BBoundary Topological Superconductors
Bo-Xuan Li and Zhongbo Yan ∗ School of Physics, Sun Yat-Sen University, Guangzhou 510275, China (Dated: September 8, 2020)For strongly anisotropic time-reversal invariant (TRI) insulators in two and three dimensions, the band in-version can occur respectively at all TRI momenta of a high symmetry axis and plane. Although these classesof materials are topologically trivial as the strong and weak Z indices are all trivial, they can host an evennumber of unprotected helical gapless edge states or surface Dirac cones on some boundaries. We show in thiswork that when the gapless boundary states are gapped by s ± -wave superconductivity, a boundary time-reversalinvariant topological superconductor (BTRITSC) characterized by a Z invariant can be realized on the corre-sponding boundary. Since the dimension of the BTRITSC is lower than the bulk by one, the whole system isa second-order TRI topological superconductor. When the boundary of the BTRITSC is further cut open, Ma-jorana Kramers pairs and helical gapless Majorana modes will respectively appear at the corners and hinges ofthe considered sample in two and three dimensions. Furthermore, a magnetic field can gap the helical Majoranahinge modes of the three-dimensional second-order TRI topological superconductor and lead to the realizationof a third-order topological superconductor with Majorana corner modes. Our proposal can potentially be real-ized in insulator-superconductor heterostructures and iron-based superconductors whose normal states take thedesired inverted band structures. Topological insulators (TIs) and topological superconduc-tors (TSCs) are two classes of materials which have a non-trivial gapped band structure in the bulk and novel gaplessexcitations on the boundary[1, 2]. The band topology of TIswith time-reversal symmetry is known to be characterized bya Z invariant ν in two dimensions (2D) and four Z invari-ants ( ν ; ν ν ν ) in 3D[3–7]. If the inversion symmetry isalso preserved, these Z invariants can be simply inferredfrom the parity eigenvalues at time-reversal invariant (TRI)momenta[7], or equivalently, the number and distribution ofTRI momenta at which the band inversion occurs. In 2D,when the band inversion occurs at an odd number of TRI mo-menta, ν is necessitated to take the nontrivial value, and a TIwith an odd number of helical gapless states on each edgeis realized[8–10]. Similarly, when the band inversion occursat an odd number of TRI momenta in 3D, the strong Z in-dex ν is also necessitated to take the nontrivial value, anda strong TI with an odd number of Dirac cones on each sur-face is realized[11–14]. In contrast, when the band inversionoccurs at an even number of TRI momenta, the strong Z in-dex is trivial, but some of the three weak indices ( ν ν ν ) can still be nontrivial. For instance, when the band inversionoccurs at (mod ) TRI momenta, at least one of ( ν ν ν ) must be nontrivial according to their definition[7], leading tothe realization of a weak TI in which the surface Dirac conesonly appear selectively on certain surfaces and their numberon each surface is even rather than odd[15, 16].Because of the time-reversal symmetry, the gapless bound-ary states in TIs are spin-momentum-locked[17]. Remark-ably, this property allows the establishment of a direct con-nection between TIs and TSCs. In the pioneering work ofFu and Kane[18], it was shown that when the gapless surfaceDirac cones of a strong TI are gapped by s -wave supercon-ductivity, topological superconductivity can be realized in the π -flux vortices, manifested by the presence of Majorana zeromodes (MZMs). The vortex MZMs also follow a Z classi- fication, which means that there exists one topologically pro-tected MZM only when the number of MZMs in a π -flux vor-tex is odd[19–21]. On the other hand, the number of MZMsis directly connected to the number of surface Dirac cones,therefore, insulators with an even number of band inversionsare disfavored in this scenario. Apparently, this rules out aconsiderable amount of strongly anisotropic materials whichgenerally favor an even number of band inversions.In this work, we build a new scenario which favors suchstrongly anisotropic band-inverted insulators. Concretely, weconsider insulators with both time-reversal symmetry and in-version symmetry, whose band inversions occur at all TRI mo-menta of a high symmetry axis in 2D and a high symmetryplane in 3D. Although such band-inverted insulators are topo-logically trivial as the strong and weak Z indices are all triv-ial, they can host an even number of unprotected helical gap-less edge states in 2D and surface Dirac cones in 3D on someboundaries. The gapless boundary states are found to formfloating bands which are non-degenerate due to the breakingof inversion symmetry on the boundary. By introducing s ± -wave superconductivity rather than s -wave superconductivityto gap the gapless floating bands, we find that a TRI TSCcharacterized by a Z invariant can be realized on the cor-responding boundary, even though the band topology of thebulk is necessitated to be trivial. As the TRI TSC is realizedon the boundary, we term it boundary time-reversal invarianttopological superconductor (BTRITSC). Because the dimen-sion of the BTRITSC is lower than the bulk by one, the wholesystem is a second-order TRI TSC which harbors MajoranaKramers pairs (two MZMs related by time-reversal symme-try) at the sample corners in 2D and helical gapless Majo-rana modes at the sample hinges in 3D[22–28]. Remarkably,a magnetic field can gap the helical Majorana hinge modes in3D and lead to the realization of a third-order TSC with Ma-jorana corner modes[29, 30]. The new scenario thus unveils anew route for the realization of higher-order TSCs[31–60]. a r X i v : . [ c ond - m a t . s up r- c on ] S e p BTRITSCs in 2D.—
To illustrate the essential physics, westart with a simple 2D Bogoliubov-de Gennes (BdG) Hamil-tonian H = (cid:80) k ψ † k H BdG ( k ) ψ k , where H BdG ( k ) = (cid:18) H ( k ) − µ Σ − i Σ ∆( k ) i Σ ∆( k ) µ Σ − H ∗ ( − k ) (cid:19) ,H ( k ) = (cid:15) ( k )Σ + m ( k )Σ + λ sin k Σ + λ sin k Σ , ∆( k ) = ∆ − ∆ (cos k + cos k ) , (1)and ψ k = ( c k ,a, ↑ , c k ,b, ↑ , c k ,a, ↓ , c k ,b, ↓ , c †− k ,a, ↑ , c †− k ,b, ↑ , c †− k ,a, ↓ ,c †− k ,b, ↓ ) T . In the BdG Hamiltonian (1), Σ ij = s i ⊗ σ j , wherethe Pauli matrices s i and σ j act respectively on the spin ( ↑ , ↓ ) and orbital ( a, b ) degrees of freedom, and s and σ are × unit matrices. H ( k ) describes the normal state, with thefirst term characterizing the asymmetry of the conductionand valence bands, the second term characterizing the bandinversion, and the last two terms representing spin-orbitcoupling. Here we take (cid:15) ( k ) = (cid:15) cos k + (cid:15) cos k , m ( k ) = m − m cos k − m cos k , and λ , areassumed to be positive. ∆( k ) describes the superconductingorder parameter. In this work, we consider s ± -wave super-conductivity which can be achieved intrinsically[61–70] orextrinsically by superconducting proximity effect from aniron-based superconductor[71, 72]. Moreover, the latticeconstants are set to unity throughout for notational simplicity.It is readily verified that H has both time-reversal symme-try and inversion symmetry, with the time-reversal and inver-sion operators given by T = is ⊗ σ K and P = s ⊗ σ ,respectively, where K denotes the complex conjugation. Ac-cordingly, the bulk bands have Kramers degeneracy at ev-ery momentum, and the Z invariant characterizing the bandtopology of H ( k ) is simply given by[7] ( − ν = (cid:89) i =1 ξ ( Γ i ) , (2)where ξ ( Γ i ) denotes the parity eigenvalue of the two lowerKramers degenerate bands at the TRI momenta Γ i ( Γ i = − Γ i up to a reciprocal lattice vector). For H ( k ) , it is readily foundthat away from the critical lines m ± m ± m = 0 wherethe bulk gap gets closed, we have ( − ν = (cid:89) α = ± ,β = ± sgn ( m + αm + βm ) . (3)According to the above formula, the phase diagram can bestraightforwardly determined, as shown in Fig.1(a). In thephase diagram, the normal (or trivial) insulator (NI) phase isfurther divided into two distinct parts. The first part labeled asNI has zero (mod ) band inversion (or equivalently to say,the parity eigenvalues at the four TRI momenta take the samesign in these regimes), and the second part labeled as NI hasband inversions at two TRI momenta. We are interested in NI since gapless boundary states can appear on certain bound-aries, whereas they are completely absent in NI . (b) (c) (d) (a) / m m mm FIG. 1. (a) Phase diagram of the normal state. (b)(c)(d) Energy spec-trum under a cylinder geometry. The lattice size along the directionwith open boundary condition is L = 100 . (b) No gapless state ap-pears on the x -normal edges. (c) The in-gap dispersions (blue lines)are of double degeneracy, corresponding to the presence of two pairsof helical gapless states on each x -normal edge. The helical gaplessstates on each edge form two non-degenerate floating bands. (d) Thefloating bands remain in the gap even when the conduction-valenceasymmetry is strong enough to change the insulator to a metal. Com-mon parameters are m = 1 , m = 0 , m = 2 , λ = 0 . , λ = 1 . (cid:15) , = 0 in (b) (c), and (cid:15) = 0 , (cid:15) = 1 . in (d). In the following, we consider a specific case where the bandinversion occurs at the two TRI momenta ( k , k ) = (0 , and ( π, to illustrate the essential physics. Without loss ofgenerality, we take m = 1 as the energy unit and set m = 0 and m = 2 to realize the desired condition. To reveal theselective existence of helical gapless states on certain bound-aries, we consider that the insulator takes a cylinder geometrywith open boundary condition in one direction and periodicboundary condition in the other orthogonal direction. The re-sults shown in Figs.1(b)(c) indicate that helical gapless statesdo not appear on the x -normal edges, but appear on the x -normal edges. One can see that the helical edge states crossat both TRI momenta in the edge Brillouin zone, suggestingthe presence of two pairs of helical gapless states on each x -normal edge. However, unlike TIs, the helical gapless edgestates do not traverse the bulk gap, instead they form two float-ing bands within the gap. It is noteworthy that conduction-valence asymmetry only affects the dispersion of the float-ing bands. The floating bands exist even if the asymmetryis strong enough to change the insulator to a metal, as shownin Fig.1(d).Let us now take into account the s ± -wave superconduc-tivity. Before focusing on the boundary, we first discussthe bulk. As the TRI BdG Hamiltonian belongs to the DIIIclass, the band topology of its 2D bulk also follows a Z classification[73–75]. For the concerned spin-singlet pairing,the Z invariant is simply given by[76] N = (cid:89) s [ sgn (∆ s )] m s , (4)where m s denotes the number of TRI momenta enclosed bythe s th Fermi surface, and sgn (∆ s ) denotes the sign of thepairing on the s th Fermi surface. Because the normal statehas both time-reversal symmetry and inversion symmetry, theKramers degeneracy at every momentum forces the doubledegeneracy of Fermi surface, if any. As a result, N isnecessitated to take the trivial value . Therefore, the bulkis always topologically trivial for the concerned spin-singletpairing. However, the inversion symmetry is broken on theboundary, which consequently lifts the Kramers degeneracy.Indeed, on each x -normal edge, the floating bands shown inFigs.1(c)(d) do not have Kramers degeneracy away from thetwo TRI momenta, which, as will be shown in the following,enables the realization of 1D TRI TSC on the boundary.As the floating bands extends over the whole Brillouinzone, they can be described by a truly 1D lattice Hamiltonian.This is sharply distinct to 2D TIs in which a lattice realizationof the helical gapless edge states is known to be impossible.Without loss of generality, we take the simple conduction-valence symmetric case for illustration. In this limit, the float-ing bands shown in Fig.1(c) on one of the x -normal edges issimply described by H f ( k ) = λ sin k s . (5)In the presence of s ± -wave superconductivity, the BdGHamiltonian on the corresponding edge reads H e ( k ) = λ sin k τ ⊗ s − µτ ⊗ s +(∆ − ∆ cos k − ∆ ( k )) τ ⊗ s , (6)where the Pauli matrices τ i act on the particle-hole space, and ∆ ( k ) (cid:39) ∆ ( m − m cos k ) /m (here we provide a gen-eral expression), which is originated from the ∆ cos k termof the pairing under the open boundary condition in the x direction (see details in the Supplemental Material[77]). This1D TRI BdG Hamiltonian also follows a Z classification andthe Z invariant takes a form similar to Eq.(4)[76], N = (cid:89) s [ sgn (∆ s )] , (7)where sgn (∆ s ) denotes the sign of the pairing on the s thFermi point between and π , as illustrated in Fig.2(a). Ac-cording to Eq.(6), there are two Fermi points between and π when µ ∈ ( − λ , λ ) , which are located at k s,a =arcsin( | µ/λ | ) and k s,b = π − k s,a . Following Eq.(7), wethen have N = sgn [(∆ − m ∆ m ) − ∆ (1 − m m ) (1 − µ λ ) ] (8)in the regime µ ∈ ( − λ , λ ) . Under appropriate condition, N can take the nontrivial value − , which corresponds to E 𝑘 𝑠,𝑎 𝑘 𝑠,𝑏 𝑘 (a) 𝑥 𝑥 (b) FIG. 2. (a) The sign of pairing on the Fermi points between and π . The two solid black lines denote the floating bands formedby the gapless boundary states. The dashed blue line denotes theFermi level, and the two dashed purple lines indicate the momentaat which the pairing changes sign. The blue and red star denote theFermi points with negative and positive pairing, respectively. (b)Density profiles of four Majorana Kramers pairs are located at thefour corners of the considered square sample. The inset shows a fewenergy eigenvalues closest to zero. The parameters in (a) and (b) are m = 0 , m = 1 , m = 2 , λ = 0 . , λ = 1 . , (cid:15) , = 0 , ∆ = 0 , ∆ = 0 . . the realization of a 1D BTRITSC. It is noteworthy that when µ > λ , there is no Fermi point, so N always takes thetrivial value , indicating a trivial boundary.To further demonstrate the above analytical results, we con-sider { m , m , m , ∆ , ∆ } = { , , , , . } . Then ac-cording to Eq.(8), we have N = − in the regime µ ∈ ( − λ , λ ) . As a 1D TRI TSC is characterized by the exis-tence of one Majorana Kramers pair on each end[78–83], therealization of BTRITSCs will be manifested by the presenceof Majorana Kramers pairs at the boundary of the BTRITSCs,i.e., the corners of a square sample. As shown in Fig.2(b),the numerical result confirms the prediction. It is notewor-thy that from a bulk perspective, the presence of MajoranaKramers pairs at the corners indicates that the whole systemis a second-order TRI TSC[22, 23]. BTRITSCs in 3D.—
The generalization to 3D is straightfor-ward. We only need to generalize the normal-state Hamilto-nian into a 3D form and keep the pairing term intact. Here weconsider H ( k ) = (cid:15) ( k )Σ + m ( k )Σ + (cid:88) i λ i sin k i Σ i , (9)where (cid:15) ( k ) = (cid:80) i (cid:15) i cos k i and m ( k ) = m − (cid:80) i m i cos k i ( i runs over , and ). Similarly, without lossof generality, we consider that the band inversion occurs at thefour TRI momenta of the k = 0 plane. For such a configura-tion, both the strong and weak Z indices are trivial becausethe product of parity eigenvalues in each of the k , , = 0 /π planes gives the trivial value[7]. To realize this configuration,we take m = 0 , m = m = 1 and m = 3 . As shown inFig.3(a), this configuration realizes 2D spin-degeneracy-liftedfloating bands on the x -normal surfaces. By performing sim-ilar analysis as in 2D, we find that the floating bands of thenormal state are described by H f ( k , k ) = λ sin k s + λ sin k s . (10)In the presence of s ± -wave superconductivity, the correspond-ing surface BdG Hamiltonian takes a very simple form, whichreads H s ( k , k ) = λ sin k τ ⊗ s + λ sin k τ ⊗ s − µτ ⊗ s +(∆ − ∆ (cos k + cos k )) τ ⊗ s . (11)The band topology of H s is just characterized by the Z in-variant given in Eq.(4). As the normal-state Fermi surfaceis determined by ± (cid:113)(cid:80) j =1 , λ j sin k j = µ , and the pairingchanges sign at the nodal line determined by ∆ − ∆ (cos k +cos k ) = 0 , N can be intuitively determined by inspectingthe configuration of Fermi surface and pairing nodal line, asillustrated in Fig.3(b). When the pairing nodal line enclosesone Fermi surface, N = − , and a 2D BTRITSC is real-ized. Similarly, the realization of a 2D BTRITSC is mani-fested by the presence of helical Majorana modes[79, 84], asshown in Figs.3(c)(d). As the helical Majorana modes appearat the boundary of the z -normal surfaces, the whole system isa 3D second-order TRI TSC from a bulk perspective[24]. Effect of an external magnetic field.—
Thus far, the time-reversal symmetry, which prohibits two time-reversal partnerMajorana modes from coupling, has been assumed to be pre-served. Applying a magnetic field will generate a Zeemanterm of the form ( B τ ⊗ s + B τ ⊗ s + B τ ⊗ s ) ⊗ σ ,accordingly breaking this symmetry. Although the MajoranaKramers pairs in 2D and helical Majorana hinge modes in 3Dare no longer protected when the time-reversal symmetry isbroken, the magnetic field can induce interesting topologicalphase transitions. In 2D, the Majorana Kramers pairs can bechanged to solitary MZMs when the magnetic field exceeds acritical value[22]. Remarkably, in 3D, a magnetic field in the x - x plane can immediately gap the helical Majorana hingemodes and lead to the presence of solitary MZMs at certaininversion-related corners[35, 40]. It means that the magneticfield can change the second-order TRI TSC to a third-ordertime-reversal-symmetry-breaking TSC[29, 30]. It is notewor-thy that such a response to magnetic field is sharply distinct tosecond-order TRI TSCs realized by a combination of strongTIs and s ± -wave superconductivity[24, 85]. For the latter,counter-intuitively, the magnetic field cannot gap the helicalMajorana hinge modes, because they have a domain-wall ori-gin therein and the magnetic field cannot directly act on thedomain-wall subspace[85]. Conclusions.—
In this work, we have shown that 1D and2D TRI TSCs can be respectively realized on the boundaryof 2D and 3D trivial band-inverted insulators when their un-protected gapless boundary states are gapped by s ± -wave su-perconductivity. Because the dimension of the BTRITSCs islower than the bulk by one, the BTRITSCs open a new routefor the realization of second-order TRI TSCs. In addition, wefound that by applying a magnetic field, a third-order TSCwith Majorana corner modes can be readily induced from the (a) (b) (c) (d) 𝑘 𝑥 𝑥 E E 𝑘 𝑘 𝑥 𝑥 𝑥 FIG. 3. (a) Normal-state energy spectrum for a geometry with openboundary condition in the x direction and periodic boundary con-dition in the x and x directions. The spectrum is shown along thehigh symmetric lines of the surface Brillouin zone. (b) A configu-ration with N = − . The dashed purple line denotes the pairingnodal line, and the solid red and green lines on its two sides de-note Fermi surfaces with positive and negative pairing, respectively.(c) Superconducting-state energy spectrum for the configuration in(b). The sample takes open boundary condition in the x ( x ) and x directions and periodic boundary condition in the x ( x ) direc-tion. Because the Hamiltonian has C rotation symmetry, the energyspectra for the two cases are the same. The in-gap dispersions are offour-fold degeneracy, which correspond to four pairs of helical Ma-jorana modes. (d) The density profiles of the helical Majorana modesare localized at the hinges of the sample. The inset provides an intu-itive illustration of their distribution on the cubic sample. Commonparameters are (cid:15) , , = 0 , m = 0 , m , = 1 , m = 3 , λ , = 0 . , λ = 1 . . In (b)(c)(d), ∆ = 0 . , ∆ = 0 . and µ = 0 . . second-order TRI TSC realized in this route. Our establishednew scenario unveils that the widely-overlooked trivial band-inverted insulators can also be applied for the realization ofTSCs and concomitant Majorana modes, hopefully broaden-ing the scope of material candidates for TSCs. Acknowledgements.—
This work is supported by theStartup Grant (No. 74130- 18841219) and the National Sci-ence Foundation of China (Grant No. 11904417). ∗ [email protected][1] M. Z. Hasan and C. L. Kane, “ Colloquium : Topological insu-lators,” Rev. Mod. Phys. , 3045–3067 (2010).[2] Xiao-Liang Qi and Shou-Cheng Zhang, “Topological insulatorsand superconductors,” Rev. Mod. Phys. , 1057–1110 (2011). [3] C. L. Kane and E. J. Mele, “ Z topological order and the quan-tum spin hall effect,” Phys. Rev. Lett. , 146802 (2005).[4] Liang Fu, C. L. Kane, and E. J. Mele, “Topological insulatorsin three dimensions,” Phys. Rev. Lett. , 106803 (2007).[5] J. E. Moore and L. Balents, “Topological invariants of time-reversal-invariant band structures,” Phys. Rev. B , 121306(2007).[6] Rahul Roy, “Topological phases and the quantum spin hall ef-fect in three dimensions,” Phys. Rev. B , 195322 (2009).[7] Liang Fu and C. L. Kane, “Topological insulators with inversionsymmetry,” PRB , 045302 (2007).[8] Markus K¨onig, Steffen Wiedmann, Christoph Br¨une, AndreasRoth, Hartmut Buhmann, Laurens W. Molenkamp, Xiao-LiangQi, and Shou-Cheng Zhang, “Quantum spin hall insulatorstate in hgte quantum wells,” Science , 766–770 (2007),https://science.sciencemag.org/content/318/5851/766.full.pdf.[9] Ivan Knez, Rui-Rui Du, and Gerard Sullivan, “Evidence forhelical edge modes in inverted InAs / GaSb quantum wells,”Phys. Rev. Lett. , 136603 (2011).[10] Sanfeng Wu, Valla Fatemi, Quinn D. Gibson, Kenji Watanabe,Takashi Taniguchi, Robert J. Cava, and Pablo Jarillo-Herrero,“Observation of the quantum spin hall effect up to 100 kelvinin a monolayer crystal,” Science , 76–79 (2018).[11] Haijun Zhang, Chao-Xing Liu, Xiao-Liang Qi, Xi Dai, ZhongFang, and Shou-Cheng Zhang, “Topological insulators inbi2se3, bi2te3 and sb2te3 with a single dirac cone on the sur-face,” Nature Physics , 438–442 (2009).[12] Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai,Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z.-X. Shen, “Experimental realization of a three-dimensionaltopological insulator, bi2te3,” Science , 178–181 (2009),https://science.sciencemag.org/content/325/5937/178.full.pdf.[13] Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil,D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Obser-vation of a large-gap topological-insulator class with a singledirac cone on the surface,” Nature Physics , 398–402 (2009).[14] D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Oster-walder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer,Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation oftime-reversal-protected single-dirac-cone topological-insulatorstates in bi te and sb te ,” Phys. Rev. Lett. , 146401(2009).[15] Cheng-Cheng Liu, Jin-Jian Zhou, Yugui Yao, and Fan Zhang,“Weak topological insulators and composite weyl semimetals: β − bi X ( x = Br , i),” Phys. Rev. Lett. , 066801 (2016).[16] Ryo Noguchi, T. Takahashi, K. Kuroda, M. Ochi, T. Shirasawa,M. Sakano, C. Bareille, M. Nakayama, M. D. Watson, K. Yaji,A. Harasawa, H. Iwasawa, P. Dudin, T. K. Kim, M. Hoesch,V. Kandyba, A. Giampietri, A. Barinov, S. Shin, R. Arita,T. Sasagawa, and Takeshi Kondo, “A weak topological insula-tor state in quasi-one-dimensional bismuth iodide,” Nature ,518–522 (2019).[17] D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Oster-walder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov,H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z.Hasan, “A tunable topological insulator in the spin helical diractransport regime,” Nature , 1101–1105 (2009).[18] Liang Fu and C. L. Kane, “Superconducting proximity effectand majorana fermions at the surface of a topological insulator,”Phys. Rev. Lett. , 096407 (2008).[19] Pavan Hosur, Pouyan Ghaemi, Roger S. K. Mong, and AshvinVishwanath, “Majorana modes at the ends of superconductorvortices in doped topological insulators,” Phys. Rev. Lett. , 097001 (2011).[20] Shengshan Qin, Lunhui Hu, Xianxin Wu, Xia Dai, Chen Fang,Fu-Chun Zhang, and Jiangping Hu, “Topological vortex phasetransitions in iron-based superconductors,” Science Bulletin ,1207 – 1214 (2019).[21] Zhongbo Yan, Zhigang Wu, and Wen Huang, “Vortex end ma-jorana zero modes in superconducting dirac and weyl semimet-als,” Phys. Rev. Lett. , 257001 (2020).[22] Zhongbo Yan, Fei Song, and Zhong Wang, “Majorana cornermodes in a high-temperature platform,” Phys. Rev. Lett. ,096803 (2018).[23] Qiyue Wang, Cheng-Cheng Liu, Yuan-Ming Lu, and FanZhang, “High-temperature majorana corner states,” Phys. Rev.Lett. , 186801 (2018).[24] Rui-Xing Zhang, William S. Cole, and S. Das Sarma, “Helicalhinge majorana modes in iron-based superconductors,” Phys.Rev. Lett. , 187001 (2019).[25] Mason J. Gray, Josef Freudenstein, Shu Yang F. Zhao, RyanOConnor, Samuel Jenkins, Narendra Kumar, Marcel Hoek,Abigail Kopec, Soonsang Huh, Takashi Taniguchi, KenjiWatanabe, Ruidan Zhong, Changyoung Kim, G. D. Gu, andK. S. Burch, “Evidence for helical hinge zero modes in an fe-based superconductor,” Nano Letters , 4890–4896 (2019).[26] Yi-Ting Hsu, William S. Cole, Rui-Xing Zhang, and Jay D.Sau, “Inversion-protected higher-order topological supercon-ductivity in monolayer wte ,” Phys. Rev. Lett. , 097001(2020).[27] DinhDuy Vu, Rui-Xing Zhang, and Sankar Das Sarma, “Time-reversal-invariant c -symmetric higher-order topological super-conductors,” (2020), arXiv:2005.03679 [cond-mat.supr-con].[28] Yu-Biao Wu, Guang-Can Guo, Zhen Zheng, and Xu-BoZou, “Boundary-obstructed topological superfluids in staggeredspin-orbit coupled fermi gases,” (2020), arXiv:2007.15886[cond-mat.quant-gas].[29] Zhongbo Yan, “Higher-order topological odd-parity supercon-ductors,” Phys. Rev. Lett. , 177001 (2019).[30] Junyeong Ahn and Bohm-Jung Yang, “Higher-order topolog-ical superconductivity of spin-polarized fermions,” Phys. Rev.Research , 012060 (2020).[31] Josias Langbehn, Yang Peng, Luka Trifunovic, Felix von Op-pen, and Piet W. Brouwer, “Reflection-symmetric second-ordertopological insulators and superconductors,” Phys. Rev. Lett. , 246401 (2017).[32] Hassan Shapourian, Yuxuan Wang, and Shinsei Ryu, “Topo-logical crystalline superconductivity and second-order topolog-ical superconductivity in nodal-loop materials,” Phys. Rev. B , 094508 (2018).[33] Eslam Khalaf, “Higher-order topological insulators and super-conductors protected by inversion symmetry,” Phys. Rev. B ,205136 (2018).[34] Max Geier, Luka Trifunovic, Max Hoskam, and Piet W.Brouwer, “Second-order topological insulators and supercon-ductors with an order-two crystalline symmetry,” Phys. Rev. B , 205135 (2018).[35] Xiaoyu Zhu, “Tunable majorana corner states in a two-dimensional second-order topological superconductor inducedby magnetic fields,” Phys. Rev. B , 205134 (2018).[36] Yuxuan Wang, Mao Lin, and Taylor L. Hughes, “Weak-pairinghigher order topological superconductors,” Phys. Rev. B ,165144 (2018).[37] Chen-Hsuan Hsu, Peter Stano, Jelena Klinovaja, and DanielLoss, “Majorana kramers pairs in higher-order topological in-sulators,” Phys. Rev. Lett. , 196801 (2018).[38] Tao Liu, James Jun He, and Franco Nori, “Majorana corner states in a two-dimensional magnetic topological insulator ona high-temperature superconductor,” Phys. Rev. B , 245413(2018).[39] Zhigang Wu, Zhongbo Yan, and Wen Huang, “Higher-ordertopological superconductivity: Possible realization in fermigases and sr ruo ,” Phys. Rev. B , 020508 (2019).[40] Yanick Volpez, Daniel Loss, and Jelena Klinovaja, “Second-order topological superconductivity in π -junction rashba lay-ers,” Phys. Rev. Lett. , 126402 (2019).[41] Rui-Xing Zhang, William S. Cole, Xianxin Wu, andS. Das Sarma, “Higher-order topology and nodal topologicalsuperconductivity in fe(se,te) heterostructures,” Phys. Rev. Lett. , 167001 (2019).[42] Xianxin Wu, Xin Liu, Ronny Thomale, and Chao-Xing Liu,“High- T c Superconductor Fe(Se,Te) Monolayer: an Intrinsic,Scalable and Electrically-tunable Majorana Platform,” arXive-prints , arXiv:1905.10648 (2019), arXiv:1905.10648 [cond-mat.supr-con].[43] Chuanchang Zeng, T. D. Stanescu, Chuanwei Zhang, V. W.Scarola, and Sumanta Tewari, “Majorana corner modes withsolitons in an attractive hubbard-hofstadter model of cold atomoptical lattices,” Phys. Rev. Lett. , 060402 (2019).[44] Nick Bultinck, B. Andrei Bernevig, and Michael P. Zaletel,“Three-dimensional superconductors with hybrid higher-ordertopology,” Phys. Rev. B , 125149 (2019).[45] Sayed Ali Akbar Ghorashi, Xiang Hu, Taylor L. Hughes, andEnrico Rossi, “Second-order dirac superconductors and mag-netic field induced majorana hinge modes,” Phys. Rev. B ,020509 (2019).[46] Yang Peng and Yong Xu, “Proximity-induced majorana hingemodes in antiferromagnetic topological insulators,” Phys. Rev.B , 195431 (2019).[47] Xiaoyu Zhu, “Second-order topological superconductors withmixed pairing,” Phys. Rev. Lett. , 236401 (2019).[48] Katharina Laubscher, Daniel Loss, and Jelena Klinovaja,“Fractional topological superconductivity and parafermion cor-ner states,” Phys. Rev. Research , 032017 (2019).[49] Xiao-Hong Pan, Kai-Jie Yang, Li Chen, Gang Xu, Chao-XingLiu, and Xin Liu, “Lattice-symmetry-assisted second-ordertopological superconductors and majorana patterns,” Phys. Rev.Lett. , 156801 (2019).[50] Zhongbo Yan, “Majorana corner and hinge modes in second-order topological insulator/superconductor heterostructures,”Phys. Rev. B , 205406 (2019).[51] S. Franca, D. V. Efremov, and I. C. Fulga, “Phase-tunablesecond-order topological superconductor,” Phys. Rev. B ,075415 (2019).[52] Majid Kheirkhah, Yuki Nagai, Chun Chen, and Frank Mar-siglio, “Majorana corner flat bands in two-dimensional second-order topological superconductors,” Phys. Rev. B , 104502(2020).[53] Song-Bo Zhang and Bj¨orn Trauzettel, “Detection of second-order topological superconductors by josephson junctions,”Phys. Rev. Research , 012018 (2020).[54] Bitan Roy, “Higher-order topological superconductors in P -, T -odd quadrupolar dirac materials,” Phys. Rev. B , 220506(2020).[55] Ya-Jie Wu, Junpeng Hou, Yun-Mei Li, Xi-Wang Luo, XiaoyanShi, and Chuanwei Zhang, “In-plane zeeman-field-inducedmajorana corner and hinge modes in an s -wave superconduc-tor heterostructure,” Phys. Rev. Lett. , 227001 (2020).[56] Majid Kheirkhah, Zhongbo Yan, Yuki Nagai, and Frank Mar-siglio, “First- and second-order topological superconductivityand temperature-driven topological phase transitions in the ex- tended hubbard model with spin-orbit coupling,” Phys. Rev.Lett. , 017001 (2020).[57] Xianxin Wu, Wladimir A. Benalcazar, Yinxiang Li, RonnyThomale, Chao-Xing Liu, and Jiangping Hu, “Boundary-obstructed topological high-t c superconductivity in iron pnic-tides,” (2020), arXiv:2003.12204 [cond-mat.supr-con].[58] Apoorv Tiwari, Ammar Jahin, and Yuxuan Wang, “Chiral diracsuperconductors: Second-order and boundary-obstructed topol-ogy,” (2020), arXiv:2005.12291 [cond-mat.mes-hall].[59] Seishiro Ono, Hoi Chun Po, and Haruki Watan-abe, “Refined symmetry indicators for topolog-ical superconductors in all space groups,” Sci-ence Advances (2020), 10.1126/sciadv.aaz8367,https://advances.sciencemag.org/content/6/18/eaaz8367.full.pdf.[60] Seishiro Ono, Hoi Chun Po, and Ken Shiozaki, “ z -enrichedsymmetry indicators for topological superconductors in the1651 magnetic space groups,” (2020), arXiv:2008.05499[cond-mat.supr-con].[61] Zhijun Wang, P. Zhang, Gang Xu, L. K. Zeng, H. Miao, Xi-aoyan Xu, T. Qian, Hongming Weng, P. Richard, A. V. Fedorov,H. Ding, Xi Dai, and Zhong Fang, “Topological nature of the fese . te . superconductor,” Phys. Rev. B , 115119 (2015).[62] Xianxin Wu, Shengshan Qin, Yi Liang, Heng Fan, and Jiang-ping Hu, “Topological characters in Fe(te − x se x ) thin films,”Phys. Rev. B , 115129 (2016).[63] Gang Xu, Biao Lian, Peizhe Tang, Xiao-Liang Qi, and Shou-Cheng Zhang, “Topological superconductivity on the surfaceof fe-based superconductors,” Phys. Rev. Lett. , 047001(2016).[64] Peng Zhang, Koichiro Yaji, Takahiro Hashimoto, Yuichi Ota,Takeshi Kondo, Kozo Okazaki, Zhijun Wang, Jinsheng Wen,GD Gu, Hong Ding, et al. , “Observation of topological super-conductivity on the surface of an iron-based superconductor,”Science , 182–186 (2018).[65] Dongfei Wang, Lingyuan Kong, Peng Fan, Hui Chen, ShiyuZhu, Wenyao Liu, Lu Cao, Yujie Sun, Shixuan Du, JohnSchneeloch, et al. , “Evidence for majorana bound states in aniron-based superconductor,” Science , 333–335 (2018).[66] Lingyuan Kong, Shiyu Zhu, Michał Papaj, Hui Chen, Lu Cao,Hiroki Isobe, Yuqing Xing, Wenyao Liu, Dongfei Wang, PengFan, et al. , “Half-integer level shift of vortex bound states in aniron-based superconductor,” Nature Physics , 1–7 (2019).[67] T Machida, Y Sun, S Pyon, S Takeda, Y Kohsaka, T Hanaguri,T Sasagawa, and T Tamegai, “Zero-energy vortex bound statein the superconducting topological surface state of fe (se, te),”Nature materials , 1 (2019).[68] Qin Liu, Chen Chen, Tong Zhang, Rui Peng, Ya-Jun Yan,Chen-Hao-Ping Wen, Xia Lou, Yu-Long Huang, Jin-Peng Tian,Xiao-Li Dong, Guang-Wei Wang, Wei-Cheng Bao, Qiang-HuaWang, Zhi-Ping Yin, Zhong-Xian Zhao, and Dong-Lai Feng,“Robust and clean majorana zero mode in the vortex coreof high-temperature superconductor (li . fe . )OHFeSe ,”Phys. Rev. X , 041056 (2018).[69] C Chen, Q Liu, TZ Zhang, D Li, PP Shen, XL Dong, Z-X Zhao,T Zhang, and DL Feng, “Quantized conductance of majoranazero mode in the vortex of the topological superconductor (li0.84fe0. 16) ohfese,” Chinese Physics Letters , 057403 (2019).[70] Shiyu Zhu, Lingyuan Kong, Lu Cao, Hui Chen, Michał Papaj,Shixuan Du, Yuqing Xing, Wenyao Liu, Dongfei Wang, Cheng-min Shen, Fazhi Yang, John Schneeloch, Ruidan Zhong, GendaGu, Liang Fu, Yu-Yang Zhang, Hong Ding, and Hong-Jun Gao,“Nearly quantized conductance plateau of vortex zero mode inan iron-based superconductor,” Science , 189–192 (2020).[71] I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, “Un- conventional superconductivity with a sign reversal in the or-der parameter of lafeaso − x f x ,” Phys. Rev. Lett. , 057003(2008).[72] Fa Wang and Dung-Hai Lee, “The electron-pairing mechanismof iron-based superconductors,” Science , 200–204 (2011).[73] Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki, and An-dreas W. W. Ludwig, “Classification of topological insulatorsand superconductors in three spatial dimensions,” Phys. Rev. B , 195125 (2008).[74] Alexei Kitaev, “Periodic table for topological insulators andsuperconductors,” in AIP Conference Proceedings , Vol. 1134(AIP, 2009) pp. 22–30.[75] Arbel Haim and Yuval Oreg, “Time-reversal-invariant topolog-ical superconductivity in one and two dimensions,” Physics Re-ports , 1 – 48 (2019), time-reversal-invariant topological su-perconductivity in one and two dimensions.[76] Xiao-Liang Qi, Taylor L. Hughes, and Shou-Cheng Zhang,“Topological invariants for the fermi surface of a time-reversal-invariant superconductor,” Phys. Rev. B , 134508 (2010).[77] The supplemental material contains the derivation of boundaryHamiltonians.[78] Chris L. M. Wong and K. T. Law, “Majorana kramers doubletsin d x − y -wave superconductors with rashba spin-orbit cou-pling,” Phys. Rev. B , 184516 (2012).[79] Fan Zhang, C. L. Kane, and E. J. Mele, “Time-reversal-invariant topological superconductivity and majorana kramers pairs,” Phys. Rev. Lett. , 056402 (2013).[80] Anna Keselman, Liang Fu, Ady Stern, and Erez Berg, “Induc-ing time-reversal-invariant topological superconductivity andfermion parity pumping in quantum wires,” Phys. Rev. Lett. , 116402 (2013).[81] Arbel Haim, Anna Keselman, Erez Berg, and Yuval Oreg,“Time-reversal-invariant topological superconductivity inducedby repulsive interactions in quantum wires,” Phys. Rev. B ,220504 (2014).[82] Erikas Gaidamauskas, Jens Paaske, and Karsten Flens-berg, “Majorana bound states in two-channel time-reversal-symmetric nanowire systems,” Phys. Rev. Lett. , 126402(2014).[83] Constantin Schrade, A. A. Zyuzin, Jelena Klinovaja, andDaniel Loss, “Proximity-induced π josephson junctions intopological insulators and kramers pairs of majorana fermions,”Phys. Rev. Lett. , 237001 (2015).[84] Shusa Deng, Lorenza Viola, and Gerardo Ortiz, “Majoranamodes in time-reversal invariant s -wave topological supercon-ductors,” Phys. Rev. Lett. , 036803 (2012).[85] Majid Kheirkhah, Zhongbo Yan, and Frank Marsiglio, “Vor-tex line topology in iron-based superconductors with and with-out second-order topology,” (2020), arXiv:2007.10326 [cond-mat.supr-con]. Supplemental Material “Boundary Topological Superconductors”
Bo-Xuan Li , Zhongbo Yan , ∗ School of Physics, Sun Yat-sen University, Guangzhou, 510275, China
This supplemental material contains the derivation of boundary Hamiltonians. We start with the normal-state Hamiltonian intwo dimensions (2D), which is given by H ( k ) = ( (cid:15) cos k + (cid:15) cos k )Σ + ( m − m cos k − m cos k )Σ + λ sin k Σ + λ sin k Σ , (S1)where Σ ij = s i ⊗ σ j , with the Pauli matrices s i and σ j acting respectively on the spin ( ↑ , ↓ ) and orbital ( a, b ) degrees of freedom. s and σ are two-by-two unit matrices.Consider that the band inversion occurs at the two time-reversal invariant (TRI) momenta ( k , k ) = (0 , and ( π, , we firstmake a Taylor expansion around k = 0 to the second order of k . Accordingly, we have H ( k , k ) = [ (cid:15) ( k ) − (cid:15) k ]Σ + [ m ( k ) + m k ]Σ + λ sin k Σ + λ k Σ , (S2)where (cid:15) ( k ) = (cid:15) + (cid:15) cos k and m ( k ) = m − m − m cos k . In the following, we consider m and λ to be positive.According to the band inversion, we have m ( k ) < for an arbitrary k ∈ ( − π, π ) .To obtain the gapless states on the x -normal edges, we further consider a half-infinity sample which occupies the region ≤ x ≤ + ∞ . Because the translational symmetry is broken in the x direction, the wave vector k needs to be replaced by − i∂ x . Accordingly, the Hamiltonian becomes H ( k , − i∂ x ) = [ (cid:15) ( k ) + (cid:15) ∂ x ]Σ + [ m ( k ) − m ∂ x ]Σ + λ sin k Σ − iλ ∂ x Σ . (S3)Next, we divide the Hamiltonian into two parts, i.e., H = H + H , where H ( k , − i∂ x ) = [ m ( k ) − m ∂ x ]Σ − iλ ∂ x Σ ,H ( k , − i∂ x ) = [ (cid:15) ( k ) + (cid:15) ∂ x ]Σ + λ sin k Σ . (S4)We first solve the eigenvalue equation H ( k , − i∂ x ) ψ ( x ) = Eψ ( x ) under the boundary condition ψ (0) = ψ (+ ∞ ) = 0 . Itis readily found that there are two zero-energy solutions for an arbitrary k ∈ ( − π, π ) . The two solutions take the form ψ α =1 , ( x ) = N sin( κ x ) e − κ x e ik x χ α , (S5)with normalization given by |N | = 4 | κ ( κ + κ ) /κ | , where κ = (cid:113) − m ( k ) m − λ m , and κ = λ m . Furthermore, χ α satisfy Σ χ α = χ α , which can be chosen as χ = | s = 1 , σ = 1 (cid:105) ,χ = | s = − , σ = 1 (cid:105) . (S6)By projecting H into the subspace expanded by χ and χ , we obtain the boundary Hamiltonian, which is given by [ H e ( k )] αβ = (cid:90) + ∞ dx ψ † α ( x ) H ( k , − i∂ x ) ψ β ( x )= (cid:26) [ (cid:15) ( k ) + m ( k ) (cid:15) m ] s + λ sin k s (cid:27) αβ , (S7)or equivalently, H e ( k ) = [ (cid:15) ( k ) + m ( k ) (cid:15) m ] s + λ sin k s = [ m (cid:15) m + (cid:15) cos k − m (cid:15) m cos k ] s + λ sin k s . (S8) (a) (b) (c) E E E k k k FIG. S1. Dispersions of the helical gapless states on the x -normal edges. The blue curves correspond to numerical results, and the red curvescorrespond to analytical results. (a) (cid:15) = (cid:15) = 0 , the numerical and analytical results perfectly agree with each other. (b) (cid:15) = 0 , (cid:15) = 0 . ,the numerical and analytical results show small deviation away from time-reversal invariant momenta. (c) (cid:15) = 0 , (cid:15) = 1 . The increase ofconduction-valence asymmetry increases the deviation between numerical and analytical results, suggesting that the expansion of k to thesecond order is no longer sufficient. For the conduction-valence symmetric case, i.e., (cid:15) , = 0 , Eq.(S8) reduces to Eq.(5) of the main text, and the analytical resultsperfectly agree with the numerical results, as shown in Fig.S1(a). When conduction-valence asymmetry is present and strong,the analytical results still agree with the numerical results well, as shown in Figs.S1(b)(c).After taking into account the s ± -wave pairings, the Hamiltonian needs to be generalized as H BdG ( k ) = [( (cid:15) cos k + (cid:15) cos k ) − µ ]Σ + ( m − m cos k − m cos k )Σ + λ sin k Σ + λ sin k Σ + [∆ − ∆ (cos k + cos k )]Σ . (S9)where Σ ijk = τ i ⊗ s j ⊗ σ k , with the new Pauli matrices τ i acting on the particle-hole degrees of freedom. Similarly, we makean expansion about k , and then do the replacement k → − i∂ x and decomposition H BdG = H + H . Accordingly, we have H ( k , − i∂ x ) = [ m ( k ) − m ∂ x ]Σ − iλ ∂ x Σ ,H ( k , − i∂ x ) = [ (cid:15) ( k ) + (cid:15) ∂ x − µ ]Σ + λ sin k Σ + [∆ − ∆ cos k − (∆ + ∆ ∂ x )]Σ . (S10)Because of the increase of particle-hole degrees of freedom, now there are four zero-energy solutions of the eigenvalue equation H ( k , − i∂ x ) ψ α ( x ) = Eψ α ( x ) . The general form of ψ α ( x ) still reads ψ α =1 , , , ( x ) = N sin( κ x ) e − κ x e ik x χ α , (S11)but now χ α satisfy Σ χ α = χ α , which can be chosen as χ = | τ = 1 , s = 1 , σ = 1 (cid:105) ,χ = | τ = 1 , s = − , σ = 1 (cid:105) ,χ = | τ = − , s = 1 , σ = 1 (cid:105) ,χ = | τ = − , s = − , σ = 1 (cid:105) . (S12)By projecting H into the four-dimensional subspace expanded by χ , , , , we obtain the boundary Hamiltonian, which is givenby [ H e ( k )] αβ = (cid:90) + ∞ dx ψ † α ( x ) H ( k , − i∂ x ) ψ β ( x )= (cid:26) [ (cid:15) ( k ) + m ( k ) (cid:15) m − µ ] τ ⊗ s + λ sin k τ ⊗ s + [∆ − ∆ cos k − ∆ ( k )] τ ⊗ s (cid:27) αβ , (S13)or equivalently, H e ( k ) = (cid:18) m (cid:15) m + (cid:15) cos k − m (cid:15) m cos k − µ (cid:19) τ ⊗ s + λ sin k τ ⊗ s + [∆ − ∆ cos k − ∆ ( k )] τ ⊗ s , (S14)0 (a) (b) E E FIG. S2. Superconducting boundary bands. (a) The parameters of the 2D BdG Hamiltonian are (cid:15) , = 0 , m = 0 , m = 1 , m = 2 , λ = 0 . , λ = 1 . , µ = 0 . , ∆ = 0 and ∆ = 0 . . The lattice size along the direction with open boundary condition is latticespacings. The numerical (blue curves) and analytical (red curves) results show excellent agreement. (b) The parameters of the 3D BdGHamiltonian are (cid:15) , , = 0 , m = 0 , m , = 1 , m = 3 , λ , = 0 . , λ = 1 . , µ = 0 . , ∆ = 0 . and ∆ = 0 . . The lattice sizealong the direction with open boundary condition is lattice spacings. The numerical (blue curves) and analytical (red curves) results showperfect agreement. where, to the first two orders, ∆ ( k ) = |N | (cid:90) + ∞ dx sin( κ x ) e − κ x [∆ + ∆ ∂ x ] sin( κ x ) e − κ x = ∆ + ∆ m ( k ) m = m − m cos k m . (S15)Eq.(S14) is just the Eq.(6) of the main text. According to Eq.(S14), the energy bands are given by E ± , ± ( k ) = ± (cid:115)(cid:18) m (cid:15) m + (cid:15) cos k − m (cid:15) m cos k − µ ± λ | sin k | (cid:19) + [∆ − ∆ cos k − ∆ ( k )] ..