Fate of Majorana zero modes, critical states and non-conventional real-complex transition in non-Hermitian quasiperiodic lattices
aa r X i v : . [ c ond - m a t . d i s - nn ] S e p Fate of Majorana zero modes, critical states and non-conventional real-complextransition in non-Hermitian quasiperiodic lattices
Tong Liu, ∗ Shujie Cheng, † Hao Guo, ‡ and Gao Xianlong § Department of Applied Physics, School of Science,Nanjing University of Posts and Telecommunications, Nanjing 210003, China Department of Physics, Zhejiang Normal University, Jinhua 321004, China Department of Physics, Southeast University, Nanjing 211189, China (Dated: September 22, 2020)We study a one-dimensional p -wave superconductor subject to non-Hermitian quasiperiodic po-tentials. Although the existence of the non-Hermiticity, the Majorana zero mode is still robustagainst the disorder perturbation. The analytic topological phase boundary is verified by calcu-lating the energy gap closing point and the topological invariant. Furthermore, we investigate thelocalized properties of this model, revealing that the topological phase transition is accompaniedwith the Anderson localization phase transition, and a wide critical phase emerges with amplitudeincrements of the non-Hermitian quasiperiodic potentials. Finally, we numerically uncover a non-conventional real-complex transition of the energy spectrum, which is different from the conventional PT symmetric transition. I. INTRODUCTION
The discovery of Anderson localization is giving guid-ance to understand how the disorder affects the mobil-ity of carriers through the spatial distribution of thewave function beyond the framework of the conventionalband theory. After half a century, the Anderson local-ization phenomena were observed in a ultracold atomexperiment with correlated disordered potentials andincommensurate/quasiperiodic potentials . Nowadays,Anderson localization has been one of the important andhighly-explored research subjects in condensed matterphysics . In one-dimensional systems, many researchesshow that even though there exist particle interactions,random disordered or incommensurate disordered exter-nal potentials can form the many-body localization ,which is the many-body version of Anderson localization.A paradigmatic model to understand the Anderson lo-calization is the Aubry-Andr´e-Harper (AAH) model ,in which the increased strength of the incommensuratepotential leads to a localized transition. In some gener-alized AAH models, the rich localization phenomena canbe observed . Another interesting aspect of general-ized AAH models is the presence of the mobility edge,which characterizes the separation between the extendedand the localized region in terms of energy .It should be noted that the one-dimensional AAHmodel can be understood as the projection of the two-dimensional Hofstadter model in the one-dimensionaldirection , which supports topologically protectededge states localized at the boundary, similar to theedge states in quantum Hall insulators . Conse-quently, the topological properties of one-dimensionalquasicrystals have been gradually excavated accordingto this projection . In topology physics, the one-dimensional p -wave superconductor chain is another im-portant paradigm . A key feature of the one-dimensional p -wave superconductor is that it hosts topo- logically protected Majorana zero mode (MZM) ,which promise a platform for the error-free quantum com-putation since the qubits are immune to the weakly disor-dered perturbation . Thus, the interplay of disorderand topology in one-dimensional quasiperiodic latticeswith p -wave superconducting pairing attracted much re-search interest. In a previous research, Cai et.al. uncov-ered that the topological phase transition is accompaniedby the Anderson localization phase transition in a Her-mitian quasiperiodic chain with p -wave superconductingpairing . Further research showed that there is a criticalphase in the topologically non-trivial region .However, when the non-Hermitian and quasiperiodicpotentials are both considered, how does the topologi-cal region change compared to the Hermitian case, doesthe critical phase still exist, and does there exist real-complex transition of eigenenergies, even if the systemis not PT symmetric? In this paper, we are devoted toanswer these questions. Specifically, we deduce the phaseboundary of topological phase transition analytically andverify it from the energy gap, the spatial distributions ofthe MZMs and the topological invariant. In addition, weclarify the extended, critical and localized regions of thismodel by use of the fractal theory. Finally, we uncover areal-complex transition of the system without PT sym-metry, and give a conclusion that the extended phasecorresponds to the region where eigenenergies are totallyreal, whereas the critical and localized phase correspondto the region where eigenenergies are complex.The arrangement of the rest paper is as follows: Sec. IIdescribes the Hamiltonian of the one-dimensional p -wavesuperconductor subject to the non-Hermitian quasiperi-odic potentials and gives the definition of the inverse par-ticipation ratio; Sec. III discusses the fate of Majoranazero modes and the topological phase transition; Sec. IVdiscusses Anderson localization phase transition and thecritical phase; Sec. V discusses the non-conventional real-complex transition of the energy spectrum and presentsthe total phase diagram of the system; we make a sum-mary in Sec. VI. II. MODEL AND HAMILTONIAN
We consider the one-dimensional p -wave superconduc-tor subject to the non-Hermitian quasiperiodic poten-tials, which is described by the following Hamiltonianˆ H = L − X n =1 ( − t ˆ c † n ˆ c n +1 + ∆ˆ c n ˆ c n +1 + H.c. )+ L X n =1 V n ˆ c † n ˆ c n , (1)where ˆ c † n (ˆ c n ) is the fermion creation (annihilation) oper-ator, and L is the total number of sites. Here the nearest-neighbor hopping amplitude t and the p -wave pairingamplitude ∆ are real constants, and V n = V e i παn is the non-Hermitian quasiperiodic potential. A typi-cal choice for parameter α is α = ( √ − /
2. Forcomputational convenience, t = 1 is set as the energyunit. In the topological classification, this model be-longs to the BDI class and it does not preserve PT -symmetry . When ∆ is equal to zero, this model re-duces to the non-Hermitian AAH model , where thelocalized transition and the topological properties arewell understood. When α = 0, this Hamiltonian de-scribes the Kitaev model, where there are topologicallyprotected MZMs . When the imaginary part of thenon-Hermitian potential is omitted, the model reducesto the Hermitian non-Abelian AAH model , in whichthe topological phase transition and the Anderson local-ization transition is well studied.The Hamiltonian (1) can be diagonalized by using theBogoliubov-de Gennes (BdG) transformation:ˆ χ † m = L X n =1 [ u m,n ˆ c † n + v m,n ˆ c n ] , (2)where L denotes the total number of sites, n is the site in-dex, and u m,n , v m,n are the two components of wave func-tions. It is widely known that the particle-hole symme-try is preserved . Under this transformation, the BdGequations can be expressed as (cid:18) ˆ M ˆ∆ − ˆ∆ − ˆ M (cid:19) (cid:18) u m v m (cid:19) = E m (cid:18) u m v m (cid:19) , (3)where ˆ M ij = − t ( δ j,i +1 + δ j,i − ) + V i δ ji , ˆ∆ ij = − ∆( δ j,i +1 − δ j,i − ), u Tm = ( u m, , · · · , u m,L ) and v Tm =( v m, , · · · , v m,L ), E m is the complex eigenenergy, indexedaccording to its real part Re( E m ) and arranged in ascend-ing order with m being the energy level index.By numerically solving Eq. (3), we can obtain the en-ergy spectrum of the system and the components u m,n and v m,n of the wave functions. The inverse participa-tion ratio (IPR) is usually used to study the localization- Figure 1. (Color online) Top panel: The real part of eigen-values of Eq. (1) as a function of V under OBC. Definitely,there are stable MZMs when V < V continuously increases, the MZM eventually vanishes, andthe phase transition point is roughly located at V c = 1 + ∆.Bottom panel: Spatial distributions of φ and ψ for the lowestexcitation modes with V = 1 . V = 1 . V = 1 . φ and ψ are sym-metrically distributed at ends of the chain, which indicatesthe system is in topological phase, whereas they are locatedinside of the chain when V = 1 .
6. Other involved parametersare ∆ = 0 . L = 500. delocalization transition . For any given normal-ized wave function, the corresponding IPR is defined asIPR m = P Ln =1 (cid:16) | u m,n | + | v m,n | (cid:17)hP Ln =1 ( | u m,n | + | v m,n | ) i , (4)which measures the inverse of the number of sites beingoccupied by particles. It is well known that the IPR of anextended state scales like L − which approaches zero inthe thermodynamic limit. However, for a localized state,since only finite number of sites are occupied, the IPR isfinite even in the thermodynamic limit. The mean of IPRover all the 2 L eigenstates is dubbed the MIPR which isexpressed as MIPR = 12 L L X m =1 IPR m . (5) III. FATE OF MAJORANA ZERO MODES
In this part, we will study the fate of the MZMs andthe topological phase transition. The top panel in Fig. 1shows the real part of the energy spectrum of Eq. (1)as a function of the non-Hermitian potential strength V under the open boundary condition (OBC), with the pa-rameters ∆ = 0 . L = 500. As shown in the figure,there are stable MZMs when V < V is larger than the critical value V c , MZMs annihilateand then enter into the bulk of the system. Hence, thesystems will undergo a topological non-trivial to trivialphase transition as V increases, and the visible phasetransition point is about V c = 1 + ∆. Similar to the pre-vious works , MZMs in our system are still local-ized at ends of the system. To understand the Majoranaedge state deeply, we have to introduce the Majorana op-erators, namely λ An = ˆ c † n + ˆ c n and λ Bn = i (ˆ c † n − ˆ c n ), whichobey the relations ( λ βn ) † = λ βn and n λ βn , λ β ′ n ′ o = 2 δ nn ′ δ ββ ′ ,with β, β ′ ∈ { A, B } . Accordingly, in the Majorana pic-ture, the quasi-particle operator in Eq. (2) can be rewrit-ten as ˆ χ † m = 12 L X n =1 [ φ m,n λ An − iψ m,n λ Bn ] , (6)in which φ m,n = ( u m,n + v m,n ) and ψ m,n = ( u m,n − v m,n ).The bottom panel of Fig. 1 plots the spatial distribu-tions of φ and ψ for the lowest excitation mode un-der OBC, with ∆ = 0 . V = 1 . V = 1 . V = 1 .
4, thelowest excitation mode is just the MZM. As the corre-sponding figures show, the Majorana edge states φ and ψ are localized at ends of the system, presenting the chi-ral symmetry. On the contrary, when V = 1 .
6, the lowestexcitation mode is no longer the MZM. As a result, thevisible distributions of φ and ψ in the right bottom panelare located inside the bulk of the system. Therefore, onlyif V is less than V c , the system is topologically non-trivialand supports the MZM.Due to the bulk-edge correspondence, the topologicalproperties of non-Hermitian systems are generally pro-tected by the real gaps . In other words, topolog-ical phase transition occurs with the gap closing. Be-fore we clarify the relationship between the topologicallynon-trivial phase and the real gap, we first deduce thetopological phase transition point V c . Under the periodicboundary condition (PBC), the Hamiltonian in Eq. (1)can be rewritten asˆ H = X nn ′ (cid:20) ˆ c † n M nn ′ ˆ c n ′ + 12 (cid:16) ˆ c † n N nn ′ ˆ c † n ′ + h.c. (cid:17)(cid:21) , (7)where M is a Hermitian matrix and N is an antisymmetricmatrix, respectively expressed as M = V − t · · · − t − t V ... . . . − t − t − t V L , N = − ∆ · · · ∆∆ 0... . . . − ∆ − ∆ ∆ 0 . (8)With the above matrices, we can determine the ex-citation spectrum E m via solving the secular equationdet (cid:2) ( M + N ) ( M − N ) − E m (cid:3) = 0 . Draw on the pre-vious researches where the energy gap is closed at the topological phase transition point , we assumethat there is no exception in our model. Accordingly,the transition point V c can be solved by the equa-tion det [( M + N ) ( M − N )] = 0. Having known thatdet ( M − N ) = det ( M − N ) T = det ( M + N ), then the V c is further determined by this equation det ( M − N ) =0. Eventually, we obtain the following constraint condi-tion L Y n =1 e i παn = (cid:18) t + ∆ V (cid:19) L . (9)In the thermodynamic limit L → ∞ , V c has a real so-lution, and V c = t + ∆ (Here we have considered thecondition that α is approached by the ratio of two adja-cent Fibonacci numbers). We have noticed that this ana-lytic strategy has been used in a Hermitian non-AbelianAAH model . From our analytic result, the introducednon-Hermiticity compresses the topologically non-trivialregion. V − ∆ ∆ g ∆ = 0 . ∆ = 0 . ∆ = 0 . ∆ = 1 . Figure 2. (Color online) The real energy gap ∆ g as a functionof V − ∆ under PBC. Intuitively, the gap closes at V c = 1+∆.The size of the system is L = 1000. In order to verify the accuracy of the previous assump-tion and the analytical V c , and to understand the rela-tionship between topological phase transition and gapclosing, we numerically plot the variation of the real en-ergy gap ∆ g with respect to the non-Hermitian quasiperi-odic potential strength V under PBC, as shown in Fig. 2.It is readily seen that the real energy gap closes at V c = 1 + ∆ even though the size of the system is finite.Besides, the numerical results reflect that the assump-tion we made before are correct. Moreover, the topologi-cal properties of the system are exactly protected by thereal gaps, and the topological phase transition appearswith the gap closing.In addition to the mentioned MZM and the gap-closingpoint, the topological phase transition is more preciselycharacterized by the topological invariant Q . In a p -wavesuperconducting chain, the value of Q = ( − ν is deter-mined by the parity of the number ν of Majorana zeromodes at ends of the chain. For a periodic invariant p -wave superconducting chain, Kitaev defined the topolog-ical invariant as the Pfaffian of the Hamiltonian matrix.However, to identify the topologically non-trivial phaseof a disordered superconducting chain it is more suitableto work with the transfer matrix approach. As shownin Fig. 1, there appear a pair of Majorana zero modeswhen V < t ( u n +1 + u n − ) + V e i παn u n − ∆( v n +1 − v n − ) = Eu n , ∆( u n +1 − u n − ) − V e i παn v n − t ( v n +1 + v n − ) = Ev n . (10)For Majorana zero modes, these equations can be repre-sented in the transfer matrix form (cid:18) ψ j +1 ψ j (cid:19) = T j (cid:18) ψ j ψ j − (cid:19) where T j = (cid:18) V j ∆+ t ∆ − t ∆+ t (cid:19) . (11)If both the two eigenvalues λ and λ of the total transfermatrix T ≡ Π Lj =1 T j are less than 1 or larger than 1,the system is topological non-trivial. We set t = 1 and∆ >
0, then two eigenvalues λ and λ satisfy | λ λ | <
1. If we set | λ | < | λ | < | λ | , the topologicalproperty of the system is determined by the amplitudeof | λ | . Thus, the Lyapunov exponent is defined as R ≡ lim L →∞ L ln | λ ( V, ∆) | . =0.2 =0.5 =0.6 =1.2 Figure 3. (Color online) The topological invariant Q as afunction of V with four chosen ∆. When V < Q = −
1, which corresponds to the topologically non-trivialphase; when
V > Q = 1, which corresponds to thetopologically trivial phase. Intuitively, Q jumps at the phasetransition point, i.e., the gap closing point V c = 1 + ∆. Thesize of the system is L = 500. For 0 < ∆ <
1, we perform a transformation T j = √ ξS ˜ T j S − with S = diag( ξ / , /ξ / ) and ξ = − ∆1+∆ .Thus, the total transfer matrix T become T ( V, ∆) = ( r − ∆1 + ∆ ) L S T ( V √ − ∆ , S − . (12) From the above definition and analysis, we can obtain R ( V, ∆) = R ( V √ − ∆ , −
12 ln( 1 + ∆1 − ∆ ) . (13)When ∆ = 0, the model is reduced to the non-HermitianAAH model and the Lyapunov exponent R ( V,
0) =ln( V ), so R ( V √ − ∆ ,
0) = ln( V √ − ∆ ). According to theabove discussions, the topological transition point is at | λ | = 1, i.e., R ( V, ∆) = 0. From Eq. (13), we obtainthat the topological transition point obeys V c = 1 + ∆.In Fig. 3, we plots the variation of the topological invari-ant Q versus V for different ∆. The topological quantumnumber Q is evaluated by calculating the transfer matrixnumerically. The adopted numerical method is consis-tent with that in Refs. . When V < Q = − V > Q = 1 which corresponds to the topo-logically trivial phase. Intuitively, Q jumps at the phasetransition point, i.e., the gap-closing point V c = 1 + ∆. IV. LOCALIZED TRANSITION AND CRITICALSTATES -2 =0.5 -2 =0.8 -2 =1.2 -2 =1.5 Figure 4. (Color online) MIPR as a function of V with differ-ent ∆. The dashed lines show the sharp increase of the MIPRat phase boundaries V ec = 1 − ∆ and V c = 1 + ∆. The totalnumber of sites is set to be L = 500. Recalling the localized distributions of the lowest ex-citation modes in Fig. 1 when
V > V c , we are awarethat there is an Anderson localization phase transitionas the topological phase transition happens. Figure 4plots the variation of MIPR as a function of the poten-tial strength V with various ∆. Intuitively, the MIPRincreases steeply at V c and approaches 1. Such a phe-nomenon signals a delocalization-localization phase tran-sition, and the region where V > V c denotes the An-derson localization phase. However, the region where V < V c is not necessarily extended phase. Instead, it isdivided into two phases, i.e., the extended phase and thecritical phase, whose MIPR is greater than that of theextended phase and less than that of the localized phase,i.e., forms a platform. The extended-critical phase tran-sition point V ec is readily seen at V ec = 1 − ∆. ( ,V)=(0.5,0.2)( ,V)=(0.5,1)( ,V)=(0.5,2) Figure 5. (Color online) β min as a function of 1 /L at(∆ , V ) = (0 . , . . , . , We further validate our analysis using the fractal the-ory, which has been widely applied in the quasiperiodicmodels . The size of the system L is chosen asthe j th Fibonacci number F j . The advantage of this ar-rangement is that the golden ratio can be approximatelyreplaced by the ratio of the nearest two Fibonacci num-bers, i.e., α = ( √ − / j →∞ F j − /F j . Thena scaling index β m,n can be extracted from the on-siteprobability P m,n = u m,n + v m,n by P m,n ∼ (1 /F j ) β m,n . (14)As the fractal theorem tells, when the wave functions areextended, the maximum of P m,n scales as max ( P m,n ) ∼ (1 /F j ) , implying β min = 1. On the other hand, whenwave functions are localized, P m,n peaks at very few sitesand nearly zero at the others, suggesting max ( P m,n ) ∼ (1 /F j ) and β min = 0. As for the critical wave func-tions, the corresponding β min is located within the in-terval (0 , L = F j sites, thereare 2 F j eigenstates. Therefore, we can distinguish theextended, the critical, and the localized wave functionsby the average of β min (denoted by β min ) over all theeigenstates , and β min is expressed as β min = 12 L L X m =1 β mmin . (15)Figure 5 shows the β min as a function of 1 /L for vari-ous parameter points (∆ , V ). We find that β min tends to 1 at (∆ , V ) = (0 . , .
2) when L is infinite, suggesting thatthe system is in the extended phase. β min extrapolatesto zero at (∆ , V ) = (0 . , , V ) = (0 . , β min in the thermodynamic limit is intuitivelybetween 0 and 1. We emphasize that such an analysisstrategy works for other parameter points as well, andresults can be obtained accordingly. Hence, we can fi-nally verify that there are extended and critical phasesin the topologically non-trivial region, and that the phasetransition point is indeed at V ec = t − ∆. Meanwhile, wecan also confirm that topological phase transition is ac-companied by Anderson localized phase transition, andthe phase transition point is V c . V. NON-CONVENTIONAL REAL-COMPLEXTRANSITION -1.5 -1 -0.5 0 0.5 1 1.5-0.0500.05 V=0.2 -1.5 -1 -0.5 0 0.5 1 1.5-0.15-0.10-0.0500.050.100.15 V=0.5-1.5 -1 -0.5 0 0.5 1 1.5-0.3-0.2-0.100.10.20.3 V=1 -1.5 -1 -0.5 0 0.5 1 1.5-1-0.500.51 V=2(b)(a)(c) (d)
Figure 6. (Color online) The eigenenergies of Eq. (1) with∆ = 0 . L = 5000 under OBC. (a) V = 0 . V = 0 . V = 1 is takenfrom the critical phase, the imaginary parts of eigenenergiesare completely broadening. (d) V = 2 is taken form the lo-calized phase, the imaginary parts of eigenenergies are alsocompletely broadening. Due to the system being non-Hermitian, we turn ourattention back to the energy spectrum of the system.According to precious works, the phenomenon that real-complex transition of the energy mainly exists in a classof systems with PT symmetry . However, forour system without PT symmetry, the phenomenon ofreal-complex transition still exists. One can say that thisis a non-conventional real-complex transition, which isdifferent from the conventional PT symmetric transition.We take ∆ = 0 . L = 5000.In Fig. 6, we display the eigenenergies of Eq. (1) with var-ious V under OBC. As the figure shows, when V = 0 . Figure 7. (Color online) Phase diagram of the model in thispaper. V ec = t − ∆ (the left red dot) is the transition pointof the extended-critical transition and the real-complex tran-sition. V c = t + ∆ (the right red dot) is the transition pointof the critical-localized transition and the topological phasetransition. tended and topologically non-trivial phase. The finite“bad” imaginary energies can be interpreted as the re-sult of the finite size effect. V = 0 . V = 1is in the critical and topologically non-trivial phase, itcan be distinctly shown that the eigenenergies of the sys-tem are complex. The similar phenomenon also occurs inthe case of V = 2, in which the system is in the localizedand topologically trivial phase. We have also checkedother combinations of parameters and get the same re-sults as expected. Accordingly, we settled on such a con-clusion that only the extended phase support the fullyreal eigenenergies, providing a new result to explore therich physics of non-Hermitian systems.Synthesizing the above analyses, we finally obtain thetotal phase diagram of the system, which is shown inFig. 7. As the diagram shows, the left red dot denotes the extended-critical and the real-complex transition point V ec , satisfying V ec = t − ∆. The right red dot corre-sponds to the critical-localized and the topological phasetransition point V c , satisfying V c = t + ∆. VI. SUMMARY
In summary, we have studied the topological proper-ties and investigated the extended, critical and localizedphases of a one-dimensional p -wave superconductor sub-ject to the non-Hermitian quasiperiodic potentials. Byanalysing the energy spectrum, it is shown that there areMZMs protected by the energy gap. We demonstratethat the topological phase transition is accompanied bythe Anderson localization transition, and the analytictopological transition point is verified by calculating theenergy gap and the topological invariant. Furthermore,we find there is a critical region separated form the ex-tended region in the topologically non-trivial phase, andthe extended-critical transition point is numerically ob-tained by the MIPR and the fractal analysis. Surpris-ingly, for our system without PT symmetry, we find anon-conventional real-complex transition of the eigenen-ergies, and the energies in the extended phase are fullyreal. Unfortunately, we are failed to obtain an analyticalexpression of the real-complex transition point. However,it remains an open question to explore the relationshipbetween the extended phase and the real energy, even ifthere is no PT symmetry. ACKNOWLEDGMENTS
This work is supported by the National Natural Sci-ence Foundation of China (Grants No. 11674051, No.11835011 and No. 11774316). ∗ [email protected] † ‡ [email protected] § [email protected] P. W. Anderson, Absence of diffusion incertain randomlattices, Phys. Rev. , 1492 (1958). J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P.Lugan, D. Cl´ement, L. Sanchez-Palencia, P. Bouyer, andA. Aspect, Direct observation of Anderson localization ofmatter waves in a controlled disorder, Nature (London) , 891 (2008). G. Roati, C. D. Errico, L. Fallani, M. Fattori, C. Fort,M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio,Nature (London) , 895 (2008). A. Pal and D. A. Huse, Many-body localization phase tran-sition, Phys. Rev. B , 174411 (2010). R. Nandkishore and D. A. Huse, Many-body localizationand thermalization in quantum stastical mechanics, Annu.Rev. Comdens. Matter Phys. , 15 (2015). R. Vosk, D. A. Huse, and E. Altman, Theory of the many-body localization transition in one-dimensional systems,Phys. Rev. X , 031032 (2015). X. Li, S. Ganeshan, J. H. Pixley, and S. D. Sarma, Many-body localization and quantum nonergodicit in a modelwith a single-particle mobility edge, Phys. Rev. Lett. ,186601 (2015). M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨uschen,M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I.Bloch, Observation of many-body localization of interact-ing fermions in a quasirandom optical lattice, Science ,842 (2015). H. P. L¨uschen, P. Bordia, S. Scherg, F. Alet, E. Alt-man, U. Schneider, and I. Bloch, Observation of slow dy-namics near the many-body localization transition in one-dimensional quasiperiodic systems, Phys. Rev. Lett. ,260401 (2017). T. Kohlert, S. Scherg, X. Li, H. P. L¨uschen, S. D. Sarma, I.Bloch, and M. Aidelsburger, Observation many-body local- ization in a one-dimensional system with a single-particlemobility edge, Phys. Rev. Lett. , 170403 (2019). H. Yao, H. Khouldi, L. Bresque, and L. Sanchez-Palencia,Critical behavior and fractality in shallow one-dimensionalquasi-periodic potentials, Phys. Rev. Lett. , 070405(2019). H. Yao, T. Giamarchi, and L. Sanchez-Palencia, Lieb-Liniger bosons in a shallow quasiperiodic potential: Boseglass phase and fractal Mott Lobes, Phys. Rev. Lett. ,060401 (2020). D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Col-loquium: Many-Body localization, thermalization, and en-tanglement, Rev. Mod. Phys. , 021001 (2019). S. Aubry and G. Andr´e, Analyticity breaking and An-derson localization in incommensurate lattices, Ann. Isr.Phys. Soc. , 133 (1980). P. G. Harper, The General Motion of Conduction Electronsin a Uniform Magnetic Field, with Application to the Dia-magnetism of Metals, Proc. Phys. Soc. London Sect. A ,874 (1955). S. Das. Sarma, S. He, and X. C. Xie, Mobility Edge ina Model One-Dimensional Potential, Phys. Rev. Lett. ,2144 (1988). D. J. Thouless, Localization by a Potential with SlowlyVarying Period, Phys. Rev. Lett. , 2141 (1988). S. Das. Sarma, S. He, and X. C. Xie, Localization, mobil-ity edges, and metal-insulator transition in a class of one-dimensional slowly varying deterministic potentials, Phys.Rev. B , 5544 (1990). J. Biddle, B. Wang, D. J. Priour, and S. Das. Sarma, Lo-calization in one-dimensional incommensurate lattices be-yond the Aubry-Andr model , Phys. Rev. A , 021603(2009). J. Biddle and S. Das. Sarma, Predicted mobility edges inone-dimensional incommensurate optical lattices: An ex-actly solvable model of Anderson localization, Phys. Rev.Lett. , 070601 (2010). J. Biddle, D. J. Priour, B. Wang, and S. Das. Sarma,Localization in one-dimensional lattices with non-nearest-neighbor hopping: Generalized Anderson and Aubry-Andr´e models, Phys. Rev. B , 075105 (2011). T. Liu, H. Guo, Y. Pu, and S. Longhi, Generalized Aubry-Andr´e self-duality and Mobility edges in non-Hermitianquasi-periodic lattices, Phys. Rev. B , 024205 (2020). H. P. Luschen, S. Scherg, T. Kohlert, M. Schreiber, P.Bordia, X. Li, S. Das. Sarma, and I. Bloch, Single-ParticleMobility Edge in a One-Dimensional Quasiperiodic OpticalLattice, Phys. Rev. Lett. , 160404 (2018). T. Liu and H. Guo, Novel mobility edges in the off-diagonaldisordered tight-binding models, Phys. Rev. B , 104201(2018). Z. Xu, H. Huangfu, Y. Zhang, and S. Chen, Dynamicalobservation of mobility edges in one-dimensional incom-mensurate optical lattice, New J. Phys. , 013036 (2020). X. Li, and S. Das. Sarma, Mobility edge and intermediatephase in one-dimensional incommensurate lattice poten-tials, Phys. Rev. B , 064203 (2020). D. R. Hofstadter, Energy levels and wave functions ofBloch electrons in rational and irrational magnetic fields,Phys. Rev. B , 2239 (1976). J. Wang, X.-J. Liu, G. Xianlong, and H. Hu, Phase dia-gram of a non-Abelian Aubry-Andr´e-Harper model withp-wave superfluidity, Phys. Rev. B , 104504 (2016). D. J. Thouless, M. Kohmoto, M. P. Nightingale, andM. den. Nijs, Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett. , 405(1982). Y. Hatsugai, Chern number and edge states in the integerquantum Hall effect, Phys. Rev. Lett. , 3697 (1993). Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O.Zilberberg, Topological States and Adiabatic Pumping inQuasicrystals, Phys. Rev. Lett. , 106402 (2012). Y. E. Kraus, and O. Zilberberg, Topological Equivalencebetween the Fibonacci Quasicrystal and the Harper Model,Phys. Rev. Lett. , 116404 (2012). L.-J. Lang, X. Cai, and S. Chen, Edge states and topolog-ical phases in one-dimensional optical superlattices, Phys.Rev. Lett. , 220401 (2012). L.-J. Lang and S. Chen, Majorana fermions in density-modulated p-wave superconducting wires, Phys. Rev. B , 205135 (2012). S. Ganeshan, K. Sun, and S. Das. Sarma, Topological Zero-Energy Modes in Gapless Commensurate Aubry-Andr´e-Harper Models, Phys. Rev. Lett. , 180403 (2013). M. Z Hassan and C. L. Kane, Rev. Mod. Phys. , 3045(2010). X. L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011). A. Y. Kitaev, Phys. Usp. , 131 (2001). D. A. Ivanov, Phys. Rev. Lett. , 268 (2001). M. Stone and S.-B. Chung, Phys. Rev. B , 014505(2006). L. Fu and C. L. Kane, Phys. Rev. Lett. , 096407 (2008). R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev.Lett. , 077001 (2010). A. C. Potter and P. A. Lee, Phys. Rev. Lett. , 227003(2010). H. Menke and M. M. Hirschmann, Topological quantumwires with balanced gain and loss, Phys. Rev. B , 174506(2017). X. Cai, L.-J. Lang, S. Chen, and Y. Wang, Topologicalsuperconductor to Anderson localization transition in one-domensional incommensurate lattices, Phys. Rev. Lett. , 176403 (2013). C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu,Classification of topological quantum matter with sym-metries, Rev. Mod. Phys. , 035005 (2016). C. M. Bender and S. Boettcher, Real spectra in non-hermitian hamiltonians having PT symmetry, Phys. Rev.Lett. , 5243 (1998). B. Zhu, R. L¨u, and S. Chen, PT symmetry in thenon-Hermitian Su-Schrieffer-Heeger model with complexboundary potentials, Phys. Rev. A , 062102 (2014). C. Yuce, Topological phase in a non-Hermitian PT sym-metric system, Phys. Lett. A , 1213 (2015). Y. Liu, X.-P. Jiang, J. Cao, and S. Chen, Non-Hermitianmobility edges in one-dimensional quasicrystals withparity-time symmetry, Phys. Rev. B , 174205 (2020). L.-Z. Tang, L.-F. Zhang, G.-Q. Zhang, and D.-W. Zhang,Topological Anderson insulators in two-dimensional non-Hermitian disordered systems, Phys. Rev. A , 063612(2020). D.-W. Zhang, L.-Z. Tang, L.-J. Lang, H. Yan, and S.-L.Zhu, Non-Hermitian topological Anderson insulators, Sci.China-Phys. Mech. Astron. , 267062 (2020). W. Gou, T. Chen, D. Xie, T. Xiao, T.-S. Deng, B. Gad-way, W. Yi, and B. Yan, Tunable Nonreciprocal Quantum
Transport through a Dissipative Aharonov-Bohm Ring inUltracold Atoms, Phys. Rev. Lett. , 070402 (2020). F. Alex An, E. J. Meier, and B. Gadway, Engineeringa Flux-Dependent Mobility Edge in Disordered ZigzagChains, Phys. Rev. X , 031045 (2018). D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Non-Hermitian Boundary Modes and Topology, Phys. Rev. L , 056802 (2020). T. Liu, H.-Y. Yan, and H. Guo, Fate of topological statesand mobility edges in one-dimensional slowly varying in-commensurate potentials, Phys. Rev. B , 174207 (2017). K. Esaki, M. Sato, K. Hasebe, and M. Kohmoto, Edgestates and topological phases in non-Hermitian systems,Phys. Rev. B , 205128 (2011). E. Lieb, T. Schultz, and D. Mattis, Two soluble modelsof an antiferromagnetic chain, Ann. Phys. (N.Y.) , 407(1961). I. Snyman, J. Tworzydlo, and C. W. J. Beenakker, Cal-culation of the conductance of a graphene sheet using the Chalker-Coddington network model, Phys. Rev. B ,045118 (2008). P. Zhang and F. Nori, New J. Phys. , 043033 (2016). M. Kohmoto and D. Tobe, Localization problem in aquasiperiodic system with spin-orbit interaction, Phys.Rev. B , 134204 (2008). H. Hiramoto and M. Kohmoto, Scaling analysis ofquasiperiodic systems: Generalized Harper model, Phys.Rev. B , 8225 (1989). Y. Wang, Y. Wang, and S. Shu, Spectral statistics, finite-size scaling and multifractal analysis of quasiperiodic chainwith p-wave pairing, Eur. Phys. J. B , 254 (2016). T. Liu, P. Wang, S. Chen, and G. Xianlong, Phase diagramof a generalized off-diagonal AubryAndr´e model with p-wave pairing, J. Phys. B51