Fate of Majorana zero modes by a modified real-space-Pfaffian method and mobility edges in a one-dimensional quasiperiodic lattice
FFate of Majorana zero modes and mobility edges in a one-dimensional quasiperiodiclattice
Shujie Cheng, Yufei Zhu, Gao Xianlong, ∗ and Tong Liu † Department of Physics, Zhejiang Normal University, Jinhua 321004, China Department of Physics, University of Otago, P.O. Box 56, Dunedin 9054, New Zealand Department of Applied Physics, School of Science,Nanjing University of Posts and Telecommunications, Nanjing 210003, China (Dated: February 2, 2021)We aim to study a one-dimensional p -wave superconductor with quasiperiodic on-site potentials.From the open boundary energy spectra, we find there are Majorana zero modes, whose correspond-ing states are symmetrically distributed at ends of the systems. Further, we numerically calculatethe topological invariant by the Pfaffian, confirming that the Majorana zero modes protected bythe topology. Moreover, the topological phase transition is accompanied by the energy gap closing.In addition, we numerically find that there are mobility edges, and we qualitatively analyze theinfluence of superconducting sequence parameters and on-site potential strength on it. In general,our work enriches the study on the p -wave superconductor with quasiperiodic potentials. I. INTRODUCTION
The Marjorana Fermions are a kind of magic particle,whose antiparticles are themselves . Although havingnot captured it, researchers have found the signaturesof its presence in many systems, such as the semicon-ductor nanowires with strong spin-orbital couplings ,ultracold atoms , magnetic atom chians and theheterojunctions of normal superconductor and topolog-ical insulator . It is precisely because of its mys-tique and its application prospect in topological quantumcomputing that the Majorana Fermion has attractedextensive research interests .The Majorana Fermions are theoretically proven to ex-ist in topological superconductors with p -wave pairings,which appears in Majorna zero mode (MZM) and arelocated at both ends of the system, and are protectedby the topology . Beyond the static systems, peopleuncovered that MZM also exists in periodically drivensystems . In view of different driving engineerings,there will be Majorana π modes . As we all know,disorder will give rise to the localization phenomenon which is dangerous to the topological phases of topolog-ical superconductors . Cai et.al. discussed the in-fluence of the correlated disorder namely the quasiperi-odic disorder on the MZM. They found that with theincrease of the disorder potential, the system wouldundergo the transition from the topological non-trivialphase to the Anderson localized phase. That’s to say,the MZM keeps robustness to the weak disorder. More-over, such a transition can be characterized by the quenchdynamics and the Kibble-Zurek machanism . Wanget.al. detailedly investigated the delocalization prop-erties of the topological phase where MZM exists, andrevealed that it consists of the extended phase and thecritical phase. Non-Hermiticity usually brings some novelphenomena, such as the anomalous boundary states and the skin effect . Recent studies have shown thatthe MZM can stubbornly survive in a non-Hermitian topological superconductors , and an unconventionalreal-complex transition of the energies is investigated .We know that the Kitaev model plays an importantrole to study the topological superconductors and thetopological quantum computing. In this paper, we findthat a class of parameter-modulated quasiperiodic topo-logical superconductors have similar topological proper-ties to the Kitaev model . When the modulation param-eter is small, the topological phase boundary is almostthe same with that of Kitaev model. Besides, the modu-lation parameter can act as an new degree of freedom tomanipulate the topological superconducting phase transi-tion instead of the potential strength. The reason why westudy the quasiperiodic system is that quasiperiodic po-tentials have been realized in experiments . Moreover,quasiperiodic potentials will bring novel phenomenon.For instance, we uncover that there exists mobility edgein our system.The rest of this paper are organized as follows. InSec. II, we discussed the model and calculation methodin detail. In Sec. III We detailedly describe the topo-logical properties and the mobility edge of the system.We also analyze the competition between the modula-tion parameter and superconducting pairing parameterabout the mobility edge. We make a brief summary inSec. IV. II. MODEL AND HAMILTONIAN
Real quantum systems are more or less affected by dis-order. In this paper, we study a one-dimensional p -wavesuperconductor with quasiperiodic disordered on-site po-tentials, which is described by the following tight-bindingHamiltonianˆ H = L − (cid:88) n =1 (cid:16) − t ˆ c † n ˆ c n +1 + ∆ˆ c † n +1 ˆ c † n + h.c. (cid:17) + L (cid:88) n =1 V n ˆ c † n ˆ c n (1) a r X i v : . [ c ond - m a t . d i s - nn ] F e b where ˆ c n (ˆ c † n ) denotes the fermion annihilation (creation)operator, and L is the size of the system with n beingthe site index. The nearest neighbor tunneling strength t and the nearest superconducting pairing parameter ∆are real constants. t = 1 is set as the unit of energy.We give the representation of the quasiperiodic on-sitepotential V n V n = V − b cos(2 παn ) , (2)where V is the potential strength, b ∈ (0 ,
1) is the modu-lation parameter, and α = ( √ − / b = 0, the modelgoes back to the Kitaev model , where V = 2 t di-vides the model into two phases, namely the topologicalnon-trivial phase (V¡2t)and the topological trivial phase(V¿2t). When ∆ = 0, the model is similar to the Aubry-Andr´e model , for the reason that such a potential canbe formed by superposition of incommensurate potentialswith various frequencies V n = V tanh β · sinh β cosh β − cos(2 παn )= V tanh β ∞ (cid:88) r = −∞ e − β | r | e ir (2 παn ) = V tanh β (cid:34) ∞ (cid:88) r =1 e − βr cos [ r (2 παn )] (cid:35) , (3)where cosh β = 1 /b is the constraint condition. When b issmall, the V n can be truncated into the summation withfinite terms and it can be realized by tuning the Ramancoupling .In the particle-hole picture, the Hamiltonian is diago-nalized. In order to obtain the full energy spectrum, weshall introduce the Bogoliubov-de Gennes (BdG) trans-formation ˆ ξ † j = L (cid:88) n =1 (cid:2) u j,n ˆ c † n + v j,n ˆ c n (cid:3) , (4)where j ranges from 1 to L and The components u j,n and v j,n are real numbers. Thus, the Hamiltonian in Eq. (1)is diagonalized asˆ H = L (cid:88) j =1 E j ( ˆ ξ † j ˆ ξ j −
12 ) , (5)where (cid:15) j is the eigenenergy which can be determined bythe following BdG equations (cid:26) − t ( u n − + u n +1 ) + ∆( v n − − v n +1 ) + V n u n = E j u n ,t ( v n − + v n +1 ) + ∆( u n +1 − u n − ) − V n v n = E j v n . (6)Further, we represent the wave function as the followingform | ψ j (cid:105) = ( u j, , v j, , u j, , v j, , · · · , u j,L , v j,L ) T , (7) then, according to the BdG equations, we have the fol-lowing BdG matrix H = A B · · · · · · · · · B † A B · · · · · · B † A B · · · · · · B † A L − B · · · · · · B † A L − B · · · · · · · · · B † A L , (8)where A j = (cid:18) V j − V j (cid:19) , B = (cid:18) − t − ∆∆ t (cid:19) . (9)Intuitively, H is a 2 L × L matrix. By using theSchmidt orthogonal decomposition method to diagonal-ize the BdG matrix, we can acquire the full energy spec-trum (cid:15) j and the associated wave functions | ψ j (cid:105) directly.In the next section, we will discuss the topologicalproperties of the system, such as the topological invari-ant, Majorana zero energy modes and the correspondingstates. Moreover, we will qualitatively analyze the mo-bility edge by the inverse participation ratio. III. RESULTS AND DISCUSSIONS
The topological property of the system is directly char-acterized by a topological invariant. According to Ki-taev’s work, the Hamiltonian in Eq. (1) can be expandedin terms of Majorana operators asˆ H = i L (cid:88) (cid:96),m h (cid:96)m λ (cid:96) λ m , (10)where h lm is real antisymmetric matrix, satisfying h ∗ (cid:96)m = h (cid:96)m = − h m(cid:96) , (11)and λ (cid:96) is Majorana operator with { λ (cid:96) , λ m } = 2 δ (cid:96)m ,which is defined as λ n − ≡ ˆ c † n − + ˆ c n − = λ An ,λ n ≡ i (ˆ c † n − ˆ c n ) = λ Bn , (12)Accordingly, the represented Hamiltonian isˆ H = i (cid:34) L − (cid:88) n =1 (∆ − t ) λ An λ Bn +1 + (∆ + t ) λ Bn λ An +1 − h.c. + L (cid:88) n =1 V n ( λ An λ Bn − λ Bn λ An ) (cid:35) . (13)For a antisymmetric matrix, its Pfaffian is defined asPf( h ) = 12 L L ! (cid:88) τ ∈ S L sgn ( τ ) h τ (1) ,τ (2) · · · h τ (2 L − ,τ (2 L ) , (14)where S L denotes a series of permutations on these2 L elements with sgn ( τ ) being the sign of permutation.With the Pfaffian of the system, then the topological in-variant can be defined as M = sgn (Pf( h )) . (15) Figure 1. (Color Online) Topological invariant versus b and V for systems with α = ( √ − /
2, ∆ = 0 . t . M = − M = 1corresponds to the topological trivial phase. The blue dot-dashed denotes the phase boundary. We calculate the Pfaffian with the periodic boundarycondition, and naturally obtain the topological phase di-agram of the system, which is presented in Fig. 1. Thediagram shows that M = − M = 1 correspondsto the topological trivial phase and the blue dot-dasheddenotes the numerically obtained phase boundary. Weknow that when b = 0, our model goes back to the Ki-taev model, whose transition is located at V = 2 t . Wenotice that when b is taken at small value, the phasetransition is almost the same with that Kitaev model.This implies that our model has a certain ability to resis-tant the disordered perturbations. When b increases, thephase boundary bends in the direction of the decreasing V . It is to say that we can not only realize the topolog-ical phase transition by adjusting the potential strength V , but also manipulate the phase transition by tuningthe external parameter b . Therefore, from these two as-pects, our theoretical scheme promotes the flexibility andadjustability of Kitaev model.In topological systems, there has a special phenomenonthat topological phase transition is accompanied with thegap closing. We find that our model also has this kind ofgenerality. Figure 2 plots the energy gap ∆ g as a functionof the potential strength V with various b . The ∆ g is de-fined as the difference of the L+1th energy level and theLth energy level under periodic boundary condition, i.e. Figure 2. (Color Online) The energy gap ∆ g versus V withvarious b . Other involved parameters are α = ( √ − / . t , and L = 1000. ∆ g = E L +1 − E L . It is readily seen that the topologicalphase transition is related with the gap closing. More-over, when b is small, the gap closing point is almost at V = 2 t . As b increases, the gap closing point moves to-wards the direction of decreasing V . This feature is inaccord with the phase diagram in Fig. 1.The topological non-trivial phase implies the presenceof the Majorana zero energy modes. Figure 3(a) showsthe excitation energy spectrum as a function of the poten-tial strength V under the open boundary condition. Thespectrum reflects that the Majorana zero energy modesare only located in the topological non-trivial phase. Tosee the distributions of zero energy states, we rewrite theBdG operators as η † j = 12 L (cid:88) n =1 [Φ j,n λ An − i Ψ j,n λ Bn ] , (16)where Φ j,n = ( u j,n + v j,n ) and Ψ j,n = ( u j,n − v j,n ).Figures 3(b) and 3(c) respectively plot the spatial dis-tributions Φ and Ψ for the lowest excitation state with V = 1 .
5. Figures 3(d) and 3(e) are distributions for thelowest excitation state with V = 2. When V = 1 . V = 2, the correspondingΦ and Ψ distribute in the bulk of the system. Althoughthe topological trivial case, the system still shows thebulk-edge correspondence. It is interpreted that the sys-tem is in the topological trivial phase and the lowest ex-citation state is no longer the Majorana edge state butthe bulk state. Figure 3. (Color Online) (a) Excitation energy spectrum ofthe system under open boundary condition. (b) and (c) ((d)and (e)) are respectively the spatial distributions of Φ and Ψfor the lowest excitation state with V = 1 . t ( V = 2 t ). Otherinvolved parameters are L = 500, b = 0 . α = ( √ − / . t . We note that in the topological trivial phase, the spa-tial distributions of Φ and Ψ for the lowest excitationstate are no longer localized in the bulk but expandthroughout the whole system. We are aware that such aphenomenon in this topological superconductor has rel-evance with the mobility edge instead of the Andersonlocalization . The localization-delocalization prop-erty can be characterized by the inverse participation ra-tio (IPR). For a given normalized wave function, the as-sociated IPR is defined asIPR j = L (cid:88) n =1 (cid:0) | u j,n | + | v j,n | (cid:1) . (17)It is well know that for a extended wave function, theIPR scales like L − and it approaches 1 for a localizedwave function. We consider b = 0 . V , which are shown in Fig. 4(a), 4(b),4(c), 4(d) respectively. According to the numerical re-sults, the distinction between the extended states andthe localized states can be readily seen from the IPR(the color shows). The transition boundary in energy isjust the mobility edge. In each figure, the blue solid linedenote the mobility edge E c when ∆ = 0, which satisfies E c ≡ t/b = 2 . t and acts as a reference . Intuitively,there exists competition between b and ∆ on the mobilityedge. When ∆ is small, the corresponding mobility edgeis apparently lower than E c . When ∆ gets larger, the Figure 4. (Color Online) The excitation spectrum and IPRas a function of V with ∆ = 0 . t in (a), with ∆ = t in (b),with ∆ = 2 . t in (c), and with ∆ = 4 in (d). The blue solidline denotes the mobility edge with E c = 2 t/b when ∆ = 0.Other involved parameters are b = 0 . α = ( √ − /
2, and L = 500. resulting mobility edge become higher than E c . Turningeyes back to Fig. 4(a), we know that when V = 2 t , theIPR of the lowest excitation state approaches zero, whichsignals the extended states. The result answers why theΨ and Ψ in Fig. 3(d) and 3(e) distribute throughout thewhole system. IV. SUMMARY
Herin, a quasiperiodic p -wave superconductor has beeninvestigated. The model has similar topological charac-teristics to the Kitaev model, such as the identical topo-logical transition point at small modulation parameter,the presence of the Majorana zero modes and the as-sociated edge states. Besides, the model offers a visionthat topological phase transition in superconductors canbe realized by the modulation parameter instead of thepotential strength. Moreover, we uncovered the exis-tence of the mobility edge in this system, and we dis-cussed the competition of modulation parameter and su-perconducting pairing parameter on the mobility. In gen-eral, our theoretical work not only provides a bridge be-tween quasiperiodic p -wave superconductors and the Ki-taev model, but also has led to a further understandingof the quasiperiodic p -wave superconductors. V. ACKNOWLEDGE
Gao Xianlong and Shujie Cheng acknowledge thesupport from NSFC under Grants No.11835011 and No.11774316. Tong Liu acknowledges Natural Sci-ence Foundation of Jiangsu Province (Grant No.BK20200737) and NUPTSF (Grant No.NY220090 andNo.NY220208). ∗ [email protected] † [email protected] X. L. Qi and S. C. Zhang, Topological insulators and su-perconductors, Rev. Mod. Phys. , 1057 (2011). S.-Q. Shen, Topological Insulators (Springer-Verlag,Berlin, 2012). V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A.M. Bakkers, and L. P. Kouwenhoven, Science , 1003(2012). S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuem-meth, T. S. Jesperson, J. Nygard, P. Krogstrup, and C. M.Marcus, Nature , 206 (2016). M. T. Deng, S. Vaitiekenas, E. B. Hansen, J. Danon, M.Leijnse, K. Flensberg, J. Nygard, P. Krogstrup, and C. M.Marcus, Science , 1557 (2016). J. Chen, P. Yu, J. Stenger, M. Hocevar, D. Car, S. R.Plissard, E. P. A. M. Bakkers, T. D. Stanescu, and S. M.Frolov, Sci. Adv. , e1701476 (2017). H. 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