, P. & Ferrell, R. A. Superconductivity in a strong spin-exchange field. x – same as that in Fig. 3a c: m = −13.5 meV, m = 25 eV ⋅ Å2, v = 2.67 eV ⋅ Å, Phys. Rev. 135, A550 A563 (1964). 0 1 2. Larkin, A. & Ovchinnikov, I. Inhomogeneous state of superconductors μ = = = Δ = = = 100 meV, g 23, aL 16 Å, 0 1 meV and Bx By 2.5 Tesla (production of superconducting state in ferromagnet with fermi surfaces, examining green function). Sov. Phys.-JETP 20, 762–769 (1965). Zeeman effect, but is two orders of magnitude larger than the 3. Casalbuoni, R. & Nardulli, G. Inhomogeneous superconductivity in condensed 25 matter and qcd. Rev. Mod. Phys. 76, 263–320 (2004). Zeeman effect . It should be mentioned that the 1D Z2 topolo- = 4. Matsuda, Y. & Shimahara, H. Fulde–ferrell–larkin–ovchinnikov state in heavy gical invariants at ky Q/2 replies on the translation symmetry fermion superconductors. J. Phys. Soc. Jpn. 76, 051005 (2007). and the tLO phase is not robust against strong disorder scattering, 5. Radzihovsky, L. & Sheehy, D. E. Imbalanced feshbach-resonant fermi gases. thus requiring a relatively clean sample in experiments. We would Rep. Progress. Phys. 73, 076501 (2010). like to mention a recent experiment on the evidence of finite 6. Hatakeyama, Y. & Ikeda, R. Antiferromagnetic order oriented by fulde-ferrell- momentum pairing induced by magnetic fields in HgTe quantum larkin-ovchinnikov superconducting order. Phys. Rev. B 91, 094504 40 41 (2015). wells and 3D TI Bi2Se3 in proximity to superconductivity, and 7. Kim, D. Y. et al. Intertwined orders in heavy-fermion superconductor cecoin5. our results suggest the possibility of non-trivial topology of this Phys. Rev. X 6, 041059 (2016). finite momentum pairing. Based on the above estimate, we con- 8. Agosta, C. C. et al. Calorimetric measurements of magnetic-field-induced clude that our proposal is feasible under the current experimental inhomogeneous superconductivity above the paramagnetic limit. Phys. Rev. conditions. Lett. 118, 267001 (2017). 9. Adams, P. W., Nam, H., Shih, C. K. & Catelani, G. Zeeman-limited Our proposal is also applicable to SCing TIs in which topo- superconductivity in crystalline al films. Phys. Rev. B 95, 094520 logical surface states and bulk superconductivity can coexist. Such (2017). 42–45 materials include Cu doped Bi2Se3 , several SCing half- 10. Kinnunen, J. J., Baarsma, J. E., Martikainen, J. -P. & Törmä, P. The Heusler compounds (e.g., YPtBi, RPdBi)46,47 and Fe(Te, Se) fulde–ferrell–larkin–ovchinnikov state for ultracold fermions in lattice films48,49. We emphasize that the s-wave spin-singlet super- and harmonic potentials: a review. Rep. Progress. Phys. 81, 046401 conductivity has been demonstrated for the surface states of TI Fe (2018). fi 49,50 11. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-abelian (Te, Se) lm , which may provide a natural high-Tc SC anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 platform to realize our model. In addition, odd parity pairing has (2008). been proposed in several of the above compounds26,51, and the 12. Tanaka, Y., Sato, M. & Nagaosa, N. Symmetry and topology in influence of in-plane magnetic field will be interesting for a future superconductors–odd-frequency pairing and edge states–. J. Phys. Soc. Jpn. 81, study. Our work provides an example of the interesting interplay 011013 (2011). fi 13. Alicea, J. New directions in the pursuit of majorana fermions in solid state between nite momentum pairing and topological physics and systems. Rep. Progress. Phys. 75, 076501 (2012). similar idea also exists for exciton condensate52. 14. Elliott, S. R. & Franz, M. Colloquium. Rev. Mod. Phys. 87, 137–163 Finally, we hope to discuss how to unambiguously distinguish (2015). the tLO phase from other topological SCing phases, particularly 15. Sato, M. & Ando, Y. Topological superconductors: a review. Rep. Progress. Phys. 80, 076501 (2017). the tFF phase, in experiments. The key difference lies in the fact 16. Qi, X. -L., Hughes, T. L. & Zhang, S. -C. Chiral topological superconductor that the chiral Majorana edge mode in the tFF phase can exist at from the quantum hall state. Phys. Rev. B 82, 184516 (2010). any edge of the system while the MZM chain in the tLO phase 17. Kitaev, A. Y. Unpaired majorana fermions in quantum wires. Phys.-Uspekhi can only exist at the edge perpendicular to the in-plane magnetic 44, 131 (2001). field. As shown in Fig. 4, we compare the local density of states 18. Kitaev, A. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, fi 2–30 (2003). (LDOS) at the edge when the in-plane magnetic eld is perpen- 19. Qu, C. et al. Topological superfluids with finite-momentum pairing and dicular (blue lines) or parallel (red line) to the edge. The calcu- majorana fermions. Nat. Commun. 4, 2710 (2013). lation details of LDOS can be found in Supplementary Note 7. A 20. Zhang, W. & Yi, W. Topological fulde–ferrell–larkin–ovchinnikov states in zero-bias peak is found for the perpendicular magnetic field spin–orbit-coupled fermi gases. Nat. Commun. 4, 2711 (2013).

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The coexistence of superconductivity and topological It is a pleasure to thank Cheung Chan, Chuang Li, James Jun He, Jia-Bin Yu, Jian-Xiao order in the bi2se3 thin films. Science 336,52–55 (2012). Zhang, and Rui-Xing Zhang for the helpful discussions. C.-X.L. acknowledges the sup- 29. Chen, Y. et al. Experimental realization of a three-dimensional topological port from Office of Naval Research (Grant No. N00014-15-1-2675 and renewal No. insulator, bi2te3. Science 325, 178–181 (2009). N00014-18-1-2793), as well as the Pennsylvania State University Two-Dimensional 30. Xu, J. -P. et al. Artificial topological superconductor by the proximity effect. Crystal Consortium—Materials Innovation Platform (2DCC-MIP), which is supported Phys. Rev. Lett. 112, 217001 (2014). by NSF cooperative Agreement No. DMR-1539916. F.-C.Z. is partly supported by NSFC 31. Xu, J. -P. et al. Experimental detection of a majorana mode in the core of a grant 11674278 and National Basic Research Program of China (Grant No. magnetic vortex inside a topological insulator-superconductor bi 2 te 3/nbse 2 2014CB921203) and the Strategic Priority Research Program of Chinese Academy of heterostructure. Phys. Rev. Lett. 114, 017001 (2015). Sciences (Grant No. XDB28000000) and Beijing Municipal Science & Technology 32. Wang, J. et al. Evidence for electron-electron interaction in topological Commission project No. Z181100004218001. insulator thin films. Phys. Rev. B 83, 245438 (2011). 33. Liu, C. -X. Unconventional superconductivity in bilayer transition metal Author contributions dichalcogenides. Phys. Rev. Lett. 118, 087001 (2017). L.-H.H. performed all the calculation. C.-X.L. and F.-C.Z. supervised this project. All the 34. Barzykin, V. & Gor’kov, L. P. 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Reprints and permission information is available online at http://npg.nature.com/ 39. Sun, H. -H. et al. Majorana zero mode detected with spin selective andreev reprintsandpermissions/ reflection in the vortex of a topological superconductor. Phys. Rev. Lett. 116, 257003 (2016). Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in 40. Hart, S. et al. Controlled finite momentum pairing and spatially varying order published maps and institutional affiliations. parameter in proximitized hgte quantum wells. Nat. Phys. 13,87–93 (2017). 41. Chen, A. Q. et al. Finite momentum cooper pairing in three-dimensional topological insulator josephson junctions. Nat. Commun. 9, 3478 (2018). Open Access This article is licensed under a Creative Commons 42. Wray, L. A. et al. Observation of topological order in a superconducting doped Attribution 4.0 International License, which permits use, sharing, topological insulator. Nat. Phys. 6, 855–859 (2010). adaptation, distribution and reproduction in any medium or format, as long as you give 43. Hor, Y. S. et al. Superconductivity in cuxbi2se3 and its implications for pairing appropriate credit to the original author(s) and the source, provide a link to the Creative in the undoped topological insulator. Phys. Rev. Lett. 104, 057001 (2010). Commons license, and indicate if changes were made. The images or other third party 44. Kriener, M., Segawa, K., Ren, Z., Sasaki, S. & Ando, Y. Bulk superconducting material in this article are included in the article’s Creative Commons license, unless phase with a full energy gap in the doped topological insulator cuxbi2se3. Phys. indicated otherwise in a credit line to the material. If material is not included in the Rev. Lett. 106, 127004 (2011). article’s Creative Commons license and your intended use is not permitted by statutory 45. Matano, K., Kriener, M., Segawa, K. & Ando, Y. & Zheng, G.-q. Spin-rotation regulation or exceeds the permitted use, you will need to obtain permission directly from symmetry breaking in the superconducting state of cuxbi2se3. Nat. Phys. 12, the copyright holder. To view a copy of this license, visit http://creativecommons.org/ – 852 854 (2016). licenses/by/4.0/. 46. Lin, H. et al. Half-heusler ternary compounds as new multifunctional experimental platforms for topological quantum phenomena. Nat. Mater. 9, 546–549 (2010). © The Author(s) 2019

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