Dynamical evolution in a one-dimensional incommensurate lattice with \mathcal{PT} symmetry
aa r X i v : . [ c ond - m a t . d i s - nn ] J a n Dynamical evolution in a one-dimensional incommensurate lattice with PT symmetry Zhihao Xu
1, 2, 3, ∗ and Shu Chen
2, 4, 5 Institute of Theoretical Physics and State Key Laboratory of Quantum Opticsand Quantum Optics Devices, Shanxi University, Taiyuan 030006, China Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, P.R.China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China Yangtze River Delta Physics Research Center, Liyang, Jiangsu 213300, China
We investigate the dynamical evolution of a parity-time ( PT ) symmetric extension of Aubry-Andr´e (AA) model which exhibits the coincidence of localization-delocalization transition point with PT symmetry breaking point. One can apply the evolution of the profile of the wave packet and thelong-time survival probability to distinguish the localization regimes in PT symmetric AA model.The results of the mean displacement show that when the system is in the PT symmetry unbrokenregime, the wave packet spreading is ballistic which is different from that in PT symmetry brokenregime. Furthermore, we discuss the distinctive features of the Loschmidt echo with the post-quenchparameter being localized in different PT symmetric regimes. PACS numbers:
I. INTRODUCTION
Exploration localization induced by disorder is a long-standing research topic in condensed matter physics. TheAnderson localization induced by random disorder firstproposed by P. W. Anderson since 1958 has found itsway across a wide range of different fields, such as coldatomic gases , quantum optics , acoustic waves and electronic systems . In comparison with the randomdisorder cases, the quasicrystal systems constitute an in-termediate phase between periodic lattices and disordermedia, which displays a long-range order but no period-icity. A paradigmatic example of a one-dimensional qua-sicrystal system is the Aubry-Andr´e (AA) model whichhas attracted increasing interest in recent years . Atypical feature of the AA model is that, the system un-dergoes a metal-insulator transition when the amplitudeof the quasicrystal potential exceeds a finite critical valuewhich is determined by the self-duality property . TheAA model has been experimentally realized by the coldatomic technique in bichromatic optical lattices .On the other hand, thanks to the impressive progressin controlling quantum matter in the last decades, therealizations of the real-time dynamics of quantum sys-tems have achieved on various experimental platformssuch as ultra-cold atoms in optical lattices, trapped ionsand photonic lattices, and have been researched the in-accessible dynamical phenomena. Especially, dynamicalevolution of a wave packet in a disordered system has ob-tained considerable interest. One tries to understand therelation between the energy spectrum and the dynamicalpropagation of the wave packet . Dynamical obser-vation of wave packet evolution and many body localiza-tion in one-dimensional incommensurate optical latticeshas been also reported in recent works . The dynam-ical phase transition based on Loschmidt echo is anothernovel topic attracting wide attention . A dynamical phase transition occurs, when the quench process goesacross the critical point. It corresponds to the Loschmidtecho vanishing which has been successfully applied in theAA model and its extensions .Recently, great interest has been devoted to discuss theinterplay of non-Hermiticity and disorder which bringsnew perspective of the localization properties .Non-Hermitian models are found in a variety of theopen physical systems exchanging energy or particleswith the environment. For a non-Hermitian system,the non-Hermiticity is generally obtained by introduc-ing nonreciprocal hopping terms or gain and loss po-tentials. According to the random matrix theory,the spectral statistics of the non-Hermitian disordersystems exhibits different properties from the Hermi-tian ones . The Hatano-Nelson model describ-ing the interplay of the nonreciprocal hopping and ran-dom disorder exhibits a finite localization-delocalizationtransition . Non-Hermitian extensions of the AAmodels realized by introducing non-reciprocal hopping or PT symmetric potential have been investigated in vari-ous references . For a PT symmetric extensionof the AA model, one can find the coincidence of localiza-tion transition point with PT symmetry breaking pointby both numerical and analytical calculations .The analytical results show that the localization lengthin the insulator phase is independent of energy which issimilar to the Hermitian AA case. On the other hand, theenergy spectrum is gapless in the metallic phase which isunlike the Hermitian AA model . Since the anomalousenergy spectrum of the PT symmetric AA model, someinteresting questions arise here: What is the features ofthe dynamical evolution of the wave packet in the one-dimensional incommensurate lattice with PT symmetry?Whether the Loschmit echo method is still suitable fordetecting the localization transition for PT symmetricAA model? -3 -3 =0.6=0.8 -3 =1.2=1.5 -3 =1.0 (a) (b) (c) FIG. 1: (Color online) Scaling of MIPR for the PT symmetricAA model in (a) λ < J , (b) λ > J , and λ = J , respectively.The dashed line indicates a power-law fitting. Here, J = 1. In this work, we address these questions by study-ing the dynamical evolution of the PT symmetric AAmodel in different localization regimes. We find thatthe propagation of the profile of the wave packet andthe long-time survival probability exhibit distinctive fea-tures in different localization regimes. While the be-haviors of the evolution of the mean displacement aredependent on the breaking of the PT symmetry in thenon-Hermitian AA model. The Loschmidt echo dynamicswith the post-quench parameter localized in PT symme-try unbroken regime can be used to detect localizationtransition which is similar to the Hermitian case. How-ever, for the post-quench parameter localized in PT sym-metry broken regime, our results indicate the detectionof the dynamical phase transition is unavailable by usingLoschmidt echo dynamics. II. MODEL AND HAMILTONIAN
We consider a non-Hermitian extension of the AAmodel with complex incommensurate lattice subjectedto PT symmetry, which can be described by the Hamil-tonian ˆ H = − X j J (ˆ c † j ˆ c j +1 + H.c. ) + X j λ j ˆ n j , (1)where ˆ c j is the annihilation operator of fermions at the j -th site, and J denotes the strength of the hopping term.The on-site PT symmetric potential can be written as λ j = λ exp ( − i παj ) (2)with the amplitude λ and α being irrational. The poten-tial is the complexification of the standard AA modelwith on-site potential λ j = 2 λ ′ cos (2 παj + ϕ ) taking ϕ = ih , and the limits λ ′ → h → ∞ , keeping λ ′ exp ( h ) = λ finite. Numerical and analytical results exhibit a metal-insulator phase transition at λ = J forthe non-Hermitian system which also corresponds to PT symmetric breaking . When λ < J , all the energyspectra are real values and all eigenstates are extended,while for λ > J the energies become complex and alleigenstates are localized with the Lyapunov exponent ξ = ln ( λ/J ) independent of energy.To characterize the localized properties of the PT sym-metric AA model, we can investigate the fractal dimen-sion of the wave functions β defined by MIPR ∝ L − β where the mean inverse participation ratio (MIPR)MIPR = 1 L L X n =1 L X j =1 | ψ n,j | | ψ n,j | (3)with ψ n,j being the amplitude of the eigenstate | ψ n i ofthe eigenvalue E n at the j th site and L is the size ofthe lattice. It is known that β = 0 for localized regime, β = 0 for extended regime, and 0 < β < PT symmetric AA model in differentlocalized regimes. As shown in Fig. 1(a) for λ/J = 0 . . β = 1 and the MIPRstend to 0 with the increase of L in the extended regime.In the localized regime, taking λ/J = 1 . . L with β = 0. For λ/J = 1 [Fig. 1(c)],when L → ∞ , the MIPR approaches to 0 with β ≈ . λ/J = 1. As a comparison,we briefly recall the main conclusions for the HermitianAA model, corresponding to λ j = 2 λ ′ cos (2 παj ) with α being irrational. The system exhibits a transition fromdelocalized phase for λ ′ < J with β = 0 to localizedregion for λ ′ > J with β = 1, and for λ ′ = J , the systemis localized in the multifractal phase with β = 0 . .We can see that the PT symmetric AA model exhibitsthe similar localization properties as the Hermitian one.However, their dynamical evolution of both cases showsome distinctive behaviors.In this paper, we study the dynamical behaviors of anon-Hermitian AA model with PT symmetric potentialsdescribed by the Hamiltonian (1) in real space on a ring,and take J = 1 as the energy unit. The irrational number α = ( √ − / F µ − /F µ with F µ being the µ th Fibonacci numberdefined by F µ = F µ − + F µ − with F = F = 1 yieldingthe size of the lattice L = F µ . III. WAVE PACKET DYNAMICS
We first investigate the expansion dynamics of thewave packet | Ψ(0) i = | j i initially localized at the cen-ter of the lattice j governed by the non-Hermitian AAmodel described by the Hamiltonian (1). The evolution FIG. 2: (Color online) (a)-(c) Evolutions of the profile of thewave function ρ j ( t ) with (a) λ = 0 .
6, (b) λ = 1 .
0, and λ = 1 . P ( r ) for non-Hermitian case with λ = 0 . . t = 800 and λ = 1 . t = 400 marked by the solid lines and Hermitian case with λ ′ = 0 .
6, 1 .
0, and 1 . t = 8000 marked by the dashed lines,respectively. Here, J = 1 and L = F = 377. wave function at time t can be written as | Ψ( t ) i = 1 √N e − i ˆ Ht | Ψ(0) i , (4)with N being normalization coefficient of the time-evolution wave function | Ψ( t ) i .To obtain an intuitive picture, we study the profile ofthe wave function at time t given by ρ j ( t ) = h Ψ j ( t ) | Ψ j ( t ) i shown in Figs. 2(a)-(c) for different λ with L = F =377. For the extended case [Fig. 2(a) with λ = 0 . λ to the localized regime, e.g., λ = 1 . .To further observe the dynamical behaviors of the non-Hermitian system in different phases, we define the long- -4 -3 -2 -1 -4 -3 -2 -1 -4 -3 -2 -1 -4 -3 -2 -1 -4 -3 -2 -1 -4 -3 -2 -1 (a2) (b1) (c1)(b2) (c2)(a1) FIG. 3: (Color online) Time evolution of the mean displace-ment σ ( t ) for the non-Hermitian cases marked by the solidlines and the Hermitian cases marked by the dashed lines.The black dotted lines represent the power-law fitting curves.(a1) and (a2) are for λ = λ ′ = 0 .
6. (b1) and (b2) are for λ = λ ′ = 1 .
0. (c1) and (c2) are for λ = λ ′ = 1 .
5. Here, J = 1and L = F = 377. time survival probability P ( r ) = lim t →∞ X | j − j |≤ r/ h Ψ j ( t ) | Ψ j ( t ) i , (5)which represents the normalized probability of detectingthe wave packet in sites within the region ( − r/ , r/ . It is proportional to ( r/L ) ˜ β with ˜ β = β . Figure 2(d) shows P ( r ) as the function of r/L for both non-Hermitian and Hermitian cases with L = F = 377 in the long time limit. When thestrength of the modulation is localized in the extendedregime, P ( r ) of both cases linearly increase with r , i.e. , P ( r ) ≈ r/L , since the probability of finding the normal-ized wave packet at each site is the same for both cases.For λ = λ ′ = 1, P ( r )s of both cases show the multifractalfeature with ˜ β ∈ (0 , λ ′ = 1 . P ( r ) is finite at r = 0, and presents an exponen-tial rise, and rapidly reaches to ( r/L ) = 1. However, forthe non-Hermitian case with λ = 1 .
5, the value of P ( r )approaches to zero in r/L ≪ r , and rapidly reaches to( r/L ) = 1. The results indicate that the long-time sur-vival probability P ( r ) can be applied to non-Hermitiancase to distinguish the long-time dynamical behaviors ofthe wave packet in different localization regimes. Spe-cially, P ( r ) of the non-Hermitian case in the localizedregime presents a vacuum space in the small r limit whichis different from the Hermitian one.The wave packet spreading dynamics can be describedby the evolution of the mean displacement givenby σ ( t ) = vuut L X j =1 h Ψ j ( t ) | ( j − j ) | Ψ j ( t ) i . (6)In general, for a long expansion time in a Hermitian case,the mean displacement obeys the power-law σ ( t ) ∼ t ˜ γ .Apart from ballistic spread for ˜ γ = 1 and localizationor diffusive transport for ˜ γ = 0 or 1 /
2, subdiffusion for0 < ˜ γ < / / < ˜ γ < γ >
1, the initiallocalized wave packet exhibits a hyperdiffusion transportfor a certain time scale. Figure 3 shows the time evo-lution of σ ( t ) with an initial localized wave packet | j i at the center of the lattice j . The evolution of σ ( t ) indifferent localization regimes for both the non-Hermitianand Hermitian cases is marked by the solid lines andthe dashed lines, respectively. During the first stage ofevolution shown in Figs. 3(a1)-(c1) with λ = λ ′ = 0 . .
0, and 1 .
5, respectively, the wave packet spreading isballistic with the same power-law indices ˜ γ = 1 indepen-dent of the hermiticity of the system and the strengthof the modulation. The longer time evolution of σ ( t )is shown in Figs. 3(a2)-(c2). During the second stage,as shown in Fig. 3(a2) with λ = λ ′ = 0 .
6, both casespresent the ballistic diffusion with ˜ γ = 1. When theon-site potential amplitude is localized in a multifractalregime, the diffusion exponent ˜ γ = 0 . γ = 1)for the non-Hermitian one with λ = 1. For λ = λ ′ > λ = λ ′ = 1 . σ ( t ) exhibits an oscillating character-istic and the diffusion exponent ˜ γ = 0. However, thenon-Hermitian case is quite different. After the ballis-tic spreading stage, σ ( t ) enters into a temporary stagewhose behavior is similar to that in the second stage ofthe Hermitian case. It then presents a hyperdiffusionwith ˜ γ ≈ . The diffu-sion exponent ˜ γ > λ . At last, the wave packet seems to be frozenwith ˜ γ = 0. According to our results, we can see thatthe evolution of the mean displacement σ ( t ) shows a bal-listic spreading for the non-Hermitian AA model (1) in PT symmetry unbroken regime independent of its local-ization properties, which is different from the Hermitiancase. However, when the non-Hermitian AA model en-ters into the localized regime corresponding to PT sym-metry broken regime, σ ( t ) successively undergoes ballis-tic diffusion to localization to hyperdiffusion and back tolocalization. The results of σ ( t ) suggest that the dynam-ics in PT symmetry unbroken regime and broken regimedisplay distinctive behaviors for the non-Hermitian AAmodel with PT symmetry. IV. LOSCHMIDT ECHO DYNAMICS
The Loschmidt echo plays a significant role in charac-terizing the dynamical signature of the quantum phasetransition. It was shown that the Loschmidt echo evo-lution could characterize the localization-delocalizationtransition in a Hermitian AA model . When the ini-tial and post-quench systems are located in the same lo-calization regime, the Loschmidt echo will oscillate with-out decaying to zero in a long time. However, if they arelocated in different localization regimes, the Loschmidtecho will decay and reach nearby zero at some time in-tervals. However, for a PT symmetric AA model, thebehavior of the Loschmidt echo dynamics is still puzzled.In this section, we focus on the quench dynamics ofthe non-Hermitian AA model described by the Hamilto-nian (1). The system is initially prepared in an eigen-state | Ψ( t ) i of the Hamiltonian ˆ H ( λ i ) at time t with h Ψ( t ) | Ψ( t ) i = 1, and then suddenly quenched to the fi-nal Hamiltonian ˆ H ( λ f ). We define the return amplitude G ( t ; λ i , λ f ) = h Ψ( t ) | Ψ( t ) i (7)where | Ψ( t ) i = 1 √N e − i ( t − t ) ˆ H ( λ f ) | Ψ( t ) i , (8)with N being normalization coefficient of the time-evolution wave function | Ψ( t ) i and setting t = 0. Thebehavior of the return probability (Loschmidt echo) canbe described by L ( t ; λ i , λ f ) = | G ( t ; λ i , λ f ) | , (9)where the superscripts i and f are corresponding to be-fore and after the quench process, respectively. The dis-tinctive dynamics in different PT symmetric regimes sug-gest us to study the quench processes in λ f ≤ J and λ f > J , respectively. A. Quench processes with λ f ≤ J We first consider the case of the parameter after thequench process localized in λ f ≤ J region which is a PT symmetry unbroken regime with all energy spectra beingreal. For the limiting case of the quench process from λ i → ∞ to λ f = 0, the initial state is localized in a singlesite, | ψ λ i →∞ m i = P j δ jm ˆ c † j | i = | m i , and the correspond-ing eigenenergy E m = λ i exp ( − i παm ). By performinga quench process to λ f = 0, the eigenstates for the case of λ f = 0 can be written as | ψ λ f =0 k i = 1 / √ L P j e ikj ˆ c † j | i = | k i , with the wave vector k = 2 πl/L ( l = 1 , , · · · , L ).And the corresponding eigenvalue is E k = 2 J cos k . Thereturn amplitude is G ( t ; ∞ ,
0) = h m | e − iH ( λ f ) t | m i = X k h m | e − iH ( λ f ) t | k ih k | m i = 1 L X k e − i Jt cos ( k ) . (10) (a)(b) (c) FIG. 4: (Color online) Evolution of Loschmidt echo with L = F = 610. The initial state is chosen to be the eigenstateof the energy with the lowest real part at different λ i , andthe final Hamiltonian with the parameter (a) λ f = 0, (b) λ f = 0 .
6, and (c) λ f = 1, respectively. The inset of (a) showsthe evolution of dynamical free energy f ( t ) for different λ i and λ f = 0. Here, J = 1. (a)(b)(c) FIG. 5: (Color online) Evolution of Loschmidt echo with L = F = 377 and λ f = 0 .
6. The initial states are chosen to bethe ˜ n -th eigenstates with the corresponding the real part ofthe energies in ascending order for (a) λ i = 0 .
8, (b) λ i = 1 . λ i = 1 .
5, respectively. Here, J = 1. In the thermodynamic limit ( L → ∞ ), G ( t ) = J (2 Jt )with J ( x ) being the zero-order Bessel function of the firstkind. It is clear the zeros of the Loschmidt echo t ∗ occurat the half of the zeros of J ( x ). The emergence of thezeros of the Loschmidt echo implies the initial and finalstates are localized in different localized regimes, accord-ing to previous results in the Hermitian cases. Figure 4(a)shows the evolution of Loschmidt echo with λ f = 0 anddifferent λ i > J . In this case, the Loschmidt echoes L ( t )for λ i = 100, 10, 5, and 3 oscillate the same frequencyas the analytical result. It implies that the frequenciesof the Loschmidt echo are not sensitive to the initial pa-rameter λ i as long as λ i is large enough. To see the zerosof L ( t ) more clearly, we introduce the dynamical free en-ergy f ( t ) = − ln L ( t ), which is shown in the inset of Fig.4(a). It will be divergent at the dynamical phase transi-tion time t ∗ . In the large λ i limit, f ( t ) exhibits obviouspeaks at t = t ∗ almost completely overlap to the analyt-ical result. With the decrease of λ i , the change of thepeak positions of f ( t ) is tiny, while the peak amplitudesof f ( t ) decrease.The analytical result for the limit case of the quenchprocess from λ i = ∞ to λ f = 0 exhibits the nearly zero ofthe Loschmidt echo. Now, we consider the general casesthat λ i and λ f deviate from the limit case for λ f ≤ J .Figures 4(b) and (c) show the evolution of Loschmidtecho with different λ i , the parameter of the final Hamil-tonian deviating from zero, and the initial state beingchosen to be the eigenstate of the energy with the lowestreal part at different λ i . In Fig. 4(b), we choose λ f = 0 . λ i = 0 . . L ( t ) oscillates and has a positive lower boundnever approaching zero during the evolution. However, if λ i > J , L ( t ) approaches to zero after some time intervals[see Fig. 4(b) for λ f = 1 . λ f = 1 where thesystem is localized in the multifractal phase. As seen inFig. 4(c), the long time evolution of the Loschmidt echoapproaches zero for the initial states localized in eitherextended ( λ i = 0 . .
8) or localized ( λ i = 1 . .
5) regime.To indicate our result is independent of the ini-tial eigenstate’s choice, we consider the different eigen-states as the initial states to calculate the evolutionof Loschmidt echo. As a concrete example, we choose λ f = 0 .
6, and the initial states are the ˜ n -th eigenstates ofthe corresponding eigenenergies in ascending order with˜ n = 100, 150, 200, 250, and 300 as shown in Fig. 5.For the case of λ i = 0 . λ i = 1 . . λ f = 0 . λ ≤ J region where all (a) (b)(c) (d) FIG. 6: (Color online) (a) and (b) Evolution of Loschmidtecho for large λ f with L = F = 610, and the initial statebeing chosen to be ground state at λ i = 0. (a) L ( t ; 0 , λ f )versus t . (b) L ( t ; 0 , λ f ) versus the rescaled time λ f t . (c)Evolution of Loschmidt echo for finite λ f with L = F =377, and the initial state being chosen to be the wave packetlocalized at the center of the lattice. (d) Log-log plot of 1 −L ( t ; λ i , λ f ) as the function of time t for λ f = 1 . L = F = 377, and the initial state being chosen to be the wavepacket localized at the center of the lattice. The black dashedline represents a power-law fitting. Here, J = 1. the eignenergies are real, the behaviors of the Loschmidtecho are similar to the Hermitian cases that the dynami-cal signature of localized transition can be characterizedby the emergence of zero points in the evolution of theLoschmidt echo. B. Quench processes with λ f > J In this subsection, we consider the case of the finalHamiltonian with the parameter λ f > J , where all theeigenstates are localized and the PT symmetry is bro-ken. First, an analytical calculation is considered thatthe quench process with λ i = 0 and λ f = ∞ . For λ i = 0,the system is initially prepared in a plane wave state | k i with the eigenvalue E k = 2 J cos k . By performinga sudden quench to ˆ H ( λ f ) in the limit of λ f → ∞ , theeigenstates of ˆ H ( λ f ) are localized in a single site | m i ,and the corresponding eigenenergy is E m = λ f e − i παm .The evolution wave function can be written as | Ψ( t ; λ i = 0 , λ f → ∞ ) i = 1 √N e − i ˆ H ( λ f ) t | k i = 1 √N L X m e ikm e − iλ f te − i παm | m i . (11) In the large L → ∞ limit, we can obtain G ( t ; 0 , ∞ ) = 1 √N ∞ X ν = −∞ ( − ν J ν ( λ f t )I ν ( λ f t ) , (12)where N = I (2 λ f t ) with J ν ( x ) being the Bessel func-tion of the first kind of the index ν and I ν ( x ) beingthe modified Bessel function of the first kind of ν order.Figures 6(a) and (b) show the evolution of Loschmidtecho with the the initial state chosen to be groundstate of ˆ H ( λ i = 0) and large λ f as a function of t ,and the rescaled time λ f t , respectively. The Loschmidtechoes for λ f = 10, 30, and 100 rapidly decay to zerovalue with time, and the larger λ f is the faster L ( t )decays. They nearly overlap with the analytical result L ( t ; 0 , ∞ ) = 1 / N | P ν ( − ν J ν ( λ f t )I ν ( λ f t ) | by rescalingthe time λ f t shown in Fig. 6(b).Another limit is also considered that is the parame-ter λ i → ∞ of the Hamiltonian (1) quenched to finite λ f > J . As an example, we set the initial state to bethe wave packet localized at the center of the lattice | j i ,which is shown in Figs. 6(c) and (d). It is known thatin λ f → ∞ limit, L ( t ; ∞ , ∞ ) = 1, due to the initialstate being the eigenstate of the post-quench Hamilto-nian. When λ f deviates from the infinite value, the ini-tial state | j i is no longer the eigenstate of ˆ H ( λ f ), andthe excited single-channel has a superposition of vari-ous eigenstates. No matter how small is the superposi-tion with the initial condition, the mode with the largestimaginary part among them will dominate after a finiteevolution time, and a non-Hermitian jump process oc-curs which will induce an evident change of the returnamplitude. Hence, for a finite λ f , the Loschmidt echo nolonger keeps uniform, and will present complex features.As seen in Fig. 6(c), the Loschmidt echo first decays to afinite value L c in a short time ˜ t corresponding to a finiteoverlap between the evolution wave function | Ψ( t ) i and | j i . L ( t ; λ i , λ f ) then displays an oscillation around thelimited value L c in a certain time interval ∆˜ t . L c and ∆˜ t depend on λ f . In the large λ f limit, the value of L c tendsto 1 and ∆˜ t →
0. With the decrease of λ f , L c decreases,while ∆˜ t increases in the large λ f case and decreases fora finite λ f . Finally, after the temporary localization pro-cess, the Loschmidt echo evolution displays a λ f depen-dent damping to zero which shows that the overlap of thefinal state and | j i tends to zero, and corresponds to theemergence of a non-Hermitian jump process. Figure 6(d)shows the log-log plot of 1 −L ( t ; λ i , λ f ) as the function oftime t for λ f = 1 .
5. We can see that the Loschmidt echopresents similar behaviors as the mean displacement σ ( t )in this limit, and the same pow-law indices are found inthe corresponding evolution stages.The results of two limits suggest that the existenceof zero points during the Loschmidt echo evolutionseems no longer a dynamical signature of localization-delocalization transition for the case of λ f > J in whichis PT symmetry broken regime. To clarify such conjec-ture, we calculate the evolution of the Loschmidt echo (a)(b)(c) FIG. 7: (Color online) Evolution of Loschmidt echo with L = F = 377 and λ f = 1 .
5. The initial states are chosen to bethe ˜ n -th eigenstates with the corresponding the real part ofthe energies in ascending order for (a) λ i = 0 .
6, (b) λ i = 1 . λ i = 1 .
2, respectively. Here, J = 1. with λ f = 1 . λ i and ˜ n with the real part ofthe energies in ascending order, which is shown in Fig.7. When λ i is localized in the extended ( λ i = 0 .
6) orthe multifractal regime ( λ i = 1 . λ i and ˜ n . However, for the initial state being aneigenstate of ˆ H ( λ i ) with λ i > J , e.g., λ i = 1 . V. CONCLUSION
In this paper, we study the dynamics evolution of thenon-Hermitian AA model with PT symmetry. The PT symmetric AA model exhibits a PT symmetry brokenpoint at λ = J . When λ < J , all the eigenenergiesare real and the corresponding eigenstates are extended.While for λ > J , the complex energies emerges and thecorresponding states are localized. The states at thetransition point are multifractal. We can apply the evolu-tion of the profile of the wave function and the long-timesurvival probability to distinguish the localization prop-erties of the system. The evolution of the mean displace-ment displays distinctive behaviors in different PT sym-metric regimes which is also available for the evolution ofthe Loschmidt echo. According to our calculation, whenthe post-quench parameter is localized in PT symmetryunbroken regime, the behaviors of the Loschmidt echoare similar to the Hermitian cases that the dynamicalsignature of localization-delocalization transition can becharacterized by the emergence of a series of zero pointsin the evolution of the Loschmidt echo. However, whenthe post-quench parameter is localized in PT symme-try broken regime, the dynamical detection of the tran-sition point by the Loschmidt echo method is unavail-able. The similar conclusions for Loschmidt echo in someother PT symmetric systems without disorder have beenreported . Our results can be easily examined in afinite silicon waveguide lattice, where the PT symmetricpotential can be implemented with carefully controlledlossy silicon waveguides. Acknowledgments
Z. Xu is supported by the NSFC (Grant No. 11604188)and STIP of Higher Education Institutions in Shanxi un-der Grant No. 2019L0097. S. Chen is supported bythe National Key Research and Development Program ofChina (2016YFA0300600 and 2016YFA0302104), NSFCunder Grants No.11974413 and the Strategic Priority Re-search Program of Chinese Academy of Sciences underGrant No. XDB33000000. This work is also supportedby NSF for Shanxi Province Grant No.1331KSC. ∗ Electronic address: [email protected] P. W. Anderson, Phys. Rev. , 1492 (1958). J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P.Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, andA. Aspect, Nature , 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). S. S. Kondov, W. R. McGehee, J. J. Zirbel, B. DeMarco,Science , 66 (2011). F. Jendrzejewski, A. Bernard, K. M¨uller, P. Cheinet, V.Josse, M. Piraud, L. Pezz´e, L. Sanchez-Palencia, A. As-pect, and P. Bouyer, Nat. Phys. , 398 (2012). G. Semeghini, M. Landini, P. Castilho, S. Roy, G. Spag-nolli, A. Trenkwalder, M. Fattori, M. Inguscio, and G.Modugno, Nat. Phys. , 554 (2015). M. Pasek, G. Orso, and D. Delande, Phys. Rev. Lett. ,170403 (2017). C. Hainaut, A. Ran¸con, J.-F. Cl´ement, I. Manai, P. Szrift-giser, D. Delande, J. C. Garreau, and R. Chicireanu, NewJ. Phys. , 035008 (2019). J. Richard, L. -K. Lim, V. Denechaud, V. V. Volchkov, B.Lecoutre, M. Mukhtar, F. Jendrzejewski, A. Aspect, A.Signoles, L. Sanchez-Palencia, and V. Josse, Phys. Rev.Lett. , 100403 (2019). T. Sperling, W. B¨uhrer, C. M. Aegerter, and G. Maret,Nat. Photon. , 48 (2013). D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini,Nature (London) , 671 (1997). M. St¨orzer, P. Gross, C. M. Aegerter, and G. Maret, Phys.Rev. Lett. , 063904 (2006). T. Schwartz, G. Bartal, S. Fishman, and B. Segev, Nature(London) , 52 (2007). Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti,D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. , 013906 (2008). H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, andB. A. van Tiggelen, Nat. Phys. , 845 (2008). S. Katsumoto, F. Komori, N. Sano, and S. Kobayashi, J.Phys. Soc. Jpn , 2259 (1987). S. Aubry and C. Andr´e, Ann. Isr. Phys. Soc. , 133 (1980). D. J. Thouless, J. Phys. C , 77 (1972). J. B. Sokoloff, Phys. Rep. , 189 (1985). D. Hofstadter, Phys. Rev. B , 2239 (1976). C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari,Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino,T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, andP. Kim, Nature (London), , 598 (2013). D. R. Grempel, S. Fishman, and R. E. Prange, Phys. Rev.Lett. , 833 (1982). M. Kohmoto, Phys. Rev. Lett. , 1198 (1983). S. Das Sarma, S. He, and X. C. Xie, Phys. Rev. Lett. ,2144 (1988). Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti,N. Davidson, and Y. Silberberg, Phys. Rev. Lett. ,013901 (2009). J. Biddle, B. Wang, D. J. Priour, Jr., and S. Das Sarma,Phys. Rev. A , 021603 (2009). J. Biddle and S. Das Sarma, Phys. Rev. Lett. , 070601(2010). M. Pouranvari, Phys. Rev. B , 155121 (2019). C. Aulbach, A. Wobst, G. -L. Ingold, P. H¨anggi, and I.Varga, New J. Phys. , 70 (2004). M. Modugno, New J. Phys. , 033023 (2009). M. Larcher, M. Modugno, and F. Dalfovo, Phys. Rev. A , 013624 (2011). G.-L. Ingold, A. Wobst, Ch. Aulbach, P. H¨anggi, Eur.Phys. J. B , 175 (2002). Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O.Zilberberg, Phys. Rev. Lett. , 106402 (2012). L.-J. Lang, X. Cai, and S. Chen, Phys. Rev. Lett. ,220401 (2012). J. R. M. Silva, M. S. Vasconcelos, D. H. A. L. Anselmo,and V. D. Mello, J. Phys. Cond. Matt. , 505405 (2019). Z. Xu, L. Li, and S. Chen, Phys. Rev. Lett. , 215301(2013). Z. Xu and S. Chen, Phys. Rev. B , 045110 (2013). S.-L. Zhu, Z.-D. Wang, Y.-H. Chan, and L.-M. Duan,Phys. Rev. Lett. , 075303 (2013). D. M. Basko, I. L. Aleiner, B. L. Altshuler, Ann. Phys. , 1126 (2006). M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨uschen,M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I.Bloch, Science , 842 (2015). S. Iyer, V. Oganesyan, G. Refael, and D. A. Huse, Phys.Rev. B , 134202 (2013). P. Bordia, H. P. L¨uschen, S. S. Hodgman, M. Schreiber,I. Bloch, and U. Schneider, Phys. Rev. Lett. , 140401(2016). M. Kohmoto, L. P. Kadanoff, and C. Tang, Phys. Rev.Lett. , 1870 (1983). S. Ostlund, R. Pandit, D. Rand, H. J. Schellnhuber, andE. D. Siggia, Phys. Rev. Lett. , 1873 (1983). M. Kohmoto and J. R. Banavar, Phys. Rev. B , 563(1986). M. Kohmoto, B. Sutherland, and C. Tang, Phys. Rev. B , 1020 (1987). S. Abe and H. Hiramoto, Phys. Rev. A , 3549 (1987). D. E. Katsanos, S. N. Evangelou, and S.J. Xiong, Phys.Rev. B , (1995). T. Geisel, R. Ketzmerick, and G. Petschel, Phys. Rev.Lett. , 1651 (1991). R. Ketzmerick, G. Petschel, and T. Geisel, Phys. Rev.Lett. , 695 (1992). B. Huckestein and L. Schweitzer, Phys. Rev. Lett. , 713(1994). R. Ketzmerick, K. Kruse, S. Kraut, and T. Geisel, Phys.Rev. Lett. , 1959 (2017). Z. Zhang, P. Tong, J. Gong, and B. Li, Phys. Rev. Lett. , 070603 (2012). S. Dadras, A. Gresch, C. Groiseau, S. Wimberger, and G.S. Summy, Phys. Rev. Lett. , 070402 (2018). A. Sinha, M. M. Rams, and J. Dziarmaga, Phys. Rev. B , 094203 (2019). X. Deng, S. Ray, S. Sinha, G. V. Shlyapnikov, and L.Santos, Phys. Rev. Lett. , 025301 (2019). N. Silberstein, J. Behrends, M. Goldstein, and R. Ilan,Phys. Rev. B , 245147 (2020). H. P. L¨uschen, S. Scherg, T. Kohlert, M. Schreiber, P.Bordia, X. Li, S. Das Sarma, and I. Bloch, Phys. Rev.Lett. , 160404 (2018). T. Kohlert, S. Scherg, X. Li, H. P. L¨uschen, S. Das Sarma,I. Bloch, and M. Aidelsburger, Phys. Rev. Lett. ,170403 (2019). Z. Xu, H. Huangfu, Y. Zhang, and S. Chen, New J. Phys. , 013036 (2020). M. Heyl, A. Polkovnikov, and S. Kehrein, Phys. Rev. Lett. , 135704 (2013). C. Karrasch and D. Schuricht, Phys. Rev. B , 195104(2013). E. Canovi, P. Werner, and M. Eckstein, Phys. Rev. Lett. , 265702 (2014). R. A. Jalabert and H. M. Pastawski, Phys. Rev. Lett. ,2490 (2001). F. M. Cucchietti, D. A. R. Dalvit, J. P. Paz, and W. H.Zurek, Phys. Rev. Lett. , 210403 (2003). T. Gorin, T. Prosen, T. H. Seligman, and M. Znidaric,Phys. Rep. , 33 (2006). H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P.Sun, Phys. Rev. Lett. , 140604 (2006). R. Jafari and H. Johannesson, Phys. Rev. Lett. ,015701 (2017). J. C. Budich and M. Heyl, Phys. Rev. B , 085416(2016). S. Vajna and B. D´ora, Phys. Rev. B , 161105(R) (2014). S. Vajna and B. D´ora, Phys. Rev. B , 155127 (2015). S. Sharma, U. Divakaran, A. Polkovnikov, and A. Dutta,Phys. Rev. B , 144306 (2016). U. Bhattacharya and A. Dutta, Phys. Rev. B , 184307(2017). C. Yang, Y. Wang, P. Wang, X. Gao, and S. Chen, Phys.Rev. B , 184201 (2017). D.M. Kennes, C. Karrasch, A.J. Millis, Phys. Rev. B ,081106 (2020). N. Szpak and R. Sch¨utzhold, arXiv:1901.05941. P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C.Hempel, B. P. Lanyon, M. Heyl, R. Blatt, and C. F. Roos,Phys. Rev. Lett. , 080501 (2017). J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P.Becker, H. Kaplan, A. V. Gorshkov, Z.-X. Gong, and C.Monroe, Nature , 601 (2017). N. Fl¨aschner, D. Vogel, M. Tarnowski, B. S. Rem, D.-S.L¨uhmann, M. Heyl, J. C. Budich, L. Mathey, K. Seng-stock, and C. Weitenberg, Nat. Phys. , 265 (2018). X.-Y. Guo, C. Yang, Y. Zeng, Y. Peng, H.-K. Li, H. Deng,Y.-R. Jin, S. Chen, D. Zheng, and H. Fan, Phys. Rev.Applied , 044080 (2019). K. Wang, X. Qiu, L. Xiao, X. Zhan, Z. Bian, W. Yi, andP. Xue, Phys. Rev. Lett. , 020501 (2019). T. Tian, H.-X. Yang, L.-Y. Qiu, H.-Y. Liang, Y.-B. Yang,Y. Xu, and L.-M. Duan, Phys. Rev. Lett. , 043001(2020). K. Xu, Z.-H. Sun, W. Liu, Y.-R. Zhang, H. Li, H: Dong,W. Ren, P. Zhang, F. Nori, D. Zheng, H. Fan, and H.Wang, Science Adv. , eaba4935 (2020). H. Yin, S. Chen, X. Gao, and P. Wang, Phys. Rev. A ,033624 (2018). T. Liu and H. Guo, Phys. Rev. B , 104307 (2019). Y. Liu, X.-P. Jiang, J. Cao, S. Chen, Phys. Rev. B ,174205 (2020). S. Peotta, F. Brange, A. Deger, T. Ojanen, and C. Flindt,arXiv:2011.13612. X. Tong, Y. Meng, X. Jiang, C. Lee, G. Neto, and GaoX., arXiv:2012.07001. N. Hatano and D. R. Nelson, Phys. Rev. Lett. , 570(1996). N. Hatano and D. R. Nelson, Phys. Rev. B , 8384(1998). A. V. Kolesnikov and K. B. Efetov, Phys. Rev. Lett. ,5600 (2000). Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Hi-gashikawa, and M. Ueda, Phys. Rev. X 8, 031079 (2018). A. F. Tzortzakakis, K. G. Makris, and E. N. Economou,Phys. Rev. B , 014202 (2020). Y. Huang and B. I. Shklovskii, Phys. Rev. B , 014204(2020). Y. Huang and B. I. Shklovskii, Phys. Rev. B , 064212(2020). Q. -B. Zeng and Y. Xu, Phys. Rev. Research , 033052(2020). Q. -B. Zeng, S. Chen, and R. L¨u, Phys. Rev. A , 062118(2017). T. Liu, H. Guo, Y. Pu, and S. Longhi, Phys. Rev. B , 024205 (2020). Y. Liu, Q. Zhou, and S. Chen, arXiv:2009.07605.
Y. Liu, Y. Wang, X. -J. Liu, Q. Zhou, and S. Chen,arXiv:2009.02012.
Y. Liu, Y. Wang, Z. Zheng, and S. Chen,arXiv:2012.10029.
S. Longhi, Phys. Rev. Lett. , 237601 (2019).
S. Longhi, Phys. Rev. B , 125157 (2019).
S. Longhi, Opt. Lett. , 1190 (2019). C. H. Liang, D. D. Scott, and Y. N. Joglekar, Phys. Rev.A , 030102 (2014). C. Mej´ıa-Cortes and M. I. Molina, Phys. Rev. A ,033815 (2015). A. K. Harter, T. E. Lee, and Y. N. Joglekar, Phys. Rev.A , 062101 (2016). N. X. A. Rivolta, H. Benisty, and B. Maes, Phys. Rev. A , 023864 (2017). H. Jiang, L.-J. Lang, C. Yang, S.-L. Zhu, and S. Chen,Phys. Rev. B , 054301 (2019).
D. -W. Zhang, L. -Z. Tang, L. -J. Lang, H. Yan, S. -L.Zhu, Sci. China-Phys. Mech. Astron. , 267062 (2020). J. Claes, T. L. Hughes, arXiv:2007.03738
L. -J. Zhai, S. Yin, G. -Y. Huang, Phys. Rev. B ,064206 (2020).
N. Okuma, M. Sato, arXiv:2008.06498.
A. F. Tzortzakakis, K. G. Makris, S. Rotter, and E. N.Economou, Phys. Rev. A , 033504 (2020).
C. Wang, and X. R. Wang, Phys. Rev. B , 165114(2020).
R. Hamazaki, K. Kawabata, and M. Ueda, Phys. Rev.Lett. , 090603 (2019).
C. -H. Liu and S. Chen, arXiv:2012.13583.
A. F. Tzortzakakis, K. G. Makris, A. Szameit, and E. N.Economou, arXiv:2007.08825.
I. I. Yusipov, T. V. Laptyeva, and M. V. Ivanchenko,Phys. Rev. B , 020301(R) (2018). M. Balasubrahmaniyam, S. Mondal, S. and S. Mujumdar,Phys. Rev. Lett. , 123901 (2020).
I. Y. Goldsheid and B. A. Khoruzhenko, Phys. Rev. Lett. L. G. Molinari, J. Phys. A: Math. Theor. H. Markum, R. Pullirsch, and T. Wettig, Phys. Rev. Lett. , 484 (1999). J. T. Chalker and B. Mehlig, Phys. Rev. Lett. , 3367(1998). X. Deng, S. Ray, S. Sinha, G. V. Shlyapnikov, and L.Santos, Phys. Rev. Lett. , 025301 (2019).
Z. Zhang, P. Tong, J. Gong, and B. Li, Phys. Rev. Lett. , 070603 (2012).
C. M. Dai, W. Wang, and X. X. Yi, Phys. Rev. A ,013635 (2018). M. Znojil, Ann. Phys. (NY) , 162 (2017).
D. Krejcirik, P. Siegl, M. Tater, and J. Viola, J. Math.Phys. , 103513 (2015). K. Wang, X. Qiu, L. Xiao, X. Zhan, Z. Bian, W. Yi, andP. Xue, Phys. Rev. Lett.122