Topological superconductivity in proximity to type-II superconductors
TTopological superconductivity in proximity to type-II superconductors
Alexander Nikolaenko
1, 2 and Falko Pientka
3, 2, ∗ Karazin Kharkiv National University, Kharkiv 61022, Ukraine Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Str. 38, 01187 Dresden, Germany Institut f¨ur Theoretische Physik, Goethe-Universit¨at, 60438 Frankfurt am Main, Germany (Dated: October 1, 2020)One-dimensional systems proximity-coupled to a superconductor can be driven into a topologicalsuperconducting phase by an external magnetic field. Here, we investigate the effect of vortices cre-ated by the magnetic field in a type-II superconductor providing the proximity effect. We identifydifferent ways in which the topological protection of Majorana modes can be compromised and dis-cuss strategies to circumvent these detrimental effects. Our findings are also relevant to topologicalphases of proximitized quantum Hall edge states.
Introduction. — Topological superconductors arethought to exist in numerous different platforms, rang-ing from one-dimensional materials such as atomicchains [1, 2] and semiconductor wires [3–6] to emergentone-dimensional systems such as edge modes of quantumHall systems [7–9] and two-dimensional topological insu-lators [10–12] or Josephson junctions [13–15]. The greatvariety of proposals originates in the conceptual simplic-ity of topological superconductors: any one-dimensionalspinless superconductor is topological. While pairing canbe reliably be induced in one-dimensional systems viathe proximity effect of a parent s -wave superconductor[16], the technological challenge is to break the spindegeneracy in a controlled way that does not impair theinduced superconductivity. Experimental progress so farhas mostly been based on thin superconducting filmsthat can withstand moderate in-plane magnetic fields[6].An alternative avenue are type-II superconductorswhich sustain sizable magnetic fields even in the bulk byallowing for vortices. They are indispensable for topo-logical superconductors platforms based on the quan-tum Hall effect [9] and, in particular, for realizationsof parafermions in proximity-coupled fractional quantumHall systems [7, 8]. Type-II superconductors can evenbe beneficial for nanowire realizations as they allow forout-of-plane magnetic fields, which enables more flexi-ble designs of nanowire networks needed for topologicalquantum computation [17]. Here, we elucidate the ef-fect of vortices inside an s -wave superconductor on theinduced topological phase in a proximity coupled one-dimensional system [see setup in Fig. 1(a)]. We focus, inparticular, on the decay length of Majorana end statesas a measure of the topological protection. Theoretical model. —Our starting point is the Hamilto- ∗ [email protected] nian H = H d + H s + H t , where H d = (cid:88) l,σ ( (cid:15) d − µ ) d † l,σ d l,σ − (cid:88) l,σ ( w l,l +1 d † l,σ d l +1 ,σ + h . c . ) − K (cid:88) l,σ,σ (cid:48) S l d † l,σ σ σ,σ (cid:48) d l,σ (cid:48) (1)describes a one-dimensional chain with on-site energy (cid:15) d , chemical potential µ and nearest-neighbor hoppingstrength w . The operator d l annihilates a fermion in thechain at site R l = ( x l , y c ), where y c is fixed and l runsfrom 1 to L c . We assume the wire to have helical mag-netic order with an exchange splitting K and a spin tex-ture S l = S (cos α l , sin α l ,
0) and α l = 2 k h x l . The chain isplaced on top of a two-dimensional superconductor withnearest-neighbor hopping of strength w s and attractiveinteraction of strength V described by the Hamiltonian H s = − (cid:88) i,σ µc † i,σ c i,σ − (cid:88) (cid:104) i,j (cid:105) ,σ w sij c † i,σ c j,σ + V (cid:88) i c † i ↑ c i ↑ c † i ↓ c i ↓ . (2)The operator c j annihilates a fermion in the supercon-ductor at site R i = ( x i , y i ) defined on a square latticeof size L x × L y . The superconductor and the chain arecoupled via tunneling at sites R l H t = − t (cid:88) l,σ ( c † l,σ d l,σ + d † l,σ c l,σ ) . (3)In the presence of a magnetic field perpendicular to thesuperconductor, the hopping amplitudes acquire Peierlsphases, w sij = w s exp(2 πi (cid:82) R j R i (cid:126)A · d(cid:126)r/ Φ ), where Φ = h/e is the flux quantum and a similar relation for w l,l +1 . Weassume periodic boundary conditions for the supercon-ductor and open boundary conditions for the chain un-less stated otherwise. Throughout the paper we chose w s = 1, µ = 0 . K = 1, S = 2, and (cid:15) d = 3.While the Hamiltonian describes a wire with a spinhelix, it can be mapped to a ferromagnetic wire (or awire with a large Zeeman splitting due to an externalfield) in proximity to a superconductor with spin-orbit a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p (a)(b)FIG. 1. (a) A wire is placed on top of a superconductor ina perpendicular magnetic field. The magnetic field createsvortices, where the superconducting order parameter is sup-pressed as shown by the color scale on the top surface of thesuperconductor. (b) The vortex cores are encircled by super-currents, whose magnitude increases towards the vortex core.The interaction is chosen to be V = − . w s . The size ofthe magnetic unit cell is N x × N y = 32 ×
16. The number ofmagnetic unit cells in each direction is M x = 16 and M y = 1although this choice does not affect the results strongly. coupling. This can be seen by rotating all spins ontoa single axis and performing the unitary transformation d j,σ → exp( − iα j / σ z ) d j,σ , c j,σ → exp( − iα j / σ z ) c j,σ .As a result, the chain fermions have spins aligned along x and the hopping amplitudes in the superconductor trans-form as w sij → exp( ik h ( x i − x j ) σ z ) w ij (and equivalentlyfor w ij ), which couples the spin to orbital motion alongthe x direction. We do not expect qualitative changeswhen a more realistic type of spin-orbit coupling is as-sumed because the topological phase is predominantlyaffected by the spin-orbit component along the chain.We approach the problem in several steps. We firstdetermine the gap in the isolated superconductor H s ina magnetic field self-consistently within mean-field the-ory. From this we can obtain the superconductor Green’sfunction in real space. Finally, we can obtain the spec-trum and the Majorana wavefunction from the Green’sfunction of the full system that accounts for the couplingbetween chain and superconductor via a Dyson equation. Notice that this approach ignores a possible suppressionof the superconducting gap due to the coupling to thechain, which is valid in the weak-coupling limit.The superconducting Hamiltonian H s can be describedin mean-field theory by the Bogoliubov-de Gennes equa-tion [18] H s (cid:18) u n v n (cid:19) = (cid:18) h ∆∆ ∗ − h ∗ (cid:19) (cid:18) u n v n (cid:19) = E n (cid:18) u n v n (cid:19) . (4)The Hamiltonian is spin degenerate and we can thereforesuppress the spin indices. The Nambu vector ( u, v ) T hasdimension 2 L x L y , ∆ is a diagonal matrix, and h ij = − δ ij µ − w sij . The lattice translation symmetry is bro-ken by the magnetic field, however, one can constructmagnetic translation operators that commute with theHamiltonian [19]. The magnetic unit cell is larger by afactor N x × N y , where N x N y a B = Φ , and the totalsize of the lattice is chosen to be multiple of the unit cell L x × L y = M x N x × M y N y . The magnetic translationoperators give rise to a magnetic Bloch theorem (cid:18) u n,k ( (cid:126)r + (cid:126)R ) v n,k ( (cid:126)r + (cid:126)R ) (cid:19) = e i(cid:126)k (cid:126)R (cid:32) e iχ ( (cid:126)r, (cid:126)R ) / u n,k ( (cid:126)r ) e − iχ ( (cid:126)r, (cid:126)R ) / v n,k ( (cid:126)r ) (cid:33) , (5)where (cid:126)R = mN x (cid:126)e x + nN y (cid:126)e y is a unit cell vector and χ ( (cid:126)r, (cid:126)R ) = − π/ Φ (cid:126)A ( (cid:126)R ) · (cid:126)r = 4 πn x/N x in the gauge (cid:126)A = − By(cid:126)e x . Hence, the self-consistency equation takes theform (see App. A) [20]∆( (cid:126)r + (cid:126)R ) = VM x M y e iχ ( (cid:126)r, (cid:126)R ) (cid:88) n,k u n,k ( (cid:126)r ) v ∗ n,k ( (cid:126)r ) f ( E n,k )(6)with f the Fermi distribution.We now return to the full system of the chain coupledto the superconductor. The Green’s function of the chainsatisfies the Dyson equation g d = (1 − Σ g d ) − g d (7)where the self-energy Σ = T g s T describes the tunnel-ing to the superconductor. Here g d = ( E − H d ) − and g s = ( E − H s ) − are the real-space Nambu Green’s func-tions of the chain and superconductor in the absence ofa coupling and T is the tunneling matrix, which equals tτ z on the sites covered by the chain and zero otherwise.The spectrum corresponds to poles of the Green’s func-tion and can hence be obtained fromDet(1 − T g s T g d ) = 0 . (8)Notice that T g s T depends only on the superconductorGreen’s function at the lattice sites covered by the chain.To find the wavefunction of zero modes, it is thereforesufficient to consider a reduced Green function ˜ g s , whichis given by g s projected to the chain sites [21]. The wave-function of a zero mode is then given by the kernel of g − = (cid:18) (˜ g ss ) − tτ z tτ z ( g dd ) − (cid:19) (9)evaluated at zero energy. The local density of states(LDOS) is simply n ( r, E ) = − (1 /π )Im g ee ( E, r ), where g ee is the electronic part of the Green’s function. Topological phase. — The perpendicular magnetic fieldintroduces vortices into the superconductor. To obtainthe spatial dependence of the superconducting order pa-rameter, we solve the self-consistency equation (6). Thetopological phase in the chain can be affected both by thesuppression of the pairing strength as well as by the gra-dient of the superconducting phase due to supercurrentsencircling the vortices [22]. The phase gradient betweenneighboring sites i and j can be written in a gauge in-variant form as∆ θ i,j = φ j − φ i + 4 π Φ (cid:90) R j R i d r · A ( R i ) (10)where φ i = arg ∆ i . The supercurrent flowing betweensites i and j is related to the phase gradient by j i,j =(2 en s /m ) sin ∆ θ i,j , where n s is the geometric mean of thecondensate density at the two sites. Figure 1(b) showsthe real-space image of the vectorial supercurrent at eachsite in the magnetic unit cell. As each vortex carries aflux h/ e , there are two vortices in each unit cell. The su-percurrent decreases further away from the vortex coresand it essentially zero half-way between two vortices.Before we consider the effect of vortices on the topo-logical phase, we first consider the case of zero orbitalmagnetic field as a reference. In this case, the Hamil-tonian possess both particle-hole and time-reversal sym-metry there are different topological phases characterizedby a Z number [23]. The corresponding phase diagramcalculated for a chain with periodic boundary conditionsis shown in Fig. 2(a) (see App. B). The parameters arechosen such that the chain Hamiltonian H d has a singleband that crosses zero energy. Accordingly, at small cou-pling to the superconductor, the chain is typically in thetopological phase. At strong coupling t (cid:38) w s the systemeventually becomes trivial. This can be understood asfollows: in the limit of strong coupling, the chain fermions d l on each site dimerize with the neighboring fermions inthe superconductor c l , pushing the spin-polarized statesaway from the Fermi level. As the model has a chiralsymmetry in the absence of vortices, another phase tran-sition occurs inside the topological phase between phaseswith topological index ν = ± (a)(b)FIG. 2. (a) Phase diagram for the wire coupled to a homoge-neous superconductor. The gray area corresponds to the triv-ial phase, yellow to the topological phase with Z = 1 and cyanto the topological phase with Z = −
1. We chose V = − . N y = 64. The wire and the superconductor are periodicin x -direction. (b) Dependence of the Majorana coherencelength on the vertical hopping t between the wire and thesuperconductor. The blue line corresponds to a homogeneoussuperconductor and the green line to the superconductor withvortices. We chose V = − . k h = 0 .
4, and y c = 2. Thesizes of the systems are: N x × N y = 32 × M x × M y = 16 × L = 300. The inset shows a semi logarithmic plot of the Ma-jorana wavefunction for the case of a superconductor withvortices and t = 0 . larger coupling strengths, the phase transition betweenthe two topological phases with ν = ± Z number (see App. B). The crit-ical point is broadened into a gapless phase (cf. Refs. 22and 24).Placing the wire closer to the vortices increases themaximal phase gradient along the wire as can be seenfrom Fig. 1(b). As a result the size of the gapless phasegrows as the wire approaches the vortices. This trend isvisible in Fig. 3, which compares the inverse coherence FIG. 3. The coherence length for different locations of thewire relative to the vortex cores inside the superconductor,where y c = 0 refers to the position half-way between twovortices. The inset shows the LDOS at site x = 1, y c = 1for a homogeneous superconductor (orange line) and a su-perconductor with vortices (blue line) with t = 0 .
5. Otherparameters are chosen to be the same as in Fig. 2(b). length for different positions of the wire. Besides thereduced topological phase space, the protection of theMajorana modes inside the topological phase is decreasedas the induced coherence length increases. Importantly,however, the phase gradient is basically zero when thewire is right between two vortices, y c = 0, [see Fig. 1(a)]and hence the corresponding phase diagram and inducedcoherence length is similar to the case without vortices.In addition to introducing phase gradients, vortices candegrade the topological protection of Majorana states byreducing the superconducting gap. In a tunneling ex-periment measuring the LDOS [25], this results in anenhanced spectral weight at low energies. The inset ofFig. 3 compares the LDOS in the topological phase withand without vortices. While the zero-energy peak re-mains largely unaffected by the magnetic flux, the fi-nite bias signatures are changed drastically. Most impor-tantly, the coherence peaks of the superconducting sub-strate at E (cid:39) . y c = 0, where thephase gradient is essentially absent and the suppressionof the superconducting order parameter is minimal. Inthe vicinity of the phase transitions, where the coher-ence length is long, the presence of vortices has negligi-ble effect. Deep inside the topological phase, however,the localization is degraded by the vortices, as the de-cay constant ξ − is considerably smaller than in the casewithout orbital magnetic field. This suggests that hy- FIG. 4. The inverse Majorana decay length when the super-conductor is divided into several segments. The plot comparesa homogeneous superconductor without vortices (blue line),with a superconductor with vortices in one segment (red line),four segments (purple line) and twenty segments (green line).The inset shows a schematic representation of the wire abovea superconductor with several isolated parts. The parame-ters are y c = 0, V = − . k h = 0 . N x × N y = 64 × M x × M y = 20 ×
16, and L = 900. bridization between Majorana and vortex states delocal-izes the Majorana states. This is further corroborated bya comparison with the results in Fig. 2(b) and 3, whichwere calculated using a stronger attractive interactionbut otherwise unchanged parameters. There is almostno deviation between the induced coherence length at y c = 0 in the presence of vortices in Fig. 3 (green curve)and the induced coherence length for a vortex-free super-conductor in Fig. 2(b) (blue curve). The reason for thisis that the coherence length in the superconducting sub-strate is short and the vortices have very little overlapwith each other or with the wire. When the interactionstrength is reduced, as in Fig. 4, the vortex size grows andthe Majorana decay length is enhanced by hybridizationwith vortex states even at y c = 0.The increase of the induced coherence length due tovortices can be understood as a consequence of the hy-bridization between Caroli-de Gennes-Matricon states indifferent vortices, which leads to the formation of a bandof subgap states in the superconducting substrate. TheMajorana end states in the chain hybridize with the low-energy extended states in the vortex lattice which in-creases their localization length. Here, the extended na-ture of the vortex states is crucial, as a coupling to lo-calized subgap states cannot lead to delocalization of theMajorana modes. In order to corroborate this interpre-tation, we have divided the superconductor into severalisolated strips by eliminating hoppings w s along a seriesof cuts running perpendicular to the chain (see Fig. 4).The coherence length shown in Fig. 4 monotonically de-creases as the number of segments is increases while allother parameters, including the chain length, are keptthe same. In the maximal case of N segm = 20 segments,when each segment has the length of a magnetic unit cell,the coherence length is comparable to the case withoutvortices over a large range of coupling strengths t . At theoptimal value t (cid:39) .
6, the segmentation of the supercon-ductor leads to a three-fold reduction of the coherencelength.
Conclusions. — Sizable external magnetic fields are arequirement for some of the most promising topologicalsuperconductor realizations to generate one-dimensionalhelical liquids. While superconducting thin films arelimited to field strengths of the order of 1 − [1] 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 , 602607 (2014).[2] M. Ruby, F. Pientka, Y. Peng, F. von Oppen, B. W.Heinrich, and K. J. Franke, Phys. Rev. Lett. , 197204(2015).[3] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. , 177002 (2010).[4] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev.Lett. , 077001 (2010).[5] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P.A. M. Bakkers, and L. P. Kouwenhoven, Science ,10031007 (2012).[6] S. M. Albrecht, A. P. Higginbotham, M. Madsen,F. Kuemmeth, T. S. Jespersen, J. Nyg˚ard, P. Krogstrup,and C. M. Marcus, Nature , 206209 (2016).[7] N. H. Lindner, E. Berg, G. Refael, and A. Stern, Phys.Rev. X , 041002 (2012).[8] D. J. Clarke, J. Alicea, and K. Shtengel, Nat. Commun. , 1348 (2013).[9] G.-H. Lee, K.-F. Huang, D. K. Efetov, D. S. Wei, S. Hart,T. Taniguchi, K. Watanabe, A. Yacoby, and P. Kim,Nature Physics , 693698 (2017).[10] L. Fu and C. L. Kane, Phys. Rev. B , 161408 (2009).[11] E. Bocquillon, R. S. Deacon, J. Wiedenmann, P. Leub-ner, T. M. Klapwijk, C. Br¨une, K. Ishibashi, H. Buh-mann, and L. W. Molenkamp, Nature Nanotechnology , 137143 (2017).[12] R. S. Deacon, J. Wiedenmann, E. Bocquillon, F. Dom-nguez, T. M. Klapwijk, P. Leubner, C. Brne, E. M. Han-kiewicz, S. Tarucha, K. Ishibashi, H. Buhmann, andL. W. Molenkamp, Phys. Rev. X , 021011 (2017).[13] F. Pientka, A. Keselman, E. Berg, A. Yacoby, A. Stern,and B. I. Halperin, Phys. Rev. X , 021032 (2017).[14] H. Ren, F. Pientka, S. Hart, A. T. Pierce, M. Kosowsky,L. Lunczer, R. Schlereth, B. Scharf, E. M. Hankiewicz,L. W. Molenkamp, B. I. Halperin, and A. Yacoby, Nature , 9398 (2019).[15] A. Fornieri, A. M. Whiticar, F. Setiawan, E. Por-tols, A. C. C. Drachmann, A. Keselman, S. Gronin,C. Thomas, T. Wang, R. Kallaher, G. C. Gardner, E. Berg, M. J. Manfra, A. Stern, C. M. Marcus, andF. Nichele, Nature , 8992 (2019).[16] S. D. Franceschi, L. Kouwenhoven, C. Sch¨onenberger,and W. Wernsdorfer, Nat. Nano. , 703711 (2010).[17] T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonder-son, M. B. Haings, C. Nayak, J. Alicea, K. Flensberg,S. Plugge, Y. Oreg, C. M. Marcus, and M. H. Freed-man, Phys. Rev. B , 235305 (2017).[18] P. Degennes, Superconductivity Of Metals And Alloys (Basic Books, 1994).[19] B. A. Bernevig and T. L. Hughes,
Topological insulatorsand topological supercon- ductors (Princeton UniversityPress, 2013).[20] Y.-D. Zhu, F. C. Zhang, and M. Sigrist, Phys. Rev. B , 1105 (1995).[21] Y. Peng, F. Pientka, L. I. Glazman, and F. von Oppen,Phys. Rev. Lett. , 106801 (2015).[22] A. Romito, J. Alicea, G. Refael, and F. von Oppen,Phys. Rev. B , 020502 (2012).[23] S. Tewari and J. D. Sau, Phys. Rev. Lett. , 150408(2012).[24] F. Pientka, L. I. Glazman, and F. von Oppen, Phys.Rev. B , 155420 (2013).[25] M. Ruby, F. Pientka, Y. Peng, F. von Oppen, B. W.Heinrich, and K. J. Franke, Phys. Rev. Lett. , 087001(2015).[26] B. Nijholt and A. R. Akhmerov, Phys. Rev. B , 235434(2016). Appendix A: Self-consistency equation in the magnetic unit cell
The mean-field superconductor Hamiltonian H mf = (cid:88) i ∆ i c † i ↑ c † i ↓ + ∆ ∗ i c i ↓ c i ↑ , ∆ i = V (cid:104) c i ↓ c i ↑ (cid:105) (A1)can be written in the Nambu basis ( c ↑ , c ↓ , c †↓ , − c †↑ ) T as H s = H ij H ij ∗ − H ∗ ij
00 ∆ ∗ − H ∗ ij (A2)Due to the spin symmetry of the Hamiltonian, eigenstates at energy E take the form( u ( E ) , v ∗ ( − E ) , v ( E ) , − u ∗ ( − E )) T (A3)and the self-consistency relation reads ∆ i = V (cid:88) E v ∗ i ( E ) u i ( E ) f ( E ) . (A4)A typical spatial profile of the pairing strength ∆ i in a magnetic field is shown in Fig. 5. The current that flows fromsite i to j can be derived from the continuity equation as j i → j = iet (cid:126) (cid:88) n (cid:2) ( e − iχ ij u n,i u ∗ n,j − e iχ ij u ∗ n,i u n,j ) f ( E n ) + ( e − iχ ij v ∗ n,i v n,j − e iχ ij v n,i v ∗ n,j ) f ( − E n ) (cid:3) (A5)where χ ij = 2 π/ Φ (cid:82) ji A ( (cid:126)r ) d(cid:126)r .The magnetic translational operators, which commute with Hamiltonian and themselves in the gauge (cid:126)A = − By(cid:126)e x look as follows: ( T Mx ) N x = (cid:88) m,n c † m + N x ,n c m,n ( T My ) N y = (cid:88) m,n c † m,n + N y c m,n e − πmNx . (A6) FIG. 5. Spatial profile of ∆ i in a magnetic unit cell for V = − . w s , N x × N y = 64 × The magnetic unit cell has the size N x × N y , where N x N y a B = Φ . Given the translational operators, we canformulate the magnetic Bloch theorem: (cid:18) u k ( (cid:126)r + (cid:126)R ) v k ( (cid:126)r + (cid:126)R ) (cid:19) = e i(cid:126)k (cid:126)R (cid:32) e iχ ( (cid:126)r, (cid:126)R ) / u k ( (cid:126)r ) e − iχ ( (cid:126)r, (cid:126)R ) / v k ( (cid:126)r ) (cid:33) , (A7)where χ ( (cid:126)r, (cid:126)R ) = 4 πn x/N x , (cid:126)R = mN x (cid:126)e x + nN y (cid:126)e y , and (cid:126)k = 2 πl x N x M x (cid:126)e x + 2 πl y N y M y (cid:126)e y , l i = 0 , , ..M i − , i = { x, y } (A8)The parameters M x and M y denote the number of unit cells in x and y directions. Now we can partially diagonalizethe original Hamiltonian, and perform the calculation for M x × M y matrices of 2 N x N y size, instead of one big matrixof 2 N x M x N y M y size. The self-consistency equation changes in the following way:∆( (cid:126)r + (cid:126)R ) = V (cid:88) n u n ( (cid:126)r + (cid:126)R ) v ∗ n ( (cid:126)r + (cid:126)R ) f ( E n ) = VM x M y e iχ ( (cid:126)r, (cid:126)R ) (cid:88) n,k u n,k ( (cid:126)r ) v ∗ n,k ( (cid:126)r ) f ( E n,k ) (A9)∆( (cid:126)r ) = VM x M y (cid:88) n,k u n,k ( (cid:126)r ) v ∗ n,k ( (cid:126)r ) f ( E n,k ) . (A10)With the help of the Bloch basis | (cid:126)r k , µ (cid:105) = 1( M x M y ) / (cid:88) (cid:126)R e i(cid:126)k · (cid:126)R + iχ µ ( (cid:126)r, (cid:126)R ) / | (cid:126)r + (cid:126)R, µ (cid:105) (A11)we can obtain real space Green function in the magnetic unit cells G ( (cid:126)r + (cid:126)R, (cid:126)r (cid:48) + (cid:126)R (cid:48) , µ, ν ) = 1 M x M y (cid:88) (cid:126)k e i(cid:126)k · ( (cid:126)R − (cid:126)R (cid:48) ) e i ( χ µ ( (cid:126)r, (cid:126)R ) − χ ν ( (cid:126)r (cid:48) , (cid:126)R (cid:48) )) / G ( (cid:126)r, (cid:126)r (cid:48) , µ, ν, (cid:126)k ) (A12) Appendix B: Topological quantum numbers
In the presence of time-reversal symmetry, the topological Z number can be computed by off-diagonalizing theHamiltonian in the particle-hole space using a unitary transformation H = (cid:18) A ( k ) A ∗ ( k ) 0 (cid:19) . The Z number is simply W = πi (cid:82) π ∇ k log det A ( k ) dk . In magnetic field, time-reversal symmetry is broken bysupercurrents. However, the particle-hole operator P = Λ K = iσ y τ y K remains a symmetry. Therefore, it is possibleto define a Z quantum number as Z = sign { Pf(Λ H ( E = 0 , k = 0)) } sign { Pf(Λ H ( E = 0 , k = π )) } In Fig. 6, the Z phase diagrams for a wire placed at y c = 0 and y c = 2 are shown. Phase diagram for y c = 2 Phase diagram for y c = 0FIG. 6. The phase diagram for y = 0 almost coincides with phase diagram for homegeneous superconductor, while case with yy