Non-Hermitian mobility edges in one-dimensional quasicrystals with parity-time symmetry
NNon-Hermitian mobility edges in one-dimensional quasicrystals with parity-timesymmetry
Yanxia Liu, ∗ Xiang-Ping Jiang,
1, 2, ∗ Junpeng Cao,
1, 2, 3 and Shu Chen
1, 2, 4, † Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China Songshan lake Materials Laboratory, Dongguan, Guangdong 523808, China Yangtze River Delta Physics Research Center, Liyang, Jiangsu 213300, China (Dated: February 19, 2020)We investigate localization-delocalization transition in one-dimensional non-Hermitian quasiperi-odic lattices with exponential short-range hopping, which possess parity-time ( PT ) symmetry. Thelocalization transition induced by the non-Hermitian quasiperiodic potential is found to occur atthe PT -symmetry-breaking point. Our results also demonstrate the existence of energy dependentmobility edges, which separate the extended states from localized states and are only associated withthe real part of eigen-energies. The level statistics and Loschmidt echo dynamics are also studied. I. INTRODUCTION
Ever since the seminal work of Anderson , Andersonlocalization has become a fundamental paradigm for thestudy of localization induced by random disorder in con-densed matter physics. While all eigenstates are local-ized in the presence of infinitesimal disorder strengthsin one- and two-dimensional noninteracting systems, lo-calized and extended states can coexist at different en-ergies in three dimensions with a single-particle mobil-ity edge (SPME) , i.e., a critical energy separating lo-calized and delocalized energy eigenstates. As an inter-mediate case between the disordered and periodic sys-tems, quasicrystals display very different behaviors andmay support a localization-delocalization transition evenin one dimension. A well known example is given by aone-dimensional (1D) quasiperiodic system described bythe Aubry-Andr´e (AA) model , which undergoes a lo-calization transition when the strength of the quasiperi-odical potential exceeds a critical point determined bythe self-duality condition. The AA model has beenexperimentally realized in bichromatic optical lattices .By introducing short-range or long-range hopping pro-cesses, some modified AA models may support energy-dependent mobility edges , which were found to ap-pear in other quasiperiodic models . Experimentalobservation of mobility edge and many body localizationin 1D quasiperiodic optical lattices was also reported inrecent works .Recently, there has been growing interest in non-Hermitian Hamiltonians from theory to experiment .In general, the non-Hermiticity is achieved by introduc-ing nonreciprocal hopping processes or gain and lossterms, which may induce exotic phenomena without Her-mitian counterparts, such as parity-time ( PT ) phasetransitions , exceptional points and non-Hermitianskin effect . Interplay of non-Hermiticity and dis-order was studied in terms of the Hatano-Nelson typemodels , in which the nonreciprocal hopping in-troduced in the 1D Anderson model leads to a finite localization-delocalization transition. Effects of non-Hermiticity on quasiperiodic lattices have been studiedin different contexts . However, the non-Hermitianeffect on the mobility edges in quasicrystals is still lack-ing. Since the eigenvalues of a non-Hermitian systemare generally complex, particularly interesting questionsarise here: whether there exist mobility edges in thenon-Hermitian quasiperiodic lattices with short-range orlong-range hopping? If so, how we characterize the non-Hermitian mobility edge?In this work, we address these questions by studyinga non-Hermitian extension of AA model with exponen-tially decaying short-range hopping and PT -symmetry.By analyzing the spatial distribution of wavefunctionsand spectral information, we find that the increase ofquasiperiodic potential strength can lead a localizationtransition at the PT -symmetry breaking point, and un-veil that there exists an intermediate regime with mo-bility edges, which separate the extended states from lo-calized states and are only relevant to real part of spec-trum. We also analyze the level statistics and study theLoschmidt echo dynamics of the system. II. GENERALIZED NON-HERMITIAN AAMODEL
We consider a 1D tight binding model with short rangehopping terms and a non-Hermitian quasiperiodic poten-tial, described by Eu n = (cid:88) n (cid:48) (cid:54) = n te − p | n − n (cid:48) | u n (cid:48) + V n u n , (1)where t is the hopping amplitude with the exponentiallydecaying parameter p > V n isgiven by V n = V cos(2 παn + φ ) . (2)Here V is the potential strength, α is an irrational num-ber and φ = θ + ih describes a complex phase factor. a r X i v : . [ c ond - m a t . d i s - nn ] F e b FIG. 1. Energy eigenvalues and eigenstates of Eq.1 with lattice sites L = 1597, α = ( √ − / p = 1 . h = 0 . V /t = 0 . , . , .
9, and 3 .
0. Distributions of eigenstates corresponding todifferent eigenvalues for the system with
V /t = 1 .
9: in (b1) Re( E ) > Re( E c ) and the corresponding state is an extended stateabove the mobility edge, in (b2) Re( E ) ≈ Re( E c ) and the state is a critical state near the mobility edge, in (b3) Re( E ) < Re( E c )and the state is a localized state below the mobility edge. (c) The shading of real energy curves indicates the magnitude ofthe IPR for the corresponding eigenstates, and the black solid line represents the boundary given by Eq.(6), which separateslocalized and extended states. (d) The corresponding imaginary energies of (c). When h = 0, the model reduces to the Hermitian modelstudied in Ref. , which is an exponential hopping gen-eralization of the AA model. The AA model only includesa nearest-neighbor hopping term with the hopping am-plitude t = te − p , (3)and manifests a localization-delocalization transition forall eigenstates at the self-dual point V = 2 t . For a finite p >
0, the generalized AA model has energy dependentmobility edges given by cosh( p ) = E + tV which was deter-mined by a generalized self-dual transformation . Wenote that the transition point and mobility edges are in-dependent of the value of phase factor θ in the Hermitianlimit.Now we consider the non-Hermitian case with h (cid:54) = 0.Particularly, we shall consider the case with θ = 0, forwhich we have V n = V ∗− n and the non-Hermitian modelfulfills PT symmetry. In the following, we shall studythe PT -symmetric generalized AA (GAA) model with V n = V cos(2 παn + ih ) , (4)and take α = ( √ − / α taking other values of irra-tional numbers. Due to the existence of PT symmetry,one may expect that all eigenvalues of the GAA modelare real if the PT symmetry is unbroken. In Fig.1 (a),we display all the eigenvalues of the system with p = 1 . h = 0 . V in the complex space of energies.For convenience, here we take t as the unit of energy,and the periodic boundary condition (PBC) is consid-ered. It is shown that all the eigenvalues are real when V /t = 0 .
4. Further increasing the potential strength V and exceeding a certain threshold V c /t = 0 . E c becomecomplex accompanying with the breakdown of PT sym-metry, whereas above E c remain real, as shown in Fig.1 (a) for V /t = 1 . .
9. When V exceeds the secondthreshold V c /t = 2 .
02, all eigenvalues are complex, asshown in Fig.1 (a) for
V /t = 3.By inspecting the spatial distribution of the eigen-states, we find that all the states with complex eigen-values are localized states, whereas the states with realeigenvalues are extended states distributing over thewhole lattice. This suggests the localization transitionis simultaneously accompanied by the PT -symmetry-breaking transition. In Fig.1 (b), we display the distri-butions of wavefuntions with the real part of eigenvaluesRe(E) above, close and below the critical value Re( E c )for the system with V /t = 1 .
9. It is clear that the statewith Re(E) above the critical value is an extended stateand the state below the critical value is a localized state.This indicates clearly that there exists a regime where thelocalized and extended states coexist and are separatedby mobility edges, when V is in the region V c < V < V c .Next we determine the mobility edges numerically. Inorder to characterize the localization properties of aneigenstate, we calculate the inverse participation ratio(IPR) defined as IPR ( i ) = (cid:80) n | u in | ( (cid:80) n | u in | ) , (5)where the superscript i denotes the i th eigenstate of thesystem, and n labels the lattice coordinate. Here thecorresponding complex energy E i is indexed accordingto their real part Re( E i ) in ascending order. For a fulllocalized eigenstate, the IPR is finite and IPR (cid:39)
1. Foran extended state, the IPR (cid:39) /L and approaches zerowhen L tends to infinity. In Fig.1(c) and (d), we plot thereal parts and imaginary parts of eigenvalues as well asthe IPR of the corresponding wavefunctions versus thepotential strength V , respectively. The black solid linein the Fig.1(c) marks the transition points, which sepa-rate the extended and localized states, with the values numericalresults R e ( E ) /t V/t -2024 0 0.5 1 1.5-2024 0 0.5 1 1.5-2024 p=1.5h=1.5p=1.4h=1.5 p=1.1h=1.5 p=1.5h=0.2p=1.5h=0.5p=2.0h=1.5 p=0.8h=1.5p=1.5h=3.5 (a3) (a4)(b4)(b3)(a1)(b1) (b2)(a2) FIG. 2. Numerical results of mobility edges obtained fromIPRs and spectrum (red circles) for systems with L = 1597and different parameters (a1)-(a4) p = 1 . h = 3 .
5, 1 . .
5, and 0 .
2, (b1)-(b4) h = 1 . p = 2 .
0, 1 .
4, 1 . .
8, respectively. The black solid lines are obtained by usingEq.(6). of IPR above which approaching zero and below beingfinite. Such a line gives the mobility edge and is foundto be well described by a simple relationcosh( p ) = Re( E ) + tV e h . (6)Despite lack of exact proof, the above analytical relationfor mobility edge boundary agrees well with numericalresults from IPR and spectrum calculations. As shownin Fig.2 (a1)-(a4), the numerical results of mobility edgesfor systems with p = 1 . h are well describedby Eq.(6). In Fig.2 (b1)-(b4), we display the numeri-cal results for systems with fixed h = 1 . p . It is shown that Eq.(6) agrees with numerical resultsfor systems with p = 2 . .
4, and deviation can beobserved for p = 1 .
1. From our numerical results, wefind that Eq.(6) fails to describe SPMEs of systems with p < p is small, the effect oflong-range hopping becomes more important. Althoughthese systems still support mobility edges, we are notable to get a simple analytical expression for them.We have become aware of the existence of mobilityedges in the non-Hermitian GAA model. To distinguishthe region with SPMEs from the extended and localizedregions, it is convenient to consider the normalized par-ticipation ratio (NPR) defined as ,NPR ( i ) = (cid:34) L (cid:88) n | u in | (cid:35) − , (7)which is a complementary quantity for the IPR. Takingaverage over all eigenstates, we can get the averaged NPR( (cid:104) NPR (cid:105) ) and IPR ( (cid:104)
IPR (cid:105) ), which provide complete com-plementary information for the extended, intermediate,and localized phases. We calculate the NPR and IPR forall eigenstates of the non-Hermitian GAA model and dis-play their average values in Fig.3(a), which shows clearlythe existence of three distinct phases depending on thestrength of the quasiperiodic potential
V /t for the given FIG. 3. Mobility edges for the non-Hermitian GAA modelwith lattice sites L = 1597, α = ( √ − /
2, and p = 1 . h = 0 .
5. (b) IPR of all eigenstates for the sys-tem with h = 0 .
5. Here eigenstates numbers are ordered byRe( E ). The white lines mark out the SPME. (c) Phase di-agram in the parameter space spanned by V /t and h . Theblue solid lines are the phase boundaries separated the in-termediate regime from the extended and localized regimes,which can be obtained numerically by using Eq.(6). parameters p = 1 . h = 0 .
5. When the potentialstrength is smaller than the threshold V c /t = 0 . (cid:104) IPR (cid:105) and a finite (cid:104)
NPR (cid:105) . When the potential strengthexceeds the second threshold V c /t = 2 .
02, all eigen-states are localized, as indicated by a finite (cid:104)
IPR (cid:105) and avanishing (cid:104)
IPR (cid:105) . When the potential strength lies in be-tween two thresholds, an intermediate regime with bothfinite (cid:104)
IPR (cid:105) and (cid:104)
NPR (cid:105) is characterized by the coexis-tence of extended and localized states, which can also beread out from the distribution of IPRs for all eigenstatesas shown in Fig.3(b).We display the average IPR in the two-dimensional pa-rameter space
V /t versus h in Fig.3(c), in which the bluesolid lines distinguish the extended, intermediate and lo-calized regime, respectively. When gradually increasing h , the intermediate regime with SPME diminishes. Onthe other hand, if we fix h and increase p , the interme-diate regime with SPME also diminishes. Particularly,when p → ∞ , our model reduces to the non-HermitianAA model with only nearest-neighboring hopping ,and Eq.(6) reduces to V e h = 2 t , (8)indicating the absence of mobility edge. III. LEVEL STATISTICS AND LOSCHMIDTECHO DYNAMICS
The level statistics provides a powerful tool to char-acterize the localization transition in Hermitian disordersystems . For our non-Hermitian model, the eigen-values in the localized regime are complex. The nearest-neighboring level spacing statistics for non-Hermitian dis-order systems has been investigated in terms of non-Hermitian random-matrix theory . According toEq.(6), the mobility edge is only associated with the real
V/t r Re(E)>E c Re(E) FIG. 4. The average of adjacent gap ratio (cid:104) r (cid:105) for systems with L = 1597, α = ( √ − / p = 1 . h = 0 . V versus V /t . part of complex energies, and it is reasonable to count thereal part of level spacings. So, we calculate the adjacentgap ratio r of ordering Re( E ), and is given as r n = min( s n , s n − )max( s n , s n − ) , (9)with s n the level spacing between the real part of the n th and ( n − r n isintroduced as (cid:104) r (cid:105) = 1 L (cid:88) n r n . (10)In Fig.4, we show the real level statistics across the local-ization transition. The average value (cid:104) r (cid:105) approaches tozero in the delocalized phase, whereas approaches 0 . (cid:104) r (cid:105) presents a steplike growth fromzero to 0 . (cid:104) r (cid:105) approaches to thevalue in the extended or localized regime, respectively, asshown in Fig.4.Loschmidt echo is an important quantity for describingquench dynamics , which measures the overlap of aninitial quantum state and its time-evolution state after aquench process. The behavior of Loschmidt echo is re-lated to both the initial state and post-quench states.It was shown that the Loschmidt echo dynamics cancharacterize the localization-delocalization transition instandard AA model , and was applied to study the dy-namical observation of mobility edges in 1D incommensu-rate optical lattices . Here, we explore the Loschmidt-echo characteristic of our non-Hermitian quasiperiodic t L t L FIG. 5. Evolution of Loschmidt echo. The initial state is cho-sen to be the state corresponding to minimum and maximumof real part of eigenvalues of the initial Hamiltonian with (a)and (e) V i = 0 . 2, (b) and (f) V i = 1 . 4, (c) and (g) V i = 2 . V f are shown by different colors. Herewe have set the energy unit t = 1. system. The system is initially prepared in the eigen-state | φ i (cid:105) of an initial Hamiltonian H i with tunable pa-rameter V = V i . Then the potential strength is suddenlyswitched to a new value V f , resulting in a final state | φ f ( t ) (cid:105) = e − itH f | φ i (cid:105) , (11)where e − itH f is the evolution operator after quenchingand (cid:126) = 1 is set for convenience. We need to emphasizethat for the final system with real eigenvalues, the finalstate oscillates over time, and for the final system withcomplex eigenvalues, the final state becomes a steadystate for a long time, which is similar to the imaginarytime evolution for finding the ground state of a Hermitiansystem. The difference is that for the non-Hermitian sys-tem the steady state is an eigenstate of the final systemwith the maximum eigenvalue of imaginary part, insteadof the ground state. The form of Loschmidt echo is L ( t ) = |(cid:104) φ i | φ f ( t ) (cid:105)| (cid:104) φ i | φ i (cid:105) (cid:104) φ f ( t ) | φ f ( t ) (cid:105) , (12)where the denominator is introduced to make sure thatthe initial and final state are normalized. The dynamicsof non-Hermitian system is a kind of non-unitary dynam-ics, due to the existence of complex eigenvalues.Fig.5(a) and 5(e) show the quench dynamics for ini-tial states prepared as eigenstates of the system in theextended regime with V i = 0 . 2, corresponding to min-imum and maximum eigenvalues, respectively. For thefinal systems with V f = 0 . V f = 0 . 7, they locate inthe same regime as the initial system with all eigenval-ues being real, and L ( t ) oscillates with a positive lowerbound, which never approaches zero during the evolu-tion process. When the final system locates in the mixedregime with V f = 1 . V f = 2 . 0, respectively, both thereal and complex eigenvalues coexist, and L ( t ) oscillatesat short time but approaches zero at long times. Whenthe final system is in the localized regime with V f = 2 . L ( t ) exhibits similar behaviour as in the mixing regime.Fig.5(b) and 5(f) show the quench dynamics for ini-tial states prepared in the mixing regime with V i = 1 . V i = 2 . L ( t ) always approaches zero at long times for the finialsystems in different regimes. Our results demonstratethat Loschimt echo exhibits different dynamical behav-iors for systems with initial states in different regimes. IV. SUMMARY In summary, we studied localization transition inducedby non-Hermitian quasiperiodic potential in 1D PT - symmetric quasicrystals, described by the non-HermitianGAA model with exponential hopping. Our resultsdemonstrate that there exist three different regimes, i.e.,extended, mixed and localized phases. While all theeigenstates are either extended or localized in the ex-tended or localized regime, the extended and localizedstates coexist in the mixed regime and are separated byenergy dependent mobility edges. By analyzing the dis-tribution of wavefunctions and corresponding eigenen-ergies, we found that the localization transition is al-ways accompanied by the PT -symmetry breaking tran-sition and the mobility edges only depend on the realpart of energies. We also investigated the level statis-tics and Loschmidt echo dynamics in our non-Hermitianquasiperiodic systems and unveiled that they display dif-ferent behaviors in different regimes. ACKNOWLEDGMENTS The work is supported by NSFC under GrantsNo.11974413 and the National Key Research and De-velopment Program of China (2016YFA0300600 and2016YFA0302104). ∗ These authors contributed equally to this work. † Corresponding author: [email protected] P. W. Anderson, Absence of diffusion in certain randomlattices, Phys. Rev. , 1492(1958). 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