Dynamics of a quantum phase transition in the Aubry-André-Harper model with p -wave superconductivity
Xianqi Tong, Yeming Meng, Xunda Jiang, Chaohong Lee, Gentil Dias de Moraes Neto, Gao Xianlong
DDynamics of a quantum phase transition in the Aubry-Andr´e-Harper model with p -wave superconductivity Xianqi Tong, Yeming Meng, Xunda Jiang, Chaohong Lee,
2, 3
Gentil Dias de Moraes Neto, and Gao Xianlong Department of Physics, Zhejiang Normal University, Jinhua 321004, People’s Republic of China Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing & School of Physics and Astronomy,Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China State Key Laboratory of Optoelectronic Materials and Technologies,Sun Yat-Sen University (Guangzhou Campus), Guangzhou 510275, China (Dated: December 15, 2020)We investigate the nonequilibrium dynamics of the one-dimension Aubry-Andr´e-Harper modelwith p -wave superconductivity by changing the potential strength with slow and sudden quench.Firstly, we study the slow quench dynamics from localized phase to critical phase by linearly decreas-ing the potential strength V . The localization length is finite and its scaling obeys the Kibble-Zurekmechanism. The results show that the second-order phase transition line shares the same criticalexponent zν , giving the correlation length ν = 0 .
997 and dynamical exponent z = 1 . V = 0 and V = ∞ , we analytically study the sudden quench dynamics via the Loschmidt echo. Theresults suggest that, if the initial state and the post-quench Hamiltonian are in different phases, theLoschmidt echo vanishes at some time intervals. Furthermore, we found that, if the initial valueis in the critical phase, the direction of the quench is the same as one of the two limits mentionedbefore, and similar behaviors will occur. PACS numbers: Valid PACS appear here
I. INTRODUCTION
In recent years, extensive researches have been car-ried to unravel the behavior of quasiperiodic (QP)structures . QP system, being aperiodic but determin-istic, lacks translational invariance but shows long-rangeorder leading to a rich critical behavior. The criticalproperties are different or can be regarded as interme-diate from those of ordinary (periodic) and disordered(random) systems. For instance, the spatial modulationof the parameters can change the universality class ofa quantum phase transition (QPT), i.e. the critical ex-ponents that characterized the equilibrium properties ofthe physical observables at the transition point. Fur-thermore, one-dimensional (1D) QP systems, known asthe Aubry-Andr´e-Harper (AAH) model, show Andersonlocalization transition at a finite strength of the QP dis-order that differs from the original 1D random model.In the AAH model, the states at the critical pointare neither extended nor localized but critical, charac-terized by power-law localization, and fractal-like spec-trum and wave functions. In an interacting system, themany-body localization with random or QP case exhibitquite different behaviors . Furthermore, the quantumphase transitions of QP system related to quantum mag-netism described by spin Hamiltonians and respec-tive fermionic counterpart , were studied extensively.In particular, the anisotropic XY chain in a transversemagnetic field , that maps via Jordan-Wigner trans-formation, to the AAH model with p -wave superconduct-ing (SC) pairing terms , and contains the quantumIsing and XY chains as limiting cases, has drawn atten- tion for a rich phase diagram, as depicted in Fig. 1. Theanisotropy (SC pairing) destroys the self-duality of theisotropic XY model and stabilizes the critical phase sand-wiched between extended and localized phases.Although, the phase diagram of the AAH model withSC pairing is well understood, it lacks the thorough inves-tigation of the critical behavior and the nonequilibriumdynamics. Specifically, in a continuous phase transition,the correlation length ξ and corresponding gap ∆ divergeat the transition as ξ ≈ (cid:15) − ν and ∆ ≈ ξ − z , where (cid:15) is thedistance from the critical point of the disorder strength, ν and z are the correlation length and dynamical criti-cal exponents. To the best of our knowledge, there is noreport in literature about the critical exponents of theAAH model with SC pairing.In this context, it is important to determine thenonequilibrium dynamical signatures of a quantumphase transition (QPT) , which has also being ex-plored in QP system both experimentally andtheoretically . It is useful to discriminate be-tween two limiting processes of slowly and instanta-neously changing of the parameters. Driving the param-eter across the second-order phase transition is usuallydescribed by the Kibble Zurek mechanism (KZM) .The essence of the KZM is the breaking of the adiabatic-ity for crossing the critical point of a QPT, which leadsto the corresponding excitations following a power lawrelation with respect to the quench rate. For the dy-namical quantum phase transition (DQPT), the quantumsystem is quenched out of equilibrium by suddenly chang-ing the parameters of the Hamiltonian. For the suddenquench dynamics, Loschmidt echo is an important quan- a r X i v : . [ c ond - m a t . d i s - nn ] D ec tity, which measures the overlap between the initial stateand the time-evolved state . Many theoretical workshave demonstrated that the Loschmidt echo plays a sig-nificant role in characterizing the nonequilibrium dynam-ical signature of the quantum phase transition .Recently, thanks to the developments of the quantumsimulation techniques, DQPT can be directly detectedin a string of ions simulating the interacting transversefield Ising model .However, the time evolution of the Loschmidt echo andthe KZM requires an in-depth investigation in a 1D QPsystem which exists phase transitions among localizedphase, critical phase, and extended phase. Here, we payattention to such a quantum disordered system describedby the AAH model with p -wave SC paring .The rest of the paper is organized as follows. InSect. II, we explicitly write down the Schr¨odinger equa-tion of the 1D QP system. In Sect. III, we calculate thecritical exponents and verify the KZM hypothesis. InSect. IV, we discuss the sudden quench dynamics of thequantum phase transition between different phases andgive the analytical expressions of two limits cases. Sec-tion V is devoted to conclusion. II. MODEL HAMILTONIAN
The generalized 1D AAH model with p -wave SC paringis described by the following Hamiltonian H = N (cid:88) j =1 V j c † j c j + N − (cid:88) j =1 ( − Jc † j +1 c j + ∆ c † j +1 c † j + H.c. ) , (1)where c j ( c † j ) is the fermionic annihilation (creation) op-erator at the j -th site. Here V j = V cos(2 παj + φ ) is theincommensurate potential with α = ( √ − / V is the strength of the incom-mensurate potential, and the random phase φ ∈ [0 , π )is introduced as a pseudorandom potential. J is thenearest-neighbor hopping amplitude and we set J = 1as energy unit throughout this paper. ∆ is the ampli-tude of the p -wave SC paring. The phase diagram of thissystem has three different phases shown in Fig. 1: lo-calized phase, critical phase and extended phase, whichare marked by green, white and blue, respectively. For V = 2 | J +∆ | , the system undergoes a second-order phasetransition from critical phase to localized phase . For V = 2 | J − ∆ | , the system has a phase transition fromcritical phase to extended phase . Firstly, we needto rewrite the Hamiltonian by using the Bogoliubov-deGennes (BdG) transformation, η † n = N (cid:88) j =1 [ u n,j c † j + v n,j c j ] , (2)where n = 1 , ..., N, the Bogoliubov modes ( u n,j , v n,j )are the eigenstates of the Hamiltonian and u n,j , v n,j are Localized ExtendedExtended Critical 𝑉 / 𝐽 ∆/𝐽 Figure 1. (Color online) Sketch of the phase diagram of theAAH model with p -wave superconducting paring order pa-rameter ∆ and the disorder strength V . Three differentphases, that is, extended phase, critical phase and localizedphase are shown up in different parameter regimes. The linebetween critical phase and localized phase is a second-orderphase transition line. The vertical line at ∆ = 0 . chosen be real, so the Hamiltonian can be diagonalized as H = N (cid:88) n =1 ε n ( η † n η n −
12 ) , (3)with ε n being the spectrum of quasiparticles. For the n -th Bogoliubov modes, we have the following BdG equa-tions: − Ju j − + ∆ v j − + V j u j − Ju j +1 − ∆ v j +1 = εu j , − ∆ u j − + Jv j − − V j v j + ∆ u j +1 + Jv j +1 = εv j . (4)The wave function is expressed as | Ψ n (cid:105) = [ u n, , v n, , u n, , v n, , ..., u n,N , v n,N ] T , (5)then for the Schr¨odinger equation H | Ψ n (cid:105) = ε n | Ψ n (cid:105) , theHamiltonian can be written as a 2 N × N matrix: H = A B . . . . . . . . . CB † A B . . . . . . B † A B . . . . . . B † A N − B . . . . . . B † A N − BC † . . . . . . . . . B † A N , (6) (a) N r N Figure 2. (a) The localization length ξ as a function of thedistance from the critical point (cid:15) = V − V c . Here ξ was calcu-lated by using the ground state of the corresponding Hamilto-nian. The linear fit ξ ∼ (cid:15) − ν yields correlation-length exponent ν = 0 . ± . r whichis the sum of two lowest eigenenergies at the critical point asa function of N . Fitting ∆ r ∼ N − z yields a dynamical expo-nent z = 1 . ± . V c =3, and the lattice size N =987 in (a). Averaging is done over 200 random values of φ . where A = (cid:18) V j − V j (cid:19) , (7) B = (cid:18) − J − ∆∆ J (cid:19) , (8)and C = (cid:18) − J ∆ − ∆ J (cid:19) . (9)Here, we assume the Hamiltonian with periodic bound-ary condition, hence α can be approximated by a rationalnumber with L in the denominator. Dependence of L implies an order L = F m , α = F m − /F m , where F m is aFibonacci number. III. KIBBLE-ZUREK MECHANISM
When V is gradually decreased to approach thecritical point, correlation length will diverge as: ξ ≈ (cid:15) − ν , (cid:15) = V − V c , (10)where (cid:15) is the distance from the critical point and ν =0 . ± .
006 is a correlation-length exponent extractedfrom Fig. 2(a). We set ∆ = 0 . V c = 3.The dynamical exponent z can be determined by thescaling of system size N and the relevant gap, i.e., ∆ r = F ( t ) f f f f Q = 10 Q = 50 Q = 100 Q = 300 0.00 0.03 0.050.50.81.0 F ( t ) Figure 3. The fidelity as a function of (cid:15) for four different τ Q =10 , , , f = 0 .
9. The blue circle,orange square, green triangle and red hexagon represent fourcorresponding “frozen out” instants. Inset: Enlarged view of (cid:15) between 0 and 0.05. The parameters are chosen as ∆ = 0 . V c = 3 and N = 987. ε + ε , which is the sum of energies of the two positivelowest energy quasiparticles ∆ r ∼ N − z . (11)We use the linear fit to log-log plot of Fig. 2(b) whichyields z = 1 . ± . zν determine how the gap vanishes with the distance fromthe critical point. These critical exponents can be ob-tained from the study of the the fidelity susceptibility and scaling analysis of superfluid fraction for differentlattice sizes . The whole results are also true for otherpoints on the second-order phase transition line, exceptfor the limited conditions of ∆ = 0 , −
1. When ∆=0, theAubry-Andr´e model with p -wave superconductivity willreturn to the Aubry-Andr´e model . When ∆ = −
1, themodel will return to quasiperiodic Ising model .The initial state is deeply prepared in the localizedstate, and the potential V is slowly changed across thecritical point between the critical and the localized phase.Near the critical point, (cid:15) can be approximated by alinear quench: (cid:15) ≈ − tτ Q , (12)here τ Q is the quench time. When the state is far awayfrom the critical point, the state is adiabatically evolv-ing. Then, the state crosses the adiabatic region to thediabatic region at a time point when its reaction time τ ∼ r ( t ) ∼ | tτ Q | − zν equals the time scale | (cid:15)/ ˙ (cid:15) | . Thusthere exists an intersection in which two timescales areequal, t = ± ˆ t , where ˆ t ∼ τ zν/ (1+ zν ) Q . (13)The time-dependent state is still at the ground state until t = − ˆ t and ˆ (cid:15) = ˆ tτ Q ∼ τ − / (1+ zν ) Q , with localization lengthˆ ξ ∼ ˆ (cid:15) − ν ∼ τ ν zν Q . (14)In zero-order approximation, the two time points ± ˆ t di-vide the whole evolution into three regimes. Initially,when t < − ˆ t , the state can adjust to the change of theHamiltonian. However, at t = − ˆ t this tracking will cease,and the wave-packet does not follow the instantaneousground state until ˆ t with a finite localization length ˆ ξ .Afterwards, it is the initial state for the adiabatic pro-cess that begins at ˆ t which is similar to the one “frozenout” at − ˆ t .We should remember that such a “frozen out” instantis only a feasible hypothesis. However, it is very helpful todeduce the scaling law. Actually, a realistic system doesnot exist a sudden change at a certain moment duringthe evolution, which is a process from the adiabatic tothe diabatic regime. Therefore, we can numerically testthe KZM hypothesis by solving the critical dynamics,and estimate the frozen instant when the adiabaticitybreaking. In this connection, although there is no uniqueway to quantify adiabatic loss, we use the fidelity F ( t ), F ( t ) = | (cid:104) ψ ( t ) | Ψ ( t ) (cid:105) | , (15)to describe the loss of adiabaticity, which provides a goodapproximation . Here, (cid:104) ψ ( t ) | is the time-evolved state,and | Ψ ( t ) (cid:105) is the instantaneous ground state. In Fig. 3,we plot the time-dependent fidelity F ( t ) as a functionof (cid:15) for four different quench rates, and the fidelity F ( t )decreases dramatically at the critical point ? From this,we can get the estimated values of the “frozen out” in-stants. The blue circle, orange square, green triangle,red hexagon represent the instants with different τ Q . Itis clearly shown that the corresponding “frozen out” in-stants is closer to the critical point as τ Q increasing. Wechoose one value represented by the straight line f = 0 . A. KZ POWER LAWS
In order to test the KZ scaling, we use smooth tanh-profile (cid:15) ( t ) = − tanh ( t/τ Q ) starting from − τ Q for thesake of suppressing excitation derived from the initialdiscontinuity of the time derivative ˙ (cid:15) at − τ Q .When the system’s evolution crosses the adiabatic areaat − ˆ t , then in the diabatic area, the localization lengthˆ ξ does not change under the zero-order approximationuntil the time at ˆ t . In Fig. 4, we plot ˆ ξ estimated by thedispersion of the probability distribution as a functionof τ Q at the critical point (cid:15) = 0. The power law fittingimplies z = 1 .
364 for ν = 1. And z = 1 .
361 for ν = 0 . z = 1 .
361 extracted from ˆ ξ in Fig. 4 and z (cid:39) .
373 from Fig. 2(b) differ by 1%.Similarly, the critical exponent ν (cid:39) .
997 is also 0 . Q Figure 4. The width of the wave packet as a function ofthe quench time τ Q at the critical point. The fitted straightline gives ˆ ξ = τ . ± . Q ; cf. Eq. (14). The parameters arechosen as N=987, ∆ = 0 . V = 3, and φ = 0. ( t ) (a)(a)(a)(a)(a)(a)(a)(a)(a)(a) (b) Figure 5. In (a), the width of the wave packet ˆ ξ ( t ) as a func-tion of the scaled time − ˆ t and ˆ t which represents the impulseregime. In (b), the scaled width of the wave packet ˆ ξ ( t ) / ˆ ξ and scaled time all collapse to their respective scaling func-tion. The parameters are chosen as Fig. 4. away from the value ν = 1. The difference is almostthe same as the system error. Therefore, within a smallerror range, our numerical results are consistent with thepredicted results.In the impulse area, ˆ ξ is the relevant scale of length.When τ Q → ∞ , the adiabatic limited is recovered. ˆ ξ diverges in the limit and becomes the only relevant scalesin the long-wavelength regime. This logic proves the KZscaling hypothesis for a correlation length ˆ ξ ( t ) in thediabatic regime: ˆ ξ ( t ) = ˆ ξF ξ ( t/ ˆ t ) , (16)where F ξ is not a universal function as shown in Fig. 5. IV. LOSCHMIDT ECHO
In the following section, we discuss another nonequilib-rium dynamics by suddenly quenching the on-site poten-tial V , not only between the localization phase and criti-cal phase separated by the second-order phase transitionline, but also between the critical phase and extendedphase.By preparing the initial state as the eigenstate of theHamiltonian H ( V i ), and then suddenly quenching theHamiltonian to H ( V f ), we calculate the return proba-bility (Loschmidt echo) : L ( t, V i , V f ) = | G ( t, V i , V f ) | , (17)where G ( t, V i , V f ) is the return amplitude (a type ofLoschmidt echo amplitude): G ( t, V i , V f ) = (cid:104) ψ ( V i ) | e − itH ( V f ) | ψ ( V i ) (cid:105) , (18)where ψ ( V i ) is the eigenstate of the initial Hamiltonian H ( V i ), and V i ( V f ) represents the strength of the ini-tial (final) incommensurate potential. The initial state ischosen to be the ground state of the initial Hamiltonian,and the results are also true for all the other eigenstates.Then, we illustrate whether the zero points of theLoschmidt echo can be regarded as the signature ofthe phase transition among the localized phase, criticalphase, and extended phase. To give a more intuitive ex-planation, we should consider two limiting cases. Forthese two cases, the initial value of V i is set to 0( ∞ ) and V f = ∞ (0) which can be calculated analytically, whereasthe other cases are studied by the numerical methods.If V i = 0, the eigenvalues of the Hamiltonian is ε n = V f cos (2 παn ), and the corresponding eigenstates areplane wave states | φ k ( V i = 0) (cid:105) = e − iπ/ √ N (cid:80) Nj =1 e ikj c † j | (cid:105) .If V f = ∞ , the system is in the localized phase, theeigenstates of the Hamiltonian is the localized states | Ψ n ( V f = ∞ ) (cid:105) = (cid:80) Nj =1 δ jn c † j | (cid:105) with the eigenvalues ε n = V f cos(2 παn ). Then substituting the above re-sults into Eq. (18), we can get the analytical solution G k = J ( V f t ) [see Appendix A], where J ( V f t ) is thezero-order Bessel function. It has a number of zeros x n with n = 1 , , , ... . These zeros mean that the Loschmidtamplitude and the echo can reach zeros at times: t ∗ n = x n V f . (19)According to the DQPT theory, the appearance of thezero points in Loschmidt echo can be regarded as thecharacteristics of the DQPT and it is related to the di-vergence of the boundary partition function. Becausethe transition time t ∗ n is inversely proportional to V f , theLoschmidt echo oscillates faster with the increasing V f (see Fig. 6(a)). Then, if we rescale the time t to V f t , asshown in Fig. 6(b)-6(f), the evolution of the Loschmidtecho shows similar behaviors for the quenching processof different V f as shown in Fig .6(b)-6(c). The initialstrength V i is set to 0 and the SC paring ∆ = 0 .
5. It is ap-parent that the Loschmidt echo for V f = 2 . , . , . , . V f = 15 , , ,
60 in the local-ized phase oscillates with different frequencies. However,they are all quite similar after rescaling the time t to V f t .Except for the smaller V f = 15, in the localized phase,the numerical results almost coincide with the analyti-cal solution, shown in Fig. 6(c). Therefore, although the Figure 6. The evolution of Loschmidt echo with different t or V f t . The system size N = 987 and the SC paring ∆ = 0 . V i = 0 . (a) Loschmidt echo versus t . (b) and(c) L ( t ) versus different rescaled time V f t . (d) and (e) “dy-namic free energy” f ( t ) versus V f t . f ( t ) = − log | J ( V f t ) | isdepicted by the black dotted line. (f) The evolution of theLoschmidt echo for various V f including the extended, criti-cal, localized phases. The Loschmidt echo approaches zero atsome different time points. And it has different frequenciesfor different phases. analytical solution is under the condition of V f → ∞ ,the above results hold true for large enough V f , as shownin Fig. 6. To see the zero point in Loschmidt echo moreclearly, we calculate the “dynamical free energy”, definedas f ( t ) = − log | G ( t ) | . f ( t ) will be divergent at the timepoint t = t ∗ n . In Fig. 6(d) and Fig. 6(e), f ( t ) is plot-ted as a function of different V f t with V f in the localizedphase or in the critical phase. Obviously it reaches thepeaks at the critical times t ∗ n , especially when V f getscloser to ∞ .In Fig. 6(f), we calculate L ( t ) as a function of thescaled time V f t with a series of final value taken in dif-ferent phases. When V f < | J − ∆ | , the Loschmidt echocan not reach the zero even for long time evolution, be-cause V i and V f are in the same phase. However, whenthe final value V f is in the critical phase or in the lo-calized phase, L ( t ) shows similar oscillations with dif-ferent ∆ f and reaches zeros. And the time interval ofthe Loschmidt echo has different in approaching the zeropoints between the critical and localized phase. By notic-ing that the condition of V f → ∞ can not be met in thecritical phase, the analytical result of J ( V f t ) is no longerapplicable.Furthermore, we study the quenching process from a Figure 7. The evolution of Loschmidt echo with different V f t and t . The system size N = 987, the SC paring ∆ = 0 . V i = 2 . V i = 100( d ), ( e ), and ( f ). The initial state is the ground state of theHamiltonian. (a) Loschmidt echo versus t ; (b-c) f ( t ) versusdifferent rescaled time V f t and time t ; (d) The evolution ofthe Loschmidt echo for various t . (e-f) “dynamic free energy” f ( t ) versus t . The black dotted line corresponds to f ( t ) = − log | J (2 Jt ) | and the SC paring ∆ = 0 .
05. In Figs (d), (e)and (f), the initial states are all in the localized phase. When V f <
1, the Loschmidt echo will approach zero at the sametime interval which is independent of V f . strong disorder strength V i to V f = 0. The system is ini-tially prepared in the eigenstate of the localized phase,then is quenched into the extended regime. Similar to theabove analysis [see Appendix A], we get the Loschmidtamplitude G n = J (2 Jt ), and the zero points of theLoschmidt echo appear at times: t ∗ n = x n J , (20)which is inversely proportional to the hopping amplitude2 J , different from Eq. (19). The transition time t ∗ n isindependent of V f which means that for the different V f the dynamical free energy has almost the same behaviors.Moreover, the return amplitude is insensitive to V i , aslong as V i is large enough, even in the critical phase.In Fig. 7, the Loschmidt echo and the dynamical freeenergy f ( t ) as a function of the rescaled time V f t or time t . But different from Fig. 6, the initial system here is inthe critical phase or localized phase. In the left panel ofFig. 7, the initial state is prepared in the critical phase,and in the right panel of the Fig. 7, the initial state is setin the localized phase. Therefore, it is different from the previous analytical result. When V i = 2 . V f (cid:29)
3, we rescale t to V f t . However, the rescaling is notneeded when V f (cid:28)
1, shown in Fig. 7. In Fig. 7(a),we set V i = 2 . V f = 0 . , . , . , . ,
10, as longas V f < | J − ∆ | or V f > | J + ∆ | the Loschmidt echowill approach zero immediately, but when V f = 1 . , . L ( t ) will never approach zero duringthe time evolution. In Fig. 7(b), f ( t ) also shows similarbehavior for different V f after rescaling the time t to V f t ,due to the final value of the potential V f (cid:29) V i . But theshape of the curve is different from J (2 Jt ), because theinitial state is in the critical phase. For Fig. 7(c), thesame reason leads to mismatch between the peak shapeand J (2 Jt ). In analog to V i = 2 .
6, we set V i = 100and take a series of V f . We find that the Loschmidt echoapproaches zero when V f < | J +∆ | and it is also true for V f < | J − ∆ | in Fig. 7(d)-7(f). From Fig. 7(e), 7(f), for V f in the critical phase and extended phase, respectively, f ( t ) shows similar behaviors. In Fig. 7(f), when the SCparing ∆ = 0 .
05, the behaviors of f ( t ) with different V f almost coincide with the analysis result J (2 Jt ). Asa result, when V f approaches the limit of V f = 0, theanalytical result G n = J (2 Jt ) is a good approximation. V. CONCLUSION
In summary, we have studied the different nonequilib-rium dynamics of the 1D AAH model with p -wave su-perconductivity in two different ways. Firstly, a linearramp crossing the localization-critical phase transitionline is not adiabatic. By linearly fitting for the localiza-tion length near the critical point, we obtain the criticalexponents zν with ν = 0 .
997 which is the same as Aubry-Andr´e model, and the dynamical exponent z = 1 . .Except for the point ∆ = 0, the critical exponents arealmost the same for all the second-order phase transi-tion line V = 2 | J + ∆ | . We also tried a series of dif-ferent quenching directions. The critical exponents arethe same as what we obtained. Furthermore, we haveanalyzed the correlation length also the rescaled correla-tion length as a function of the quench time at the phasetransition point within the impulse regime between − ˆ t and ˆ t . The results are all consistent with the KZ scal-ing hypothesis. Our results indicate that KZM domi-nates the nonadiabatic dynamics of the one-dimensionalincommensurate system with the localized-critical phasetransition.Next, by using the Loschmidt echo we study the sud-den quench dynamics of the time evolution of the AAHmodel with p -wave SC pairing. The results show that theLoschmidt echo reaches zeros as long as the initial andthe final system are not in the same phase, which is alsotrue for the critical phase. Especially, if V i is in the criti-cal phase, L ( t ) and f ( t ) show similar behaviors when thechange of V has the same direction as the two limit casesmentioned before . Our research results indicate thatthe zeros of the Loschmidt echo manifest the dynamiccharacteristics in the incommensurate system, includingthe localized phase, critical phase and extended phase.Here we want to address some interesting issues to beinvestigated further. We first observe that the role playedby the incommensurability, i.e. the irrational number α on the QP potential, was only slightly explored. Ithas been known that α determines the universality classand the exotic non-power-law behavior . Finally, it isworth to study how the nonequilibrium dynamics of gen-eralized AAH models, for instance QP modulation of thehopping and on-site potential , follows the KZM scalinghypothesis and displays signatures of DQPTs capturedby the Loschmidt echo dynamics. ACKNOWLEDGMENTS
This work is supported by NSF of China under GrantNos. 11835011 and 11774316.
Appendix A
Firstly, V i is set to 0 and the system is initially pre-pared in the extended phase with periodic boundary con-dition, that is, a plane wave state is the eigenstate of theHamiltonian: | φ k ( V i = 0) (cid:105) = e − iπ/ √ N N (cid:88) j =1 e ikj c † j | (cid:105) , (A1)where the wave vector k = π ( l − N ) aN ∈ ( − πa , πa ] ( l =1 , . . . , N ) in the Brillouin zone. With the eigenvalues ε k of the initial Hamiltonian H ( V i ): ε k = 2 (cid:112) ( J cos ka ) + (∆ sin ka ) . (A2)When V f → ∞ , the eigenstates of the Hamiltonian be-come: | Ψ n ( V f = ∞ ) (cid:105) = N (cid:88) j =1 δ jn c † j | (cid:105) . (A3)Here, | Ψ n ( V f ) (cid:105) represents the n -th eigenstates of thequenched Hamiltonian. The corresponding eigenvalues ε n of the quenched Hamiltonian is: ε n = V f cos (2 παn ) . (A4)For a sudden quench, the system crosses from the initialvalue V i to final value V f . For simplicity, we use | k (cid:105) replace | φ k ( V i = 0) (cid:105) . So substituting Eqs. (A1), (A3) and(A4) into Eq. (18), the return amplitude can rewritten as G k ( t ) = (cid:104) k | e − iH ( V f ) t | k (cid:105) = (cid:88) n (cid:104) k | e − iH ( V f ) t | Ψ n ( V f ) (cid:105) (cid:104) Ψ n ( V f ) | k (cid:105) = (cid:88) n e − iε n t | (cid:104) Ψ n ( V f ) | k (cid:105) | = 1 N (cid:88) n =1 e − iV f t cos (2 παn ) . (A5)Because of the irrational number α , the phase 2 παn ( n =1 , . . . , N ) modulus 2 π is sett randomly between − π and π when we sum over from 1 to the large N . So we canapproximately replace the summation by the integration G k ( t ) ≈ π (cid:90) π − π e − iV f t cos θ dθ = J ( V f t ) , (A6)where J ( V f t ) is the zero-order Bessel function. Accord-ing to the nature of Bessel function, we know that thezero-order Bessel function J ( x ) has a series zero-point x n with n = 1 , , , ... . In the first case, the Loschmidtecho will reach zero at times: t ∗ n = x n V f . (A7)Conversely, we consider another limit, the quenchingprocess from a strong disorder strength V i → ∞ to thefinal V f = 0. By substituting Eqs. (A1), (A3) andEq. (A2) into Eq. (18), we can get the return amplitude: G n ( t ) = (cid:104) n | e − iH ( V f ) t | n (cid:105) = (cid:88) k (cid:104) n | e − it √ ( J cos ka ) +(∆ sin ka ) | k (cid:105) (cid:104) k | n (cid:105) = (cid:88) k e − it √ ( J cos ka ) +(∆ sin ka ) | (cid:104) k | n (cid:105) | = 1 N (cid:88) k e − it √ ( J cos ka ) +(∆ sin ka ) , (A8)where | n (cid:105) denotes | Ψ n ( V i = ∞ ) (cid:105) . When ∆ (cid:28) J and inthe large N limit, the sum can be transformed into anintegral. The same is true for ∆ (cid:29) JG n ( t ) = a π (cid:90) πa − πa e − iJt cos ka dk = J (2 Jt ) . (A9)Therefore, the Loschmidt echo gets zero at times: t ∗ n = x n J , (A10)which are 1 / J of the zeros of the zero-order Bessel func-tion J ( x ), different to Eq. (A7). M. Kohmoto, L. P. Kadanoff, and C. Tang, “LocalizationProblem in One Dimension: Mapping and Escape,” Phys.Rev. Lett. , 1870–1872 (1983). S. Ostlund, R. Pandit, D. Rand, H. J. Schellnhuber, andE. D. Siggia, “One-Dimensional Schr¨odinger Equation withan Almost Periodic Potential,” Phys. Rev. Lett. , 1873–1876 (1983). M. Kohmoto and J. R. Banavar, “Quasiperiodic lattice:Electronic properties, phonon properties, and diffusion,”Phys. Rev. B , 563–566 (1986). J. Q. You, J. R. Yan, T. Xie, X. Zeng, and J. X. Zhong,“Generalized Fibonacci lattices: dynamical maps, energyspectra and wavefunctions,” Journal of Physics: Con-densed Matter , 7255–7268 (1991). J. H. Han, D. J. Thouless, H. Hiramoto, and M. Kohmoto,“Critical and bicritical properties of Harper’s equationwith next-nearest-neighbor coupling,” Phys. Rev. B ,11365–11380 (1994). F. Liu, S. Ghosh, and Y. D. Chong, “Localization andadiabatic pumping in a generalized Aubry-Andr´e-Harpermodel,” Phys. Rev. B , 014108 (2015). Vedika Khemani, D. N. Sheng, and David A. Huse, “TwoUniversality Classes for the Many-Body Localization Tran-sition,” Phys. Rev. Lett. , 075702 (2017). M. M. Doria and I. I. Satija, “Quasiperiodicity and long-range order in a magnetic system,” Phys. Rev. Lett. ,444–447 (1988). V. G. Benza, “Quantum Ising Quasi-Crystal,” EurophysicsLetters (EPL) , 321–325 (1989). M. M. Doria, F. Nori, and I. I. Satija, “Thue-Morse quan-tum Ising model,” Phys. Rev. B , 6802–6806 (1989). V. G. Benza, M. Kol´a, and M. K. Ali, “Phase transitionin the generalized Fibonacci quantum Ising models,” Phys.Rev. B , 9578–9580 (1990). J. M. Luck, “Critical behavior of the aperiodic quantumIsing chain in a transverse magnetic field,” Journal of Sta-tistical Physics , 417–458 (1993). J. Hermisson, U. Grimm, and M. Baake, “Aperiodic Isingquantum chains,” Journal of Physics A: Mathematical andGeneral , 7315–7335 (1997). J. Hermisson and U. Grimm, “Surface properties of aperi-odic Ising quantum chains,” Phys. Rev. B , R673–R676(1998). J. M. Luck and T. M. Nieuwenhuizen, “A Soluble Quasi-Crystalline Magnetic Model: The XY Quantum SpinChain,” Europhysics Letters (EPL) , 257–266 (1986). I. I. Satija and M. M. Doria, “Localization and long-rangeorder in magnetic chains,” Phys. Rev. B , 9757–9759(1989). J. Hermisson, “Aperiodic and correlated disorder in XYchains: exact results,” Journal of Physics A: Mathematicaland General , 57–79 (1999). P. G. Harper, “Single Band Motion of Conduction Elec-trons in a Uniform Magnetic Field,” Proceedings of thePhysical Society. Section A , 874–878 (1955). S. Xu, X. Li, Y.-T. Hsu, B. Swingle, and S. Das Sarma,“Butterfly effect in interacting Aubry-Andre model: Ther-malization, slow scrambling, and many-body localization,”Phys. Rev. Research , 032039 (2019). D. S. Fisher, “Critical behavior of random transverse-fieldIsing spin chains,” Phys. Rev. B , 6411–6461 (1995). S. Katsura, “Statistical Mechanics of the Anisotropic Lin-ear Heisenberg Model,” Phys. Rev. , 1508–1518 (1962). E. R. Smith, “One-dimensional XY model with ran-dom coupling constants. I. Thermodynamics,” Journal ofPhysics C: Solid State Physics , 1419–1432 (1970). J. H. H. Perk, H. W. Capel, M. J. Zuilhof, and T. J.Siskens, “On a soluble model of an antiferromagnetic chainwith alternating interactions and magnetic moments,”Physica A: Statistical Mechanics and its Applications ,319 – 348 (1975). H. Nishimori, “One-dimensional XY model in lorentzianrandom field,” Physics Letters A , 239 – 243 (1984). O. Derzhko and J. Richter, “Solvable model of a randomspin-XY chain,” Phys. Rev. B , 14298–14310 (1997). I. I. Satija, “Symmetry breaking and stabilization of criti-cal phase,” Phys. Rev. B , 3511–3514 (1993). I. I. Satija and J. C. Chaves, “XY-to-Ising crossover andquadrupling of the butterfly spectrum,” Phys. Rev. B ,13239–13242 (1994). M. Heyl, A. Polkovnikov, and S. Kehrein, “DynamicalQuantum Phase Transitions in the Transverse-Field IsingModel,” Phys. Rev. Lett. , 135704 (2013). C. Karrasch and D. Schuricht, “Dynamical phase transi-tions after quenches in nonintegrable models,” Phys. Rev.B , 195104 (2013). E. Canovi, P. Werner, and M. Eckstein, “First-Order Dy-namical Phase Transitions,” Phys. Rev. Lett. , 265702(2014). F. Andraschko and J. Sirker, “Dynamical quantum phasetransitions and the Loschmidt echo: A transfer matrix ap-proach,” Phys. Rev. B , 125120 (2014). M. Heyl, “Dynamical Quantum Phase Transitions in Sys-tems with Broken-Symmetry Phases,” Phys. Rev. Lett. , 205701 (2014). M. Heyl, “Scaling and Universality at Dynamical QuantumPhase Transitions,” Phys. Rev. Lett. , 140602 (2015). R. A. Jalabert and H. M. Pastawski, “Environment-Independent Decoherence Rate in Classically Chaotic Sys-tems,” Phys. Rev. Lett. , 2490–2493 (2001). H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P.Sun, “Decay of Loschmidt Echo Enhanced by QuantumCriticality,” Phys. Rev. Lett. , 140604 (2006). R. Jafari and H. Johannesson, “Loschmidt Echo Revivals:Critical and Noncritical,” Phys. Rev. Lett. , 015701(2017). T. Gorin, T. Prosen, T. H. Seligman, and M. ˇZnidariˇc,“Dynamics of Loschmidt echoes and fidelity decay,”Physics Reports , 33 – 156 (2006). M. Fisher, “The nature of critical points, Lectures in The-oretical Physics, vol. VIIc,” (1965). J. C. Budich and M. Heyl, “Dynamical topological orderparameters far from equilibrium,” Phys. Rev. B , 085416(2016). S. Vajna and B. D´ora, “Topological classification of dy-namical phase transitions,” Phys. Rev. B , 155127(2015). S. Sharma, U. Divakaran, A. Polkovnikov, and A. Dutta,“Slow quenches in a quantum Ising chain: Dynamicalphase transitions and topology,” Phys. Rev. B , 144306(2016). U. Bhattacharya and A. Dutta, “Interconnections between equilibrium topology and dynamical quantum phase tran-sitions in a linearly ramped Haldane model,” Phys. Rev.B , 184307 (2017). M. Anquez, B. A. Robbins, H. M Bharath, M. Boguslawski,T. M. Hoang, and M. S. Chapman, “Quantum Kibble-Zurek Mechanism in a Spin-1 Bose-Einstein Condensate,”Phys. Rev. Lett. , 155301 (2016). C. Meldgin, U. Ray, P. Russ, D. Chen, D. M. Ceperley, andB. DeMarco, “Probing the Bose glass–superfluid transitionusing quantum quenches of disorder,” Nature Physics ,646–649 (2016). Alexander Keesling, Ahmed Omran, Harry Levine, HannesBernien, Hannes Pichler, Soonwon Choi, Rhine Samaj-dar, Sylvain Schwartz, Pietro Silvi, Subir Sachdev, PeterZoller, Manuel Endres, Markus Greiner, Vladan Vuleti´c,and Mikhail D. Lukin, “Quantum Kibble–Zurek mecha-nism and critical dynamics on a programmable Rydbergsimulator,” Nature , 207–211 (2019). H. Saito, Y. Kawaguchi, and M. Ueda, “Kibble-Zurekmechanism in a quenched ferromagnetic Bose-Einstein con-densate,” Phys. Rev. A , 043613 (2007). A. Sinha, M. M. Rams, and J. Dziarmaga, “Kibble-Zurek mechanism with a single particle: Dynamics of thelocalization-delocalization transition in the Aubry-Andr´emodel,” Phys. Rev. B , 094203 (2019). Victor Mukherjee, Uma Divakaran, Amit Dutta, and Dip-timan Sen, “Quenching dynamics of a quantum XY spin- chain in a transverse field,” Phys. Rev. B , 174303(2007). Chaohong Lee, “Universality and Anomalous Mean-FieldBreakdown of Symmetry-Breaking Transitions in a Cou-pled Two-Component Bose-Einstein Condensate,” Phys.Rev. Lett. , 070401 (2009). Jun Xu, Shuyuan Wu, Xizhou Qin, Jiahao Huang, Yong-guan Ke, Honghua Zhong, and Chaohong Lee, “Kibble-Zurek dynamics in an array of coupled binary Bose con-densates,” EPL (Europhysics Letters) , 50003 (2016). T. W. B. Kibble, “Topology of cosmic domains andstrings,” Journal of Physics A: Mathematical and General , 1387–1398 (1976). T. W. B. Kibble, “Some implications of a cosmologicalphase transition,” Physics Reports , 183 – 199 (1980). T. W. B. Kibble, “Phase-transition dynamics in thelab and the universe,” Physics Today , 47–52 (2007),https://doi.org/10.1063/1.2784684. N. Fl¨aschner, D. Vogel, M. Tarnowski, B. S. Rem, D.-S.L¨uhmann, M. Heyl, J. C. Budich, L. Mathey, K. Sengstock,and C. Weitenberg, “Observation of dynamical vortices af-ter quenches in a system with topology,” Nature Physics , 265–268 (2018). X. Cai, L.-J. Lang, S. Chen, and Y. Wang, “Topolog-ical Superconductor to Anderson Localization Transitionin One-Dimensional Incommensurate Lattices,” Phys. Rev.Lett. , 176403 (2013). J. Wang, X.-J. Liu, X. Gao, and H. Hu, “Phase diagramof a non-Abelian Aubry-Andr´e-Harper model with p -wavesuperfluidity,” Phys. Rev. B , 104504 (2016). Q.-B. Zeng, S. Chen, and R. L¨u, “Generalized Aubry-Andr´e-Harper model with p -wave superconducting pair-ing,” Phys. Rev. B , 125408 (2016). H.-Q. Wang, M. N. Chen, R. W. Bomantara, J. Gong,and D. Y. Xing, “Line nodes and surface Majorana flatbands in static and kicked p -wave superconducting Harper model,” Phys. Rev. B , 075136 (2017). M. Yahyavi, B. Het´enyi, and B. Tanatar, “GeneralizedAubry-Andr´e-Harper model with modulated hopping and p -wave pairing,” Phys. Rev. B , 064202 (2019). L.-J. Lang and S. Chen, “Majorana fermions in density-modulated p -wave superconducting wires,” Phys. Rev. B , 205135 (2012). J. Dziarmaga, “Dynamics of a quantum phase transition inthe random Ising model: Logarithmic dependence of thedefect density on the transition rate,” Phys. Rev. B ,064416 (2006). A. P. Young and H. Rieger, “Numerical study of the ran-dom transverse-field Ising spin chain,” Phys. Rev. B ,8486–8498 (1996). T. Caneva, R. Fazio, and G. E. Santoro, “Adiabatic quan-tum dynamics of a random Ising chain across its quantumcritical point,” Phys. Rev. B , 144427 (2007). B.-B. Wei, “Fidelity susceptibility in one-dimensional dis-ordered lattice models,” Phys. Rev. A , 042117 (2019). J. C. C. Cestari, A. Foerster, M. A. Gusm˜ao, andM. Continentino, “Critical exponents of the disorder-driven superfluid-insulator transition in one-dimensionalBose-Einstein condensates,” Phys. Rev. A , 055601(2011). E. Lieb, T. Schultz, and D. Mattis, “Two soluble modelsof an antiferromagnetic chain,” Annals of Physics , 407– 466 (1961). A. Chandran and C. R. Laumann, “Localization andSymmetry Breaking in the Quantum Quasiperiodic IsingGlass,” Phys. Rev. X , 031061 (2017). W. H. Zurek, U. Dorner, and P. Zoller, “Dynamics of aQuantum Phase Transition,” Phys. Rev. Lett. , 105701(2005). P. Zanardi and N. Paunkovi´c, “Ground state overlap andquantum phase transitions,” Phys. Rev. E , 031123(2006). M. Kolodrubetz, B. K. Clark, and D. A. Huse, “Nonequi-librium Dynamic Critical Scaling of the Quantum IsingChain,” Phys. Rev. Lett. , 015701 (2012). S. Deng, G. Ortiz, and L. Viola, “Dynamical non-ergodicscaling in continuous finite-order quantum phase transi-tions,” EPL (Europhysics Letters) , 67008 (2008). A. Francuz, J. Dziarmaga, B. Gardas, and W. H. Zurek,“Space and time renormalization in phase transition dy-namics,” Phys. Rev. B , 075134 (2016). A. Peres, “Stability of quantum motion in chaotic and reg-ular systems,” Phys. Rev. A , 1610–1615 (1984). H. Yin, S. Chen, X. Gao, and P. Wang, “Zeros ofLoschmidt echo in the presence of Anderson localization,”Phys. Rev. A , 033624 (2018). Attila Szab´o and Ulrich Schneider, “Non-power-law uni-versality in one-dimensional quasicrystals,” Phys. Rev. B , 134201 (2018). Taylor Cookmeyer, Johannes Motruk, and Joel E. Moore,“Critical properties of the ground-state localization-delocalization transition in the many-particle Aubry-Andr´e model,” Phys. Rev. B , 174203 (2020). Utkarsh Agrawal, Sarang Gopalakrishnan, and Ro-main Vasseur, “Universality and quantum criticality inquasiperiodic spin chains,” Nature Communications11