Universal presence of time-crystalline phases and period-doubling oscillations in one-dimensional Floquet topological insulators

We reported a ubiquitous presence of topological Floquet time crystal (TFTC) in one-dimensional periodically-driven systems. The rigidity and realization of spontaneous discrete time-translation symmetry (DTS) breaking in our TFTC model require necessarily coexistence of anomalous topological invariants, instead of the presence of disorders or many-body localization. We found that in a particular frequency range of the underlying drive, the anomalous Floquet phase coexistence between topological 0 and $\pi$ modes can produce the period-doubling (2T, two cycles of the drive) that breaks the DTS spontaneously, leading to the subharmonic response ($\omega/2$, half the drive frequency). The rigid period-2T oscillation is topologically-protected against perturbations due to both non-trivial opening of 0- and $\pi$-gaps in the quasienergy spectrum, thus, as a result, can be viewed as a specific"Rabi oscillation"between two Floquet eigenstates with certain quasienergy splitting $\pi/T$. Our modeling of the time-crystalline 'ground state' can be easily realized in experimental platforms such as topological photonics and ultracold fields. Also, our work can bring significant interests to explore topological phase transitions in Floquet systems and to bridge a profound connection between Floquet topological insulators and photonics and period-doubled time crystals.

Spontaneous symmetry breaking plays a profound role in modern physics, leading to a variety of condensed states of matter and fundamental particles. In analogy with conventional crystals that break the spatial translation symmetry, Wilczek [1][2][3] proposed the intriguing notion of "time crystal" that spontaneously breaks the continuous time-translation symmetry. Although the possibility of the spontaneous time-translation symmetry breaking in thermal equilibrium was ruled out by a no-go theorem [4][5][6], Floquet time crystals that break the discrete timetranslation symmetry in periodically driven systems proposed in [7][8][9][10][11] attracted intense attention. Since the external driving would heat a closed system to infinite temperature and eventually breaks the time-crystalline phase, disorder-induced many-body localization (MBL) is necessarily proposed to stabilize the long-range time-crystalline phase [8][9][10][11][12][13][14]. More interestingly, the expectation values of observables of systems exhibit later-time oscillations with multiple periods than that of the underlying drive [15][16][17].
On the other hand, Floquet time crystals do not have to require many-body localization as a prerequisite necessarily. The realizations of time crystals without MBL have also been demonstrated recently, such as a clear FTC model completely without disorders in a strongly interacting regime [22], and an experiment on nitrogen-vacancy centers in which the FTC was formed regardless of the delocalization by three-dimensional spin-dipolar interactions [16].
However, a robust FTC without many-body interaction or disorders has not been achieved [28,29].
To carry out the rigid time-crystalline phase in a clean Floquet system, we are inspired by involving the interplay between topological protection from topological insulators and photonics [30][31][32] and periodically-driven protocols from Floquet engineering [33,34]. With the assistance of Floquet engineering, we can obtain the anomalous chiral edge modes in periodically driven systems that lead to anomalous bulk-edge correspondence [35][36][37][38][39][40][41][42][43]. These anomalous Floquet phases exhibit localization and robustness as an edge state or a domain wall in the presence of perturbations (fluctuations and disorders). As a result, providing an excellent platform to realize the rigid FTCs.
In this work, we firstly reported a Floquet time crystal in the non-interacting situation that established a stable period-2T oscillation -the 'fingerprint' to experimentally observe a quantum time-crystalline phase -due to the topological protection from the nontrivial gap opening in quasienergy spectrum. Our one-dimensional model of FTC is primarily based on Floquet topological phases [35,36,43], named the zero-energy edge modes (0 modes) and πenergy edge modes (π modes), which have been extensively studied in periodically driven systems [28,[35][36][37][38][39][40]. For instance, in driven Su-Schrieffer-Heeger model as we studied previously [37], we found that in a specific drive frequency region where the two topological phases coexist with quasienergy difference given by |ε π − ε 0 | = π/T, the system exhibits a persisting period-2T oscillation which breaks the underlying discrete time-translation symmetry (T is the period of drive). We dubbed this period-doubling phenomenon in Floquet topological phases "topological Floquet time crystals (TFTCs)," since in our TFTC model, the topological protection inherited from topological phase coexistence instead of many-body interaction and disorders is of essence to stabilize the long-range subharmonic response(ω/2), and robustness against perturbations [44,45]. Clearly, our results suggest an exciting field of studying time crystals in non-interacting topological Floquet systems, which can be easily implemented experimentally, such as ultracold atoms and topological photonics. (d), a special driven SSH model is structured by both spatial and temporal dimerized, to produce the topological phase coexistence between 0 modes and π modes. (e), the quasienergy spectrum of the driven SSH model (d) is calculated from the one-cycle time evolution operator U(T) (see Appendix A1). The π-mode appears at the drive frequency region (1/3, 1), and the 0-mode appears at region (1/2, ∞). (f), the topological invariants of 0 and π modes is given by the gap invariants G 0 and G π , respectively. The two protected modes coexist in the region (1/2, 1) . (g), depicts the "Rabi" oscillation between 0 and π quasienergy levels, i.e.,  Fig.1a-d. To realize period-2T FTCs, we have to introduce two topological edge states in our driven SSH model. Fig.1b and 1c depicted the typical structure and drive configurations on how to construct the topological lattices which hold nontrivial edge states, that is, 0 modes and π modes, respectively. Zero modes of conventional SSH model (1b) have been widely observed in a number of experimental platforms [46][47][48][49], while π modes of driven SSH model (1c) was recently achieved in microwave waveguide arrays [37,50].
Intuitively, our dimerized model of interest is supposed to coexist both the topological modes by combing both the two dimerization configurations (1d). Since the quasienergy difference of the two modes is obtained as |ε π − ε 0 | = π/T, the topological coexistence can allow the superposition of two Floquet eigenstates that naturally exhibits later-time oscillations with two periods (2T) in system than that of the underlying drive [44,45]. Mathematically, our period-2T Floquet time crystal based on the driven SSH model (1d) is described by the following Hamiltonian, where † and are the creation and annihilation operators at the site i (or the i th waveguide), N is the total number of lattice sites, 0 is the constant on-site potential or chemical potential (or propagation constant). The second off-diagonal term in Eq.1 represents the nearestneighbor hopping, in which 0 is the constant coupling strength, 0 is the globally dimerized staggered coupling strength, and ( ) = δκ 1 cos(ωt + θ) is the periodically dimerized staggered coupling strength with δκ 1 being the amplitude of the time-periodic coupling, ω = 2π/T the drive frequency and θ the initial phase of the drive. All four models, as depicted in Hamiltonian H and then is presented in Fig. 1e, where the energy-related coupling parameters are normalized with bandwidth Δ. Because of the time periodicity of the drive ( ( + ) = ( )), the quasienergy lies in the spectrum ranging from π/T to −π/T, which is denoted as the Floquet-Brillouin zone [33][34][35]. We can observe that from the quasienergy spectrum, the π modes occur in the frequency region1/3 < ω/Δ < 1, while the 0 modes exist in the region ω/Δ > 1/2. Thus, we can expect that the topological phase coexistence lies in a certain region, 1/2 < ω/Δ < 1.
These edge states corresponding to the quasienergy spectrum ( Fig. 1e) are topologically protected, and we adopt the definitions of gap invariants for quantum periodically-driven systems proposed in [36,43], and calculate its topological invariants G 0 and G separately for 0-gap and π-gap in the energy-momentum space (see the details in the Appendix). Fig. 1f shows that the nontrivial invariants G 0 = 1 appears at the frequency region ω/Δ > 1/2 for 0 modes and G = 1 at the frequency region1/3 < ω/Δ < 1 for π modes, while the rest driven region is trivial with G 0 = 0 or G = 0. Indeed, these two gap invariants have been widely studied in many physical and optical systems [35][36][37][38][39]43], in which both the presence of two gap invariants indicates the anomalous bulk-edge correspondence in Floquet systems. [35,37,43]. As a result, we pointed out that both π modes and 0 modes we found in the quasienergy spectrum are topologically protected by the chiral (sublattice) symmetry. period longer than that of the drive, that is, | ± ( + 2 )⟩ = | ± ( )⟩ , because of the quasienergy difference |ε π − ε 0 | = π/T. This (quasi)energy splitting is firstly found in π-spin glass and Floquet time crystals based on MBL [8,11]. Since the Floquet phase coexistence is protected by the 0-gap and π-gap separately, the 2T periodic oscillation behaves like an isolated two-level "Rabi" oscillation ( Fig. 1g) with eliminating the gapped bulk states and suppressing the scatterings from perturbations [40]. As a consequence, the period-doubling inherits the rigidity from the topological phase coexistence of the Floquet system and breaks the discrete time-translation symmetry at particular frequencies. (we plotted the first six sites or waveguides from N=80). Only in the region of phase coexistence (Fig. 3c, at ω/Δ = 0.75), the field intensity unexpectedly oscillates along the boundary (i=1) with two periods (2T), while the rest two patterns of the evolutions (Fig. 3a, 3e) exhibit the conventional stroboscopic behavior undergoing the period of the drive. We note that in the only π-modes phase at ω/Δ = 0.37, the field intensity only propagates between the first and second sites over 24 periods (almost no energy occupied at the third site), whose nearfield evolution pattern has been recently observed by Q. Cheng et al. [37]. On the other hand, in the high-frequency approximation region, the 0-mode pattern shows its dominant effective static behavior, as shown in Fig. 3e. The second site (i = 2) has no energy occupied due to the emergent sublattice symmetry in the high-frequency limit, which requires the end states only localizes at the odd sites i = 1,3 (or even sites if counting from the other boundary).
To explore the oscillation of the system in a long-range response, we perform the Fourier transformation by taking the stable field intensity |ψ 1 (t)| 2 on the boundary in time from 10T to 24T. The corresponding Fourier spectrum at different frequencies are presented in Fig. 3b, 3d, 3f, and clearly, we found that the harmonics, subharmonics, and static response dynamically dominates in the three different topological phase regions, respectively, where only the response in the phase coexistence region is locked at half the drive frequency (1/2T) rigorously. As a result, we can attribute the period-doubling phenomenon to the topological phase coexistence in the quasienergy spectrum.  1d) is mapped into a static SSH system with no drive (Fig. 1b).
To present the Floquet phase transitions more intuitively and directly, we demonstrated the stroboscopic evolution patterns of near-field intensity distribution varying with different drive frequencies, as shown in Fig. 5. In the low-frequency region, only π mode can be excited, and the field intensity distribution exhibits harmonic behavior that oscillates with the same period as the drive frequency (Figs. 5a, 5b). As the frequency increases, the system enters the TFTC phase and shows stable period-2T oscillation (Figs. 5c-5f). At the high-frequency region, only 0 mode exists, and the driven system manifests the effective static behavior (Figs. 5g, 5h).  As a contrast, we also study the characteristic of Fourier components of the stroboscopic dynamics in the driven configuration ( ) ≠ 0, 0 = 0 (see the setup in Fig. 1c), in which only π-mode phase exists in the region 1/3 < ω/Δ < 1. The corresponding quasienergy spectrum is presented in the inset of Fig. 4b, in which the topological π modes manifest itself but without coexistence of 0-modes. Fig. 4b shows no subharmonic oscillation in the full driving range. It is worth to address that the harmonic component as a function of frequency has a sharp slope at ω/Δ = 1/3 but gradually disappears at ω/Δ = 1. This morphological feature of the harmonic component is correspondingly similar to the lineshape of the π-gap function in the quasienergy spectrum. Inspired by this speculation, we can explain the sharp and slow slopes of the subharmonic component in Fig. 4a directly rely on the lineshape feature of the emerging 0 gap at ω/Δ = 1/2 and the disappearing π gap at ω/Δ = 1, respectively.
Immunization against disorders -Finally, we assess the robustness against disorders (W) for the subharmonic response of topological Floquet time crystals. Fig. 6 demonstrates the subharmonic components in the phase coexistence region as a function of the strength of onsite disorders (Fig. 6a) and off-diagonal disorders (Fig. 6b) at the frequency ω/Δ = 0.75. We find that the spectrum peaks survive around at half the drive frequency and become more broad and diffusive when the strength of disorders W is increasingly higher than the emergent 0-and πgaps corresponding the staggered coupling strengths δκ 0 = 0.06, 1 = 0.12. It means that the stability of our TFTC is comparably rigid against disorders or imperfections in the case that the magnitude of the quasienergy gap is large enough to forbidden the disorder-induced scattering into the bulk states.
Note, that in our driven SSH model, the exact period-doubling in TFTC suffers from local perturbations (δ) on the boundaries because the chiral (sublattice) symmetry is explicitly broken [44]. In this situation, fortunately, the subharmonic response is still practically protected by two gaps, but it undergoes a slightly different stable period ( ′ ≠ 2 ) longer than that of the drive when the perturbation energy δ is added locally on the boundary (Fig. 6a). Thus, the TFTC phase with a rigid longer-period oscillation ( ′ > ) can safely remain in the presence of chiral (sublattice) symmetry, which is verified by our numerical calculations as presented in

Time-crystalline phases in other one-dimensional driven models
To demonstrate the ubiquitous existence of topological Floquet time crystal beyond the driven SSH model, here we reconstruct the coexistence of Majorana zero and pi modes in two wellknown toy models, that are, Kitaev's toy model for p-wave superconductors [51,52] and Transverse field Ising model (TFIM) for one-dimensional spin-1/2 chain [8][9][10][11], respectively, with involving the Floquet engineering protocols and SSH models. Interestingly, as we already known [51], the Kitaev model connects with our SSH model through a partial partial-hole mapping, and as well connects with the TFIM model through a non-local Jordan-Wigner transformation. Alternatively, the Floquet topological phases and invariants in these related driven models can be identified by the topological classification in terms of symmetry classes in Floquet systems [53][54][55][56].
These driven models (Eqs. 1 and 2) can be solved likewise, as we investigated in this work, and consequently, they hold typical Floquet topological excitations, such as edge modes and domain walls, which are spanned in sublattice, Nambu (particle-hole) and spin subspaces respectively. In the driven models with anomalous Floquet topological phases, therefore equivalently, the universal coexistence between zero and pi edge modes (or domain walls) can lead to the period-doubling oscillation that indicates a class of ubiquitous presence of Floquet time-crystalline period-2T oscillations in one-dimensional periodically driven systems. It is worthy to note that these three intimately-related TFTCs can be viewed as the hand-waving starting point to engage with various disorders, weak interactions, and other parameters protocols in such as topological insulators and superconductors [29,57], topological photonics and quantum synthetic materials.
Conclusion -To conclude, we reported a period-2T topological Floquet time crystal (TFTC) based on topological phase coexistence between 0 modes and π modes in a driven SSH model that exhibits a rigid and long-range subharmonic response, and we extended the ubiquitous presence of TFTCs in one-dimensional periodically-driven systems. The rigidity of perioddoubling against perturbations and disorders is topologically protected by both the nontrivial gap invariants; thus, as a consequence, it requires no many-body interactions or disorders. The mechanism of spontaneous breaking of discrete time-translation symmetry in the Floquet regime of phase coexistence can be experimentally realized in many platforms of Floquet topological insulators and topological photonics, such as cold atoms [46,58,59] and photonics/microwave platforms [37,47,48]. Also, our SSH manifestation of TFTC is a direct connection between topological insulators and Floquet (discrete) time crystals.
To extend, we would like to address two aspects. First, the TFTC phase can also be easily realized in topological superconductors, by experimentally achieving the coexistence between Floquet Majorana π modes and Majorana 0 modes [8,28]. Those experimental platforms have proposed in many literature, such as [38,39], but unfortunately, without experimentally detecting the 'fingerprint' of the novel period-2T oscillation yet. With the available technical conditions in experiment, it seems promising to realize a superconducting time crystal in near future. Second, the topological subharmonic response indicates an implementation of subharmonic radiation from the protected boundary/edge states when the interacting electromagnetic field is optically detected. The topological subharmonic radiation is an exotic optical parametric down-conversion process when the drive is viewed as the "pump" source for the artificially driven materials, which deserves more attention both from theorists and experimentalists.
Acknowledgement -We would like to thank the helpful discussions with Binghai Yan, Erez Berg, Netanel Lindner, Qingqing Cheng. This work is supported in parts by ICORE-Israel Center of Research Excellence program of the ISF, and by the Crown Photonics Center.
Y. P. and B. W. contribute equally to this work.

A1. Quasienergy spectrum from Floquet-Bloch theorem
The Floquet Hamiltonian can be expressed in the matrix form with the elements where | ⟩ denotes an orthonormal basis state of the system Hilbert space ℋ and | , ⟩⟩ denotes an orthonormal basis state of the composed Floquet-Hilbert space [60].
According to (S1), the Floquet Hamiltonian can be expressed as the matrix form where the matrix block ( ) is the ℎ Fourier component of ( ). Here, the diagonal terms, Because the spanned Floquet-Hilbert space is infinite, as shown in (S2), the practical calculation requires truncation in order to force the Hilbert into an effective finite-dimension space. The general truncation, in principle, is complicated. In our driven SSH case, however, only the first-order photon scattering is relevant, thus we truncation the space into five photon bands as calculated numerically in Fig. 2.
To explicitly explore the gap opening mechanism of the isolated modes, here let us represent our driven SSH Hamiltonian in the Bloch basis. Considering the periodic boundary condition, due to the translation symmetry, we can transform the equation (1) in the main text into the momentum space with the corresponding Bloch representation as [51] ( , ) = ( 0 + 0 + ( ) where σ x , σ y are the Pauli matrices on the basis of sublattices A and B. These Pauli operators are used to characterize the sublattice space corresponding to sublattice symmetry (i.e., chiral symmetry) after dimerization. Notice that the anti-commutation relation { ( , ), } = 0 indicates that our driven SSH modeling of period-2T Floquet time crystal possesses a sublattice ('chiral') symmetry defined by the Pauli operator [36]. In this case, the matrix blocks of the Floquet Hamiltonian in the analytical 2 × 2 matrix form can be written as  where for simplification we suppress the gauge index 0 such that = [ 0 = 0], ( ) = ( , 0) and we notice that the function η is the complex logarithm with branch cut along an axis with angle η, defined as [36] log − = , for − − 2 < < − . (S18) The Floquet operator ( ) and the effective share the same eigenstates and where = log − is the quasienergy.
Then we can get the quasienergy spectrum (here = ) of the effective Hamiltonian exhibited in Fig. 1e, and Fig. 3b in the main text. The dynamic evolution of the time-dependent system can be calculated numerically by the discretized evolution operator Up to now, we have given two equivalent methods to calculate the quasienergy spectrum by the Floquet Hamiltonian in the differential form and the effective Hamiltonian in the integral form, respectively. Even though they are comparing different in its forms, we should be able to find the unitary transformation to prove the isomorphic equivalence between their spectrum. For this reason, let us return the stroboscopic evolution operator as given in (S16) and substitute it into (S14), we obtain (S22)
(S29) When = 1, the system is topologically nontrivial, which has a topological edge mode while it is trivial for = 0.
However, for time-dependent systems, the invariant developed in the static system do not uniquely determine the number of chiral edge modes within each bulk bandgap [43]. Consider the periodically driven SSH model (Fig. 1d), where ( ) = 1 cos ( ) and = 2 / is the driving period.
A ℤ-valued bulk gap invariant G ϵ [ ] ∈ ℤ defined for the chiral gaps ϵ = 0 or can work in the driven system with chiral symmetry [36]. Note that the chiral symmetry has the constraint on the periodized evolution operator which is diagonal in the chiral basis.
( /2, ) = ( The chiral invariant for -gap is defined by We can see the chiral invariants as presented in Fig. 1f of the main text, convincing that the 0 mode exists in the high frequency region /Δ > 1/2 correspondingly with the nontrivial 0-gap invariant 0 = 1 while the mode exists in the intermediate frequency region 1/3 < /Δ < 1 with the nontrivial -gap invariant = 1, respectively.