An instanton-like excitation of a discrete time crystal
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l An instanton-like excitation of a discrete time crystal
Xiaoqin Yang and Zi Cai
1, 2, ∗ Wilczek Quantum Center and Key Laboratory of Artificial Structures and Quantum Control,School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Shanghai Research Center for Quantum Sciences, Shanghai 201315, China
Spontaneous symmetry breaking and elementary excitation are two of the pillars of condensedmatter physics that are closely related to each other. The symmetry and its spontaneous breakingnot only control the dynamics and spectrum of elementary excitations, but also determine their un-derlying structures. In this paper, we study the excitation properties of a non-equilibrium quantummatter: a discrete time crystal phase that spontaneously breaks the temporal translational sym-metry. It is shown that such an intriguing symmetry breaking allows an instanton-like excitationthat represents a tunneling between two “degenerate” time crystal phases. Furthermore, we alsoobserve a dynamical transition point at which the instanton “size” diverges, a reminiscence of thecritical slowing down phenomenon in nonequilibrium statistic physics. A phenomenological theoryhas been proposed to understand the phase dynamics of the proposed system and the experimentalrealization and detection have also been discussed.
Usually, the ground state energy of an equilibrium sys-tem does not have much to do with its observable behav-ior. What’s physically important are the properties oflow-lying excited states, which are likely to be excitedowing to weak external fields or relatively low tempera-tures. For example, the thermal and elastic properties ofa solid are determined by a few number of lattice waveexcitations known as phonons[1]. Non-equilibrium quan-tum matter fundamentally differs from its equilibriumcounterparts, and has received considerable interest invarious fields ranging from ultracold atoms[2] to solidstate physics[3] over the past decade. However, com-pared to equilibrium cases, much less is known about the“excitation” of non-equilibrium quantum matter, even itsdefinition may be questionable. For example, “exciting”such a state does not necessarily mean increasing its en-ergy. Even if the excitation can be well-defined, its prop-erties are far from clear. For example, one may wonderwhether they can be decoupled into superpositions of “el-ementary” excitations similar to equilibrium systems[1].If so, how to characterize such elementary excitations andbuild up their relationship with fundamental propertiesof non-equilibrium quantum matter, e.g. the symmetriesand their spontaneous breaking.As a prototypical example of nonequilibrium quan-tum matter, the time crystal (TC) phase has allowednew possibilities for the spontaneous symmetry break-ing (SSB) paradigm[4], and have attracted considerableinterest in its different form[5–17]. Such a state with atemporal translational symmetry breaking(TSB), despitebeing proven to be forbidden in thermodynamic equilib-rium states [18, 19], has been experimentally realized inintrinsically non-equilibrium settings with periodic driv-ing [20, 21]. Its physical observables develop persistentoscillations whose periods are an integer multiple of theHamiltonian period, and thus they spontaneously break ∗ Electronic address: [email protected] the discrete temporal translational symmetry. In equi-librium systems, the SSB is closely related to the ele-mentary excitation. In particular, it does not only af-fect the dynamics and spectrum of an elementary excita-tion, but also determines its structure. For example, fora one-dimensional (1D) system, a spontaneous discretespatial TSB allows certain soliton-like excitation: a topo-logical defect that usually carries fractionalized quantumnumber[22]. A profound question is then to understandthe relationship between the SSB and elementary exci-tation in non-equilibrium quantum matters like the timecrystal, for example, what’s kind of elementary excitationpermitted by the discrete temporal TSB?In this paper, we address this issue by studying thedynamics of a periodically driven 1D interacting bosonicmodel that can manifest in a sub-harmonic response forphysical quantities, which is an indication of a TC phase.To “excite” this TC phase, we impose a time-dependentperturbation, which transiently breaks the original timetranslational symmetry, and we then monitor the re-sponse of the physical observable. It is found that aslow perturbation (ramping) may induce an instanton-like excitation sandwiched between two “degenerate” TCphases, as shown in Fig.1. By tuning the ramping ve-locity, one can observe a transition point, at which the“size”(lifetime) of such an excitation diverges, a reminis-cence of the critical slowing down phenomenon[23, 24].A phenomenological theory has been proposed to explainour numerical results. It is worth emphasizing that thesystem we studied is a closed system, which makes its ex-citations fundamentally differ from those diffusive Gold-stone modes studied in dissipative TC phases[25].
Model and method:
We consider a 1D hard-corebosonic model with an infinite long-range interaction.The Hamiltonian reads as follows: H = − J X i (ˆ b † i ˆ b i +1 + h.c ) − V ( t ) L X ij ( − i − j ˆ n i ˆ n j (1)where J is the single-particle nearest-neighbor(NN) hop-ping amplitude, ˆ n i = ˆ b † i ˆ b i is the particle number oper-ator at site i . V ( t ) is the strength of the all-to-all in-teraction, which is time-dependent but does not decaywith distance. L is the system size of the 1D lattice,and the prefactor L in front of the interaction terms ofEq.(1) guarantees that the total interacting energy lin-early scales with the system size. In a bipartite lattice( e.g, a 1D lattice as in our case), Eq.(1) indicates thatthe interaction between a pair of bosons are attractive(repulsive) if they are located in the same (different) sub-lattice. Such an infinitely long-range interaction has beenrealized in recent high-finesse cavity experiments[26, 27]by coupling bosons to the cavity vacuum mode whoseperiod is twice of that of the optical lattice. The totalparticle number N is conserved in our system, and we fo-cus on the case of half-filling ( N = L/
2) throughout thispaper. In the equilibrium case ( V ( t ) = V ), the groundstate of the 1D Ham.(1) is always a Mott-insulator with acharge-density-wave (CDW) order for arbitrary positive V .Owing to the all-to-all coupling of the interactionin Ham.(1), in the thermodynamical limit the mean-field method is not an approximation but an exactmethod. In particular, it provides an exact descrip-tion of both the equilibrium properties and real-timedynamics of the system[28, 29] (see also SupplementaryMaterial(SM)[30]). The mean-field method allows us todecouple the all-to-all interaction by introducing an aux-iliary staggered field which is self-consistently determinedduring the time evolution, and the Ham.(1) can be ex-pressed as:¯ H ( t ) = − J X i [ b † i b i +1 + h.c ] + m ( t ) V ( t ) X i ( − i ˆ n i (2)where m ( t ) = h Ψ( t ) | L ( − i ˆ n i | Ψ( t ) i and | Ψ( t ) i is thewavefunction of the system at time t . The time evolu-tion under the Hamiltonian.(2) can be solved exactly byperforming the Jordan-Wigner transformation to trans-fer 1D hard-core bosons into spinless fermions, andthe Ham.(2) transform into a non-interacting fermionicmodel.Throughout this paper, we consider the periodicboundary condition (PBC), which allows us to performthe Fourier transformation, after which the fermionicHamiltonian turns to¯ H ( t ) = X k [ c † k c † k + π ] (cid:20) ε k m ( t ) m ( t ) ε k + π (cid:21) (cid:20) c k c k + π (cid:21) (3)where the summation is over the momentum in the firstBrillouin zone of Ham.(2) ( k ∈ [ − π , π ]), and c k ( c † k )denotes the annihilation(creation) operator of the spin-less fermion. ε k = − J cos k , thus ε k = − ε k + π . Eq.(3)indicates that the dynamics of the system can be consid-ered as a collective behavior of different k modes, each ofwhich is a two-level quantum system subjected to time-dependent field that was self-consistently determined as m ( t ) = L P k h Ψ k ( t ) | c † k c k + π | Ψ k ( t ) i . Discrete time crystal.
Despite the triviality of theground state phase diagram, the system can exhibit rich
FIG. 1: (Color online). (a) Schematic diagram of a perioddoubling dynamics in the presence of periodical driving andtwo degenerate TC phases; (b)Schematic diagram of the phaseramping protocol in our model and an instanton-like excita-tion induced by it. dynamical behavior in the presence of a time-dependent V ( t ). For instance, in a quantum quench protocol, onecan start from a ground state of Ham.(1) with V ( t = 0) = V i , suddenly change it to a different value V ( t >
0) = V f and let the system evolve under this new Hamiltonian.It has been shown that the long-time dynamics of thismodel can exhibit either persistent oscillations or ther-malization depending on the choice of initial states[31](see SM[30]). The dynamical behavior is even richer andmore interesting when we introduce periodical drivinginto Ham.(1), e.g. V ( t >
0) = V f + δ cos 2 πt with δ thedriving amplitude. We observe that[30] depending on thedifferent choices of the initial states and driving ampli-tude δ , the long-time dynamics of m ( t ) could exhibit aperiodic oscillation with a frequency that is either iden-tical to or independent of the driving frequency; the for-mer can be considered as a synchronization phenomenonwhich has been observed in periodically-driven integrablesystems[32]. In addition, it can also exhibit quasi-periodic oscillations with a multi-period structure[30].Most interesting dynamics can be observed in the in-termediate driving regime (see Fig.2 b), where m ( t ) ex-hibits a persistent oscillation whose period is twice thatof the external driving period: a signature of “discretetime crystal”[33] that the discrete time translational sym-metry in Ham.(1) H ( t ) = H ( t + T ) has been sponta-neously broken ( m ( t ) = m ( t + 2 T ) = m ( t + T ) with T = 1 is the period of driving). This phenomena isrooted in the non-linearity of the self-consistent mean-field equation of motion. However, unlike the period dou-bling phenomena in the non-linear classical systems (e.g.a driven-dissipative pendulum[34]) or the “dissipativetime crystals” in open quantum many-body systems[13],our system is a closed system, therefore there is no en-tropy generation. Ham.(1) is free from disorder, henceit is the integrability[35] rather than the many-bodylocalization[9] that prevents our model from being heatedto an infinite temperature state. -0.40.00.4 V (t) (a) m (t) (b) m (t) T =0.5 T =8.0 (c) m (t) t T =2.40 T =2.45 (d) [J -1 ] FIG. 2: (Color online). (a)Periodical driving V ( t >
0) = V f + δ cos 2 πt with a period T = 1. (b) Time crystal dynamicswith a period 2 in the absence of ramping. (c) Long-time dy-namics of m ( t ) with slow and fast ramping, which correspondto two “degenerate” TC phases. (d)Long-time dynamics of m ( t ) with an intermediate ramping rate close to the dynami-cal transition point. The parameters are chosen as V i = 10 J , V f = 3 J , δ = 0 . J and L = 5000. Instanton-like excitations:
Despite the richness of thedynamics behavior, here we will study neither the globalnon-equilibrium phase diagram of our model, nor themathematical origin behind this DTC phase. Instead, wewill use this exactly solvable model as a starting point tostudy the “excitations” of the DTC phase. Owing to thespontaneous breaking of the discrete time TTS, the TCphase are supposed to be two-fold “degenerate”, each ofwhich has a period 2 and can be connected to the otherone by shifting a half-period of the DTC along the tem-poral direction. By making an analogy with the 1D CDWsystem, we can consider the excitation as a domain wallin the form of a topological soliton[22] that separatesthese two “degenerate” states. To realize such an object,one need to “excite” the system by introducing a tempo-ral perturbation that transiently breaks the original timetranslational symmetry.Here, we introduce an additional linear ramping ofthe phase on top of the periodic driving: V ( t ) = V i + δ cos 2 π [ t + θ ( t )], where θ ( t ) = t − t i T for t ∈ [ t i , t i + T ],and θ ( t ) = 0 otherwise. t i is the initial time of the ramp-ing, and T is its duration, after which the external driv-ing accumulates an additional 2 π phase compared withthe case without ramping. Therefore, the Hamiltonianis identical the one before the ramping. We assume thatprior to the ramping ( t < t i ), the system is in one of thedegenerate DTC phase, which is “excited” by the addi-tional time-dependent perturbation once the ramping is
200 400 600 800 D T -T c0 S n (t) T =0.5 T =8.0 S n (t) t T =2.4 T =2.45 (a)(b) [J -1 ] DW size
FIG. 3: (Color online). Dynamics of the relative displacementof the peak positions S n ( t ) with (a) fast and slow and (b)intermediate ramping rates close to the dynamical transitionpoint T c = 2 .
41. The inset of Fig.2 (a) shows the dependenceof the instanton size on the ramping duration close to thedynamical transition point. Other parameters are chosen tobe the same as those in Fig.2 switched on. After the ramping ( t > t i + T ), it stilltakes some time for the system to relax before it entersinto another “stable” dynamical regime.In the following, we will study both the long-time andtransient dynamics of the system and demonstrate theirdependence on the ramping rates πT . To this end, wefix the initial states ( V i = 10 J ) and all other parameters( V f = 3 J , δ = 0 . J ) except the duration of ramping T .In the limit of T = 0, the long-time dynamics is identicalto that without ramping, because the periodic driving isabruptly changed by a phase of 2 π at t = t i , owing towhich it remains unaltered. For a rapid ramping, thesystem exhibits a similar long-time dynamics, but witha weaker oscillation amplitude, as shown in Fig.2 (c). Inthe opposite limit of slow ramping, the peak positions of m ( t ) are pinned to those of V ( t ). Thus, after the ramp-ing, the phase of V ( t ) is pushed forward by 2 π , and so is m ( t ). However, due to the period doubling feature of theDTC, m ( t ) is shifted by a half-period, and thus falls intothe other degenerate state that differs from the one beforethe ramping, although V ( t ) remains unchanged. Thus,by slowly ramping the driving phase, one can create adomain wall sandwiched between two “degenerate” timecrystal phases, whose properties will be studied later. Onthe contrary, for a fast ramping, the system cannot follow V ( t ) “adiabatically”, thus finally relaxes to a TC phasesimilar to the original one. These two distinct dynamicalbehaviors of slow and fast ramping indicate a transitionbetween them. From Fig.2 (d), we can find that thistransition occurs suddenly at T ≈ .
41 and there is nocrossover regime between them.For an ideal TC phase, the peak positions of m ( t )are pinned to those of V ( t ), whereas in realistic situa-tions, there is some relative displacement, which couldbe used to study the properties of excitations. Foreach peak of m ( t ), we define its relative displacementas S n ( t ) = | P n ( t ) − j n | , where P n ( t ) is the position ofthe n-th peak of m(t), j n is an integer number indicatingthe peak position of V ( t ) that minimizes S n ( t ). FromFig.3 (a) and (b), we can find that in the TC phase thepeak positions of m ( t ) are close to those of V ( t ), thusone can use S n ( t ) to distinguish the two “degenerate”TC phases. For solution A (B), S n ( t ) is close to 0 (1),whereas 0 < S n ( t ) < S n ( t )for various T is plotted in Fig.3, from which we can de-termine that the domain wall between two different TCphases can only be observed for T > .
41. It is also in-teresting to notice that the closer the system approachesthe transition point, the longer it takes for the system torelax. It appears that the “size” of the domain wall (or re-laxation time) diverges at the phase transition point (seethe inset in Fig.3 (a)), which reminds us of the criticalslowing down phenomena in dynamical critical systems.
Phenomenological theory of phase dynamics:
Theinstanton-like excitations observed previously could beunderstood as a macroscopic tunneling process betweentwo “degenerate” states induced by a time-dependentperturbation. Rather than the amplitude of m ( t ), wefocus on its phase degree of freedom θ ( t ). We speculatethat its dynamics can be described using a phenomeno-logical equation of motion (EOM) as: d θ ( t ) dt + γ dθ ( t ) dt + ∂U ( θ, t ) ∂θ = 0 (4)where γ is a phenomenological coefficient that character-izes the dissipation, which tends to stabilize the phasedynamics. Although the system we studied is a closedsystem without dissipation, the phase variables could ex-change energy with other degrees of freedom. Thereforeits dynamics can be dissipative. U ( θ, t ) can be consideredas an effective double-well potential imposed throughthe periodic driving. To imitate the phase ramping, wechoose U ( θ, t ) = V cos[2 π ( θ + tT )] for 0 < t < T and U ( θ, t ) = V cos[2 πθ ] otherwise. We assume that the po-tential is defined in the regime θ ∈ [ − ,
1] with PBC U ( θ = 1 , t ) = U ( θ = − , t ), within which U ( θ, t ) is adouble well potential with two minima, each of whichcorresponds to one degenerate TC phase that can be con-nected to the other by shifting a 2 π phase.We assume that initially the system is in one of the de-generate TC phases ( e.g. θ | t =0 = 0 . θ | t =0 = 0), thenwe turn on the ramping, which corresponds to a globalshift of the external potential with a constant velocity T .Once the ramping is finished, the external potential re-turns to its initial value U ( θ, t = 0) = U ( θ, t = 1), and wewill study the dynamics of θ ( t ) based on the phenomeno-logical EOM.(4). Fig.4 shows that for a quick ramping, θ has no time to follow the movement of the potential,and thus it will maintain in its original minimum. How- FIG. 4: (Color online). Dynamics of θ ( t ) predicted by theEOM.(4) in the presence of fast, intermediate and slow ramp-ing with phenomenological parameters V = π and γ = 0 . ever, a sufficiently slow shift of the potential will drive θ from 0 . − .
5. For an intermediate ramping rate, θ ( t )keeps oscillating between the two minima for a long timebefore it relaxes to one of them. The relaxation time (orthe “size” of the domain wall) depends on the rampingrate and the dissipation strength γ , which is determinedby the parameters in microscopic model. The divergenceof the relaxation time at the transition point observedin our microscopic model (Fig.3) indicates that γ → T = T c , which further implies the absence of energyexchange between the phase variable and other degreesof freedom. Experimental realization and detection:
The proposedmodel can be realized in an experimental setup simi-lar with to the one in Ref.[26]. In terms of detections,the CDW order parameter can be directly measured us-ing the superlattice band-mapping technique[36], or in-directly through the heterodyne detection[26, 27]. Foran optical lattice with J ≈ Conclusion and outlook:
In conclusion, we propose aprotocol to realize an excitation in a discrete time crys-tal. We have observed a dynamical transition betweenthe cases with fast and slow ramping, where the “size”of instanton-like excitations diverges. Future develop-ments will include the analysis of the critical behavior ofthe dynamical transition, and use them to classify dif-ferent types of TC phases and the transitions betweenthem. In addition, although it is conjectured that thedivergence of the relaxation time is related to the dissi-pationless dynamics of phase variables, the microscopicmechanism attributing to this remains unclear and willbe studied in the future. Furthermore, it is shown thatthe phase shifts induced by the ramping are always mul-tiples of 2 π , which reminds us of the quantization of par-ticle transport in Thouless pumping[37]. Therefore, onemay wonder whether there is a topological origin behindit. After all, in equilibrium physics, DW in a 1D systemis usually a topological object carrying fractionized quan-tum numbers. Finally, it is interesting to generalize ourmodel/method to the continuous time crystal ( e.g. ourmodel with δ = 0 and V i = 0 . J ), where one may expecta “phonon”-like rather than an instanton-like excitationowing to the spontaneous breaking of continuous timetranslational symmetry. If it exists, such an excitationshould differ from the diffusive Goldstone modes studiedin the dissipative time crystal[25], because our system isisolated from the environment. Acknowledgments
Acknowledgments .—We acknowledge support by theNational Key Research and Development Program ofChina (Grant No. 2016YFA0302001), NSFC of China(Grant No. 11674221, No.11574200), Shanghai Mu-nicipal Science and Technology Major Project (GrantNo.2019SHZDZX01) and the Program Professor of Spe-cial Appointment (Eastern Scholar) at Shanghai Institu-tions of Higher Learning.
Appendix A: Validity of the time-dependentself-consistent mean-field method
It is well known that for a model with infinitely long-range interaction, the mean-field method provides an ex-act treatment for the equilibrium properties in the ther-modynamic limit. In the following, we will prove thatthis is also the case for the dynamical problems.We denote F is the Fock basis of hard-core bosons (e.g. | · · · i ), and divide the evolution period [0 , t ] into Mslices with ∆ t = t/M . In the limit of M → ∞ , the pathintegral expression of the propagators can be written as: K = X { F , ··· F M − } h F | e − i ∆ tH ( t ) | F ih F | · · · e − i ∆ tH ( t M ) | F M i (A1)where | F j i is a Fock basis at the j -th time slice ( j =0 , · · · M ), with P { F j } | F j ih F j | a unit matrix. We fur-ther divide our Hamiltonian as H = ˆ T + ˆ V whereˆ T = − J P i (ˆ b † i ˆ b i +1 + h.c ), and the interacting energy can be rewritten as:ˆ V = LV ( t )[ 1 L X i ( − i ˆ n i ] . (A2)By performing Suzuki-Trotter decomposition, one canobtain that e − i ∆ tH = e − i ∆ t ˆ T e − i ∆ t ˆ V + O (∆ t ) (A3)Therefore, in the limit of ∆ t →
0, the propagator inEq.(A1) can be expressed as: K = X { F , ··· F M − } M − Y j =1 T j,j +1 e − i ∆ tV { F j } ( t j ) (A4)where T j,j +1 = h F j | e − i ∆ t ˆ T | F j +1 i , and V { F j } ( t j ) = h F j | e − i ∆ t ˆ V | F j i = e − i ∆ tLV ( t i ) S{ F j } , where S{ F j } = L P i ( − i n i | { n i }∈{ F j } .The quadratic part in Eq.(A4) can now be decoupledusing the Hubbard-Stratonovic transformation by intro-ducing auxillary fields { m j } with j = 1 , M − K = M − Y j =1 X { F j } Z dm j N T j,j +1 e − i ∆ tLV ( t j )[( m j ) − m j S{ F j } ] (A5)where N is a normalization factor to keep the identity ofthe Gaussian integral. In the thermodynamic limit L →∞ , Eq.(A5) indicates that the saddle point approxima-tion δ L [ m j ] δm j = 0 with L [ m j ] = V ( t j )[( m j ) − m j S{ F j } ],becomes exact, which means that the auxillary fields { m j } are given by the CDW order parameter of the stateas m j = S{ F j } . As a consequence, For the time evolutionof a state | ψ i in each infinitesimal time step (∆ t → e iH ( t ) δt are equal to the one af-ter mean-field approximation e i ¯ H ( t,m ( t ))∆ t with a time-dependent CDW order parameters m ( t ), which is deter-mined self-consistently during the time evolution by thesaddle point method. m (t) t V =0.75J V =10J [J -1 ] FIG. 5: (Color online). Quench dynamics of M ( t ) calculatedby the self-consistent mean-field method with different V i andthe parameters V f = 3 J , L = 5000, ∆ t = 10 − , δ = 0; Appendix B: Quantum quench dynamics withoutdriving
In this section, we study quantum quench dynamicsof the model ( δ = 0) by choosing an initial state asthe ground state of H with V ( t ) = V i , and suddenlychange the interaction strength to a different value V f and let the system evolve under this new Hamiltonian.The long-time dynamics of this model has been analyzedin Ref.[1], here we will outline the mean-field results ofRef.[1]. We plot the time evolution of the CDW orderparameter m ( t ) = L P i ( − i h n i i starting from two dif-ferent initial states in Fig.5, from which we can find twodistinct dynamical behaviors even though the evolutionHamiltonian are the same ( V f = 3 J ). For the quenchdynamics from the initial state with V i = 10 J , m ( t ) willconverge to a constant after sufficiently long time, whilefor the other case with V i = 0 . J , it persistently oscil-lates with a period that spontaneously emerges duringthe quench dynamics. m (t) (a) f ( ) m (t) (b) f() m (t) (c) f() f( ) m (t) t [J -1 ] (d) FIG. 6: (Color online). Dependence of periodically drivendynamics of M ( t ) on driving amplitudes [(a) δ = 0 . J ; (b) δ = 0 . J ; (c) δ = 8 J ;] and initial states [(d) V i = 0 . J , δ = 0 . J ]. Other parameters are chosen as V i = 10 J for(a)-(c), L = 5000 and ∆ t = 10 − J − for (a)-(d). m (t) t L=500 L=1000 L=2000 [J -1 ] (a)
950 960 970 980 990 m (t) t t=10 -3 t=10 -4 t=10 -5 [J -1 ] (b) FIG. 7: (Color online). (a) Quench dynamics of M ( t ) withdifferent system size L and the parameters δ = 0, ∆ t = 10 − ;(b) Periodically driven dynamics with different ∆ t and δ =0 . J , L = 5000; V i = 10 J and V f = 3 J for (a) and (b) Appendix C: Periodically driven dynamics
The periodically driven dynamical behavior in ourmodel is very rich, while in the main text we only focus onthe time crystal phase. Here we list several typical long-time behaviors, even though it is difficult to enumerateall the possibilities.In Fig.(7), we plot the long-time dynamics of m(t) withdifferent driving amplitude δ and initial states V i . ForFig.7 (a)-(c), we fixed the initial state as the ground stateof the Hamiltonian with V i = 10 J , while for Fig.7 (d),we change it to V i = 0 . J . As shown in Fig.7 (a), for aweak periodical driving, the long-time dynamics becomesa persistent quasi-periodic oscillation with a multi-periodstructure, which can be reflected in its Fourier spectrum f ( ω ) = T R dte iωt m ( t ). As shown in the inset of Fig.7(a), the Fourier spectrum f ( ω ) exhibits sharp peaks atdifferent characteristic frequency, while the dominant onelocates at the place exactly the same with the externaldriving frequency ( ω = 1), which indicates m ( t ) is syn-chronous with the external periodic driving. For an in-termediate driving amplitude, for instance δ = 0 . J asshown in Fig.7 (b), the peak position of f ( ω ) has beenshifted from 1 to 0 .
5, which indicates a period-doublingtime crystal phase as we studied in the main text. Whenwe further increase δ to a strongly driven regime ( e.g. δ = 8 J as shown in Fig.7 c), the frequency spectrumexhibits a broad distribution, a signature of a chaoticdynamics without a dominant time scale. If we startfrom an different initial state, for instance, the groundstate of V i = 0 . J as shown in Fig.7 (d), we can find m ( t ) exhibits an persistent oscillation with a frequency ω = 0 .
46, which has nothing to do with the externaldriving frequency.
Appendix D: Finite size effect and Convergence ofthe numerical results
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