Ученые обнаружили квантовые эффекты времени-кристалла в графене под воздействием лазерных импульсов

Abstract The Floquet formalism, developed for systems driven periodically in time, is applied to graphene, including the Rashba spin–orbit and Zeeman terms. Employing the high-frequency approximation, we studied Floquet-induced reconstruction of the Dirac spectrum, Poincaré recurrences, and the associated single-particle recurrences, as well as Lissajous curves. We observed single-particle Floquet subharmonic resonances, closed stroboscopic trajectories, and a Floquet-induced topological Lifshitz-like reconstruction of the low-energy spectrum, including anisotropic Dirac-cone deformation and Dirac-cone merging. Our main focus is on periodic time-driving via femtosecond half-cycle laser pulses. We also analyze the quantum metric—the distance between the initial and evolved wave functions—in terms of the participation rates of different bands and commensurate (non-commensurate) properties of the Floquet eigenvalues. In the weak-driving regime, we observe open curves, indicating non-periodic dynamics in the system. Within the considered single-particle Floquet framework, we identify conditions under which the stroboscopic dynamics is dominated by commensurate Floquet phases, leading to quantum Poincaré recurrences, closed Lissajous trajectories, and subharmonic spin oscillations. These effects are interpreted as Floquet time-crystal-like precursors in the sense of single-particle subharmonic dynamics. Within the presented single-particle Floquet framework, Poincaré recurrences and Lissajous resonances provide criteria for closed stroboscopic trajectories and subharmonic Floquet resonances. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction During the past decades, the periodically driven quantum systems have emerged as a benchmark of quantum chaos investigations. More recently, quantum materials driven out of equilibrium periodically in time have become a special focus of interest due to the realization of nonequilibrium band structures, i.e. Floquet-engineered states, which are not accessible in equilibrium situations [1–7]. Particularly important milestones have been reached with the realization of elusive topological insulator states in the driven graphene, which are extremely difficult to access otherwise [2, 6]. Floquet engineering in graphene has opened a new avenue for realizing light-induced topological phases and topological band structures. The state of a driven system with the absent discrete translational symmetry in the time domain has been called the time crystal [8]. Originally introduced by Wilczek, the time crystals are interpreted as a new phase of matter in which particles oscillate in a periodic pattern in time, akin to atoms in a spatial ordinary crystal lattice structure. Ambiguous mathematical definitions of the time crystals concept in those early days inspired a plethora of discussions that followed Wilczek’s work. The technical formalism of the Floquet Time Crystals has been formulated in two main equivalent forms [9], which will be explained in the text below. In this paper, we focus on periodic driving in a graphene-inspired model with intrinsic Rashba-type spin–orbit coupling. Floquet engineering in graphene and related Dirac materials has been widely studied in the context of light-induced topological phases [1, 2, 10, 11], dynamical renormalization of band parameters such as hopping amplitudes, bandwidths, and Dirac velocities, as well as the engineering of band-touching points [12, 13]. By tuning the driving parameters, one can access the regime of Floquet-induced topological transitions, in which pairs of Dirac cones carrying opposite topological charge are created or annihilated [10]. These transitions exhibit strong similarities to Lifshitz transitions induced by strain in static systems [14–18]. Closely related physics has been established for continuously driven (‘dressed’) electron systems. Pervishko et al [7] demonstrated that a strong off-resonant electromagnetic field renormalizes the Rashba spin–orbit coupling in a two-dimensional electron gas, providing all-optical control of spin dynamics and spin relaxation. Kozin et al [11] extended the dressed-state picture to a quantum ring with Rashba coupling in the regime of strong light–matter interaction, where the field reshapes the spin-split level structure and the associated persistent currents. Most relevant to the spectral part of the present work, Iorsh et al [18] predicted an optically induced Lifshitz transition in bilayer graphene, in which an off-resonant dressing field merges and annihilates low-energy band-touching points and thereby changes the Fermi-surface topology. These studies establish that high-frequency driving can renormalize spin–orbit-coupled band structures and even alter their connectivity. The present work addresses a complementary question that has not been considered in references [7, 11, 18]: instead of the properties of the dressed (effective) band structure alone, we study the stroboscopic dynamics generated by a kicked, half-cycle-pulse drive in monolayer graphene with both Rashba and Zeeman terms. Specifically, we ask when the Floquet phases of the reconstructed spectrum become commensurate, and we use quantum Poincaré recurrences and Lissajous resonances as quantitative diagnostics of the resulting closed trajectories and subharmonic response. The Dirac-cone deformation and merging that we find (section 4) can thus be viewed as a kicked, monolayer, spin–orbit-coupled counterpart of the dressing-induced band reconstruction of references [7, 11, 18], while the recurrence-Lissajous analysis have not been studied before. The surge of interest in topological materials and topological band structures is one of the most exciting developments in modern physics widely discussed in the [19–22]. Band structures in topological matter [23–27] play a central role for several reasons, e.g. robustness: protected states survive disorder and defects, which is ideal for dissipationless electronics [28–34]. Majorana zero modes store quantum information nonlocally, making them immune to local decoherence and an attractive platform for quantum computing [35–40]. Altermagnets offer spin-polarized currents with zero stray field [41–44]. There are several objectives of the present work. Our main interest lies in the reconstruction and modification of the system’s spectrum under a Floquet drive and subsequent single-particle Floquet subharmonic resonances, leading to time-crystal-like precursors. In particular, we explore Lissajous dynamics and the Quantum Poincaré Recurrences [45–48]. We also present results about the density of states, Berry curvature, frequency maps, Lifshitz transitions, and topology. We note that these results are essential to the present work. They clarify the features of the Floquet-engineered spectrum and are necessary for understanding the min findings. 2. Time crystallization The concept of time crystals is widely discussed in the literature [9, 49–54]) and here we refer only to the main facts. The symmetry of the Hamiltonian and the symmetry of the system’s state can, in general, be different. The spontaneous transition of a system to a state of lower symmetry than the Hamiltonian’s symmetry (i.e. spontaneous symmetry breaking) is well known in physics and, for instance, is the cornerstone of second-order phase transitions and electroweak interaction theories. On the other hand, this also applies to lattice crystal structures in solid-state physics, due to broken continuous translational symmetry in space, and to time crystals, due to broken continuous translational symmetry in time. Formation of a crystal lattice implies breaking of continuous spatial translation symmetry. An analogous idea applies to crystallization in the time domain. However, in the case of a Floquet system, the time crystallization is associated only with the breaking of the discrete time translational symmetry DTTS (i.e. TTSB). Let us consider a periodically driven quantum system with the Hamiltonian invariant under the time-translation operation and , meaning that discrete translation symmetry in the time domain holds. However, the system’s state may settle into a stable non-equilibrium pattern that does not fully preserve the symmetry of the driving force. A time crystal is a non-equilibrium phase of matter that exists in a periodically driven system, but spontaneously breaks the discrete time-translation symmetry (TTS). This means that the system’s state or quantum observable has a larger period (with integer ); this phenomenon is termed a subharmonic response. Similar phenomena may occur in a non-interacting system, which we refer to, in what follows, as single-particle Floquet time-crystal-like behavior and as precursors of time crystallization. In essence, we aim to explore subharmonic Floquet resonances that may occur in graphene driven periodically by half-cycle pulses. In the present work, we employ two concepts from dynamical systems theory and statistical physics: quantum Poincaré recurrences and Lissajous resonances. Quantum Poincaré recurrence refers to the tendency of a system to return to its initial state after a sufficient evolution time. However, quantum Poincaré recurrences are necessary but not a sufficient criterion for time crystalization and subharmonic response, since recurrence times may form a nonperiodic sequence. In what follows, we prove that the sufficient criteria for crystallization and subharmonic response are Lissajous resonances between two graphene modes with high participation rates in the dynamics. Historically, the concept of Floquet time crystals emerged in two equivalent forms [9]: Definition I: Let us assume that the time evolution is described by a time-dependent Hamiltonian , with a discrete-time translation symmetry such that [9]: for some . The TTSB occurs if for each , and for every state with short-ranged correlations, there exists a local operator such that where the time-evolved state is linked to the initial state via the Floquet operator , A quantum state has a short range correlations if for any local operator the following relation holds: for . Definition II: The TTSB occurs if the eigenstates of the Floquet operator cannot be short-range correlated. Basically, both definitions of TTSB mean that the periodicity of the wave function and expectation value of the local quantum operator are not the same. In its current version (cf e.g. [55] and references therein), the TTSB is being formulated in a broader context for both classical and quantum systems, and not necessarily for local operators only. An illustrative example is the anharmonic oscillator: The trajectories governed by classical Hamiltonian, equation (5), can be described via where , are slowly varying canonical variables. For more rigorous mathematical discussion about the meaning of , in the context of the Kolmogorov–Arnold–Moser theorem, we refer to works [56–58]. The convergence of the Magnus expansions provides a hint toward the distinction between the true discrete-time crystal and the pre-thermal discrete-time crystal: In the case of the true discrete time crystals, , while the nonzero indicates on the exponentially small residual time dependence in the case of pre-thermal discrete time crystals. In both cases, represents the effective Hamiltonian that captures the system’s stroboscopic dynamics for exponentially large time scales. 3. The model In the following, we apply the general concepts described above to graphene driven periodically in time by electromagnetic half-cycle pulses [59]. Such pulses can be described theoretically by a delta function in time. For in-plane pulses, the vector potential component can be written as , where fs is the full-width at half-maximum of the half-cycle pulse, and V m is the peak electric field. Then the graphene-light interaction term takes the form where , integer number counts the following periodical kicks, and is the time-period between the kicks. The full Hamiltonian for the Dirac point in graphene takes the following form: where , m s is the electronic Fermi velocity in graphene, is an external magnetic field, and , while represent matrices acting in sublattice space [60], is a two dimensional identity matrix, while refer to the usual Pauli spin operators, is the Dirac kinetic term, is the Rashba spin–orbit interaction, and is the Zeeman coupling. The time-independent part of the Hamiltonian explicitly reads: where , . We note that the Rashba SO term can is induced for example by a transverse electric field, a gate-controlled asymmetric environment, or proximity to a heavy-SOC substrate [61]. The band structure of graphene described by the unperturbed Hamiltonian in equation (11), is plotted in figure 1. In what follows, we demonstrate that nontrivial dynamical effects are associated with the in-plane component of the external magnetic field. For convenience, we express hereafter the dimensional time via the spin–orbit coupling parameter , where , (THz), (meV). Then, in the high-frequency limit, (fs), , one can obtain approximate analytical results for the Floquet operator. Considering the realistic parameter set (meV), (m s), (C), (V m), (fs), we realise that the time scale of the system is of the order of (THz), while . All results presented below are relevant to the prethermal Floquet regime, where the effective Hamiltonian description is valid for exponentially long times. In particular, heating time for exponentially slow heating in a periodically driven system can be estimated as [62, 63] where . In what follows, we are interested in the high-frequency approximation and the prethermal time is exponentially large. Hence, we restrict ourselves to the regime where the high-frequency effective Hamiltonian is expected to provide a reliable description over the time window considered. Therefore, the present single-particle model does not include thermalization or heating mechanisms. An additional degree of freedom intrinsic to graphene is the valley index associated with the nonequivalent and Dirac points. This has led to the development of valleytronics, where information is encoded in the valley degree of freedom rather than charge of spin. The time-reversal symmetry (-symmetry) in graphene can be presented by the equation where and refer to the Hamiltonians of electrons in the vicinity of and points. These Hamiltonians are related by a unitary transformation [64] where . Using (13) and (14), we can find the symmetry condition for the Hamiltonian in a single Dirac point (e.g. the point), Finally, the -symmetry condition for the Dirac point can be presented as where the anti-unitary operator of time reversal symmetry takes the form and involves rotations in both sublattice and spin spaces, and also the complex conjugation operator . Obviously, the Zeeman term in equation (11) breaks the -symmetry, which can be “restored” by simultaneous reversal of magnetic field, . In accordance with time-reversion symmetry, the Kramers theorem holds in the following form: the energy of an electron in state (, , ) is equal to the energy of electron in the state (, , ), where vector refers to the same Dirac valley. We also recall the particle-hole symmetry (-symmetry), to be used in the following as well. with . As we note from (11), the magnetic field breaks condition (17), but the -symmetry holds with the simultaneous reversal of the magnetic field. The parity symmetry is related to the space inversion where . This symmetry is also broken by the magnetic field in equation (11). Equation (14) shows that the Hamiltonians for and Dirac points are related by the unitary transformation , which indicates that the energy spectra of both Hamiltonians are the same. In turn, the eigenfunctions and of these Hamiltonians, corresponding to the same eigenvalue , are related as Considering a quantum operator , it is easy to note that the matrix elements calculated for and wave functions are related via the equation where In particular, for and the unitary operator , one finds that , which means that calculations of the observable in the valley is equivalent to the calculations of in the valley . The symmetry relations are summarized in table 1. Table 1. Summary of the symmetry relations introduced in equations (13)–(19). Here and are Pauli matrices acting in sublattice and spin spaces, respectively, is the identity matrix, and denotes complex conjugation. The valley-unitary transformation , the antiunitary time-reversal operator , the particle-hole operator , and the inversion operator [64]. We note that the above argument also applies to the Floquet dynamics. Taking into account thedefinition of the Floquet operator for kick duration : we consider the spectral decomposition Here , with . Last equality holds in our particular case and not generically, since and both commute. As and using spectral decomposition of the operator we obtain: 4. Periodic driving in time We proceed with the wave equation () governed by the time periodic Hamiltonian defined in equation (9). The solution of equation (26) can be presented in the form: Here and are the eigenstates and eigenvalues of the Floquet operator . In most cases, the initial state of the system coincides with the ground state of the unperturbed , but not necessarily. We consider the unitary transformation [5]: The Schrödinger equation for the transformed wave function reads where We note that an infinite series of delta kicks is equivalent to the harmonic polychromatic pumping, well-known from the theory of nonlinear resonance where . As one can see, equation (31) contains multiple high-frequency driving terms. Exploiting this fact, we adopt the high frequency () expansion: To evaluate the expansion terms in equation (32) we implement the Baker–Campbell–Hausdorff formula [5] and obtain: Eventually, the unitary evolution operator takes the form: We note that in the high-frequency limit and stroboscopic driving case, the effective Hamiltonian depends only on the time-independent part of the Hamiltonian and the operator part of the driving term , and is time-independent. The effective Hamiltonian can be found analytically for particular cases of and . The band structure of can be viewed as a band structure of the Floquet state. However, this statement is based on the assumption that a driven system does not relax to some equilibrium state or does not recover the initial state, i.e. the absence of the Quantum Poincaré Recurrences is required per se. In other words, the absence of the Quantum Poincaré Recurrences excludes the chance that the discrete TTSB holds. Due to the particular interest, we explore this problem for a single-layer graphene exposed to the stroboscopic driving in time as well as to an external magnetic field. Taking into account equations (11) and (33), the effective Hamiltonian takes the form: where we use the notations , , , , , and . The special case , for which the spectrum of equation (35) is independent of -dependence, can be achieved for the peak electric field magnitude (V m). Both equations (33) and (35) can be readily diagonalized using conventional tools of computer-based symbolic algebra. The resulting expressions are shown in the appendix being the corresponding eigenstate of the Hamiltonian , which carries a topological charge (also called the Chern number), extractable via an integration along any contour fully enveloping the spectral structure The evaluation of the Chern number for -point proceeds as follows. First, we note that the contour integral in equation (37) can be transformed into an area integral using Stokes’ theorem: where is the antisymmetric tensor in 2D. The resulting nonzero value is a solid-state analog of the parity anomaly in 2+1-dimensional quantum electrodynamics. It arises from the chirality of the Dirac particle together with the chosen regularization scheme. From the seminal work of Kane and Mele [68], it is known that spin–orbit coupling does not affect the topology of a single Dirac cone. Therefore, we may restrict ourselves to the case -case, without loss of generality. We begin with the case , for which the regularized Hamiltonian takes the form the regulator will ultimately be taken to zero at the end of the calculation. The eigenvalues are We restrict the consideration to the lower positive-energy Using the notation , the components of the Berry potential are calculated as which then gives for the curl It is rotationally invariant, the angular integral can be evaluated directly, giving for finite and radius of the integration circle Therefore, the contribution of the band yields in the limit the well-known half-integer Chern number corresponding to the parity anomaly of a regularized two-dimensional Dirac cone. Including the contribution from the second band gives the integer value for . For , the Dirac-cone structure is preserved and acts only as an anisotropic renormalization of the Fermi velocity with , and . The Berry curvature is therefore rescaled, but its integral over momentum space remains unchanged, leaving the topological invariant intact. At , however, the Chern number changes to . In this limit, the , making both and identically zero throughout momentum space. The evaluation for the point proceeds analogously. Owing to its opposite chirality, it contributes a Chern number with the opposite sign. Download figure: Standard image High-resolution image We note that the calculation around equations (36)–(46) is valley-resolved. The and valleys carry opposite chirality; therefore, for the valley-symmetric regularization used here, their local Dirac charges cancel in the total Chern number: The meaningful quantity in this model is therefore a valley-resolved charge, or equivalently a valley Chern diagnostic, rather than a nonzero total Hall Chern number. These different regimes can be tracked in the measurable density of states (DOS) of the Hamiltonian (35) To evaluate the functional trace we employ the Feynman identity which leads to Expanding the Green’s function appearing under the trace operator into the eigenbasis of the effective Hamiltonian yields where the trace operator Tr acts only on the matrix space. The results of the evaluations for different parametric regimes are presented in figures 3 and 4. The isoenergy contour plots are shown in figure 5. All plots are assembled from the three main elements featuring different spectral parts in 2d, cf appendix Download figure: Standard image High-resolution image Download figure: Standard image High-resolution image Download figure: Standard image High-resolution image 5. A quantum Poincaré recurrences The studies of the quantum Poincaré recurrences for the Chirikov standard map in the semiclassical regime with mixed phase space showed that the probability of Poincaré recurrences or the survival probability follows the classical until the Heisenberg time (i.e. the time scale at which the discreteness of its energy spectrum becomes important , where is the mean level spacing) [47, 48]. For times longer than the Heisenberg time , the survival probability decays as a power law, typically It is important to distinguish between commensurate and non-commensurate cases. Eigenvalues are commensurate if there exists a set of integers such that Consequently, the Eigenvalues are non-commensurate if for any . In a commensurate case, the quantum system evolves quasi-periodically, even beyond the Heisenberg time. However, since decays polynomially, returning close to the initial state requires interference among the four graphene bands. Recurrences may still appear after the time periods larger than the driving period , , . However, until recurrences appear periodically, we still have TTSB and precursors of time crystallization. On the other hand, in an incommensurate case, the evolution is not periodic, meaning that , TTSB and the effect akin to the time crystallization do not hold. Remember that the Heisenberg time is the essence of the inverse level spacing of the system. While the band structure of graphene is not akin to the Chirikov standard map, and contrary to the Chirikov standard map, we do not define the phase space of graphene; the idea of recovering the initial state is more general and applicable to graphene as well. To infer a Quantum Poincaré Recurrence, we introduce the metric distance between the time-evolved wave function and the initial state, . Quantum Poincaré Recurrence means that for any small one can find a time such that . The explicit expression for the distance in terms of the Floquet eigenstates reads: Here and are Floquet eigenstates and eigenvalues defined in equation (27). 6. An exactly solvable case The Floquet operator has been defined above as with the Boltzmann constant put to the unity . Importantly, since the Floquet operator is unitary, its eigenvalues are complex, suggesting oscillating solutions. The Floquet matrix can be expressed in terms of conventional matrices using infinite expansions. The first exponent is expanded into the simple series Using the above definition of the series is converted into The two obvious extremal regimes are at , , where the above operator reduces to the unity matrix. This seemingly trivial regime can, in fact, reflect a very subtle driving protocol, in which the time evolution of an initial state is not affected by a time-dependent potential. Second extreme case is achieved if , , where and the above operator becomes a unitary matrix , which interchanges the Dirac cones of the initial Hamiltonian at each single kick. To represent the second exponent, we can rewrite the Hamiltonian as where is a diagonal matrix with explicit form with being the eigenvalues of and the unitary matrix, diagonalizing the Hamiltonian with . The exponent can be expanded as This expression serves as an equivalent representation of the conventional spectral decomposition of the operator . Hence, the Floquet operator is brought into the form of a matrix The case with no spin–orbit coupling ( in equation (11)) is simple enough to treat it entirely analytically. The eigenvalues of the Hamiltonian appear pairwise due to particle-hole symmetry of the Hamiltonian and read with where the symbol is used as shorthand for . With the notation , the matrix then explicitly reads With that we get the following explicit Floquet matrix (, , , ): The matrix has the same block-diagonal structure as the initial Hamiltonian with . This has the practical consequence that with any given eigenstate of with , only two eigenstates of the Floquet operator have a finite overlap which considerably simplifies the evaluation of equation (27). Each of the 22 blocks, recognizable in equation (69), can be diagonalized separately, and the corresponding eigenstates determined. However, the resulting expressions are too lengthy to present here. On the other hand, we can proceed directly, without making the effort of diagonalizing , applying it subsequent times on a given initial state . Choosing as the initial state the fourth column of the matrix we directly obtain The dependence on the period length appears only in the global phase and couples unsurprisingly to the eigenvalue of the initial eigenstate equation (71). For the overlap with the initial state, we get We conclude that the mapping of the initial state upon a multiple of itself after the very first iteration (bar the phase) is possible with and , i.e. , , or at certain special points in the Brillouin zone, where . The first condition corresponds to the trivial identity evolution already evident from equation (66), whereas the second arises from the graphene-specific structure of the Floquet operator. These considerations can be applied to any arbitrary number of kicks . The state equation (72) is naturally normalized, which gives for the Quantum Poincaré recurrence in the sense of equation (59) the value which again varies across the Brillouin zone. For instance, at the manifolds within the Brillouin zone where , while at . For larger , the corresponding expressions for become more involved. Nevertheless, the recurrence structure can be evaluated explicitly for selected parameter sets. Taking and considering two special directions in momentum space, one obtains the following limiting cases: 1. For , This is the manifold within the Brillouin zone at which the system returns into itself after each single driving kick. 2. For , Equation (75) identifies a special momentum-space manifold on which the stroboscopic state is reconstructed after each driving period, up to an irrelevant overall phase. Thus, this case represents an exact one-period quantum revival in the single-particle Floquet dynamics. It is useful as a benchmark for the recurrence analysis, but it should not be interpreted as time-translation-symmetry breaking, since the recurrence period coincides with the period of the external drive. The situation described by equation (76) is different. Here the state does not return to the initial state after one kick, whereas it is reconstructed after two kicks. In this restricted analytic limit, the participating Floquet phases are commensurate and the stroboscopic recurrence time is therefore doubled with respect to the driving period. This gives a simple example of a period-doubled single-particle Floquet resonance. We emphasize, however, that equations (75) and (76) do not establish a genuine Floquet time-crystalline phase. The present calculation is performed within a non-interacting single-particle framework and the exact recurrences occur only on special momentum-space manifolds and for a tuned driving protocol. A many-body discrete time crystal would require a robust subharmonic response that persists against perturbations, disorder, interactions, and finite temperature effects. These ingredients are outside the scope of the present model. Therefore, the results obtained here should be regarded as analytically transparent signatures of commensurate Floquet dynamics and as single-particle precursors of time-crystal-like behavior, rather than as evidence for a stable many-body time-crystalline phase. This distinction is also important for the role of quantum Poincaré recurrences. A recurrence of the state is a useful diagnostic of commensurate Floquet phases, but it is not by itself sufficient to prove discrete time-translation-symmetry breaking. In the present work it is used only to identify parameter regimes in which the stroboscopic dynamics becomes closed. Whether closed dynamics really leads to the subharmonic becomes clear from the following analyzes: Let us expand the state in Floquet eigenstates with quasi-energies : with being -periodic. Observables therefore depend only on quasi-energy differences. A subharmonic response of at requires the state to have weight on at least one pair with We restrict to a single populated doublet relevant to our work: . Each amplitude acquires per period: Therefore: - After one period (): the relative phase between the cat components flips by . This is a different ray in Hilbert space. The wave function is not -periodic, not even up to a global phase. - After two periods (): the relative phase returns, , so the state returns to itself up to the overall phase . Projectively, the density matrix Thus in the ideal DTC: -periodicity of a single observable does not imply -periodicity of the state. Our statement is the converse: Thus, subharmonic response, e.g. at requires revivals of the state i.e. Poincaré recurrence is necessary for a closed (finite) dynamics. However, Poincaré recurrences are not sufficient, since they may occur non-periodically. The sufficient condition Lissajous resonance which defines the period of subharmonic response, meaning that: (a) commensurate Floquet phases can produce exact or approximate single-particle revivals; (b) if two Floquet modes dominate the initial state and their phase ratio is rational, the corresponding parametric trajectory is closed, (c) this closed trajectory gives a single-particle subharmonic Floquet resonance. 7. Time dynamics of quantum observable For exploring the TTSB, we compare the periodicity of a quantum observable, e.g. the z-component of the spin, which has a matrix form We define the time-dependent average of that operator as and compare the periodicity of with the periodicity of the Hamiltonian . In figure 6 we plot the time dependence of the Poincaré metric distance and the spin expectation value for the different sets of parameters for and points. As we see in the absence of magnetic field both spin and Poincare metric distance dynamics plotted for the and points, shown in panels (a) and (b), and in panels (e) and (f) in figure 6, consequently, manifests periodic behavior and the period substantially exceeds the driving period of the Hamiltonian. The observed effect is a signature of subharmonic Floquet resonances, akin to Floquet time–crystal–like behavior for lattice operators; cf equation (4). It is rather apparent that the TTS is broken. In order to explore manifestation of TTSB we proceed to the inequality where , for all integer , with minimal integer , and propagated wave function The inequality equation (84) shows that the observable is not simply periodic with the driving period. However, by itself this does not establish time-translation-symmetry breaking. In the present single-particle setting, we diagnose subharmonic response by checking whether the stroboscopic observable satisfies with minimal integer . We refer to such behavior as a single-particle Floquet time-crystal-like precursor. In the case of applied in-plane external magnetic field, shown in panels (c) and (d) in figure 6, the spin dynamics is not periodic at all. We do not observe a single Poincaré recurrence within the given time interval. Hypothetically, Poincaré recurrence may occur on a much larger time scale. However, we note that our discussion is limited to the prethermal time scale, and we cannot drastically exceed , see equation (12). Time dependence of and Poincaré’s metric distances manifest complex character. It contains different frequencies that depend continuously on the wave vector components . We infer those frequencies via the Fourier transform of the signal. In the absence of a magnetic field, the first Poincaré’s recurrence occurs in the system after periods of driving and . We do not observe a recurrence even at a larger time interval when the magnetic field is applied, cf panel (d) in figure 6. Download figure: Standard image High-resolution image The present calculation does not establish a many-body discrete time-crystalline phase. It identifies parameter regimes in which the single-particle Floquet phases become commensurate, leading to subharmonic spin oscillations and closed stroboscopic trajectories. Robust time-crystalline order would require a thermodynamic many-body setting and stability against perturbations, which are outside the scope of the present work. From the spectrum of the time-dependent signal , we select the dominant frequency and study its dependence on the momentum for different polarizations of the light. The results are plotted in figure 7. We observe a rather complex and asymmetric dependence of the leading frequency on the momentum components and on the light polarization: (a) when light polarization is aligned along the axis, the maximal frequency is shifted to the region of zero where we see the red high frequency island, and we see finite broadening high frequency ellipse and small islands corresponding to the finite wave vectors. If the light polarization is along the axis case (b), the maximal frequency is shifted to a nonzero . The dominant frequency map becomes more diverse when a magnetic field is applied figure 8. We proceed to the spectral analysis of the metric distance signal . The dominant frequency of the metric distance as a function of is plotted in figure 9. As we see, the structure of the leading frequency map is inverted as compared to the signal. Namely, higher-frequency regions now form islands surrounded by low-frequency rings (a) and (b) cases in figure 9. In the case of strong pulses (c), the leading frequency depends only on , and the shape is converted into stripes. Download figure: Standard image High-resolution image Download figure: Standard image High-resolution image Download figure: Standard image High-resolution image 8. The Lissajous curves Quantum Poincaré recurrence provides a useful diagnostic of exact or approximate state revival. In the present single-particle model, commensurate Floquet phases can produce both state recurrences and subharmonic observable dynamics. The single-particle Floquet time-crystal–like behavior and sub-harmonic response require periodicity in the dynamics of quantum observables with period greater than the driving period ( is an integer here). On the other hand, the quantum Poincaré recurrence is the essence of recovering the initial state and may occur non-periodically. Non-periodic recovery of the initial state or a quantum observable excludes the time-crystal–like behavior. To establish the sufficient conditions for subharmonic Floquet resonances and TTSB, we introduce quantum Lissajous resonances. We note that Lissajous resonances occur in a system of two non-interacting oscillators when the ratio of their characteristic frequencies is an integer. Concerning our problem, it means that Lissajous resonances may occur if only two bands are involved in the dynamics, and participation rates of the other two bands should be marginal. Besides, the ratio between the frequencies of participating bands should be an integer. Let us discuss this question in more detail. Suppose that the frequencies of dominant bands satisfy the relation where integers and are coprime, i.e. the only positive integer that is a divisor of both of them is 1 (e.g. and ). Then the trajectory is closed, the motion is periodic, and we have a unique period of the quantum matrix distance given by Consequently, is the lattice constant of time crystallization, and it exceeds the driving period . The key point is that in general, frequencies of dominant participating in dynamics bands depend on the wave vector . In the general case, dynamics is rather complex, and we proceed with the qualitative analysis. For further insights, we explore the quantum metric equation (59) and calculate Floquet phases , and participation rates as a function of magnetic field and wave vector . We target weak and strong electric fields of half-cycle pulses V m and V m, leading to the and cases, respectively. Our analysis shows the dominance of two participation rates in Equation (59) for . In the case of a strong electric field , the problem simplifies to one leading band Consequently quantum Poincaré recurrences occur periodically at , , where . In the case of a weak electric field two bands dominate in dynamics, and therefore we have where , . When two Floquet modes dominate the dynamics, rational ratios of the relevant Floquet phase differences produce closed Lissajous trajectories. In this restricted two-mode regime, such resonances provide a sufficient condition for closed stroboscopic trajectories and subharmonic oscillations. It is easy to see that the time-dependent dynamics in equation (88) is governed by Lagrangian [69] and The dynamics governed by equation (90) is constrained by the participation rates via the conditions . Only in very rare cases are the Lissajous curves closed, meaning that motion is periodic. Namely, in the case , the phase trajectory is an ellipse, for , the phase trajectory converges to the parabola, and the Tchebyshev polynomials for , . Already for , motion is not periodic and quantum Poincaré recurrences do not occur. After each loop phase, the incursion is of the order of . In figure 10, we plot frequencies of the dominant bands as a function of wave vector , . As we see, in certain cases we achieve the coprime condition that grants time crystallization, periodicity, and closed trajectories, as shown in figure 11. Download figure: Standard image High-resolution image Download figure: Standard image High-resolution image We note that in our formulation, closed and open Lissajous trajectories correspond to commensurate and incommensurate sets of dominant Floquet phase differences and are experimentally feasible. Namely, for tr-ARPES, the closed-trajectory regime should show quasiperiodic but repeatable stroboscopic population oscillations at selected momenta, together with sharper Floquet sideband features within the prethermal time window. In the open-trajectory regime, the same momentum-resolved population signal should show a broader, incommensurate beat spectrum and no exact recurrence over the accessible time interval. For time-resolved optical conductivity, closed trajectories should produce a small number of sharp subharmonic or beat-frequency peaks in the pump-probe response. Open trajectories should instead lead to broadened or multi-frequency spectra. We stress that the Lissajous curve is not directly measured as a geometric curve; it is reconstructed from synchronized time traces of experimentally accessible observables. 9. Analysis of closed Lissajous trajectories We now derive the conditions under which the Floquet dynamics supports closed Lissajous trajectories. The resulting commensurability map identifies regions of momentum space where two Floquet modes dominate the dynamics and their quasienergy phases satisfy a low-order rational resonance. These regions are determined by the two conditions: where is chosen close to unity and is a low-order rational. This construction identifies the regions where the stroboscopic dynamics is dominated by a closed or nearly closed Lissajous trajectory. The obtained exact expression for the Floquet operator, equation (69) provides an ideal framework for the analytical determination of Lissajous resonances for the minimal Dirac Hamiltonian. The block-diagonal structure of the Floquet operator in equation (69) implies that the positive- and negative-energy sectors evolve independently. Consequently, the dynamics can be analyzed within a single block without any loss of generality. Projecting the full Floquet operator equation (69) on positive energy band reads where , , , . Correspondingly, the trial eigenvector defined in equation (71) projects to a 2-dim vector The determinant of the Floquet operator equation (92) hence its eigenvalues are The sum of these eigenvalues which, on another hand, is the trace of the operator Hence the phase is and the phases in the eigenvalues of the Floquet operators The magnetic field merely shifts all quasienergies by . The properly normalized eigenvectors of the reduced Floquet operator read with the normalization Remarkably, the participation ratios are completely independent of the magnetic field . Crucial observation for getting the normalization in this form is which is the SU(2) property of the Floquet operator. The participation ratios are defined as Technical but direct computations yield As argued around equation (91), two remaining participation rates are zero. The commensurability condition is fulfilled, if for some integer and we have Actually, the right hand side should be , where is an integer or zero, but we restrict to zero only, which is sufficient for our goals. This directly yields If condition is fulfilled, when the Lissajous curve closes. In figure 12 the momentum-space resonance manifolds supporting closed Lissajous trajectories for several values of the driving strength at a particular magnetic field are plotted. Download figure: Standard image High-resolution image Combining it with equations (98) and (99) reveals that at resonance the two quasienergy phases themselves are indeed locked into the rational ratio The integers and classify the topology of the closed Floquet trajectories. Different rational ratios generate distinct resonance branches, producing a hierarchical family of closed Lissajous curves in parameter space. Equation (106) contains five interdependent quantities: Momentum and related angle, driving strength and magnetic field . Both sides considered separately represent oscillating functions varying between -1 and +1, and their crossing points correspond to closed Lissajous resonances. In equations (75) and (76) we identified two special points within the Brillouin zone at which the system exhibits certain periodic patterns. 1. For the solution equation (75) ( and ) the commensurability condition reduces to which selects the momenta in the Brillouin zone supporting closed Lissajous resonances 2. The Dirac point is the only point in the Brillouin zone, where the driving force is related to the strength of the magnetic field directly i.e. on the principal branch. 3. For the solution equation (76) ( and ) the commensurability condition reduces to an even simpler equation According to equation (98), this unconditionally implies that is, the Lissajous curves are closed irrespectively of the strength of the magnetic field. The commensurability condition does however require a perfect quantization of the magnetic field to where is zero or integer. On the principal branch it reduces to At this special point the commensurability condition becomes independent of crystal momentum. Consequently, the existence of closed Lissajous trajectories is controlled solely by the magnetic field. All three special cases discussed above correspond to high-symmetry points of the Brillouin zone, where the general commensurability condition simplifies considerably. In these special situations the resonance condition reduces to linear relations between the magnetic field and either the crystal momentum or the driving strength, giving rise to the fan diagrams, as the one shown in figure 13. In the generic case, however, the commensurability condition (106) remains intrinsically nonlinear, and no simple linear resonance branches are expected. Download figure: Standard image High-resolution image 10. Discussion and conclusions The main purpose of this work is to apply the Floquet formalism to periodically driven graphene as a representative solid-state system. In atomic and mesoscopic physics, well-established Floquet approaches already exist and provide a controlled framework for analyzing driven quantum systems [54]. In the present work, we employ the high-frequency approximation to investigate the corresponding Floquet dynamics in graphene. It enables circumventing the formidable task of dealing with infinite expansions while retaining only a few terms with an acceptable loss of accuracy. By applying this formalism to the graphene Hamiltonian in the presence of a Zeeman magnetic field and Rashba spin–orbit coupling, we obtained an effective Hamiltonian that resembles the original one, though with some significant differences. The effect of driving is captured by renormalizing the Hamiltonian parameters, which in turn affect its spectrum. In particular, we observe Floquet-induced reconstruction of the Dirac spectrum, behavior similar to the Lifshitz transition in the sense of Dirac-cone reconstruction. Moreover, we have demonstrated that the full Floquet framework for graphene-like systems can, in principle, be developed without relying on uncontrolled approximations or major simplifications. For a specific choice of parameters, the original Hamiltonian becomes simple enough to be diagonalized in terms of elementary functions, enabling a closed-form treatment of the program, in particular, the determination of the Floquet time-evolution operator. The true power of exact solutions has been demonstrated by identifying special points within the Brillouin zone that return the system to the same state after a driving cycle. The technology of this approach is very transparent and can be easily generalized to more sophisticated Hamiltonians, at least numerically. This is the path we intend to follow in future projects. Generally, the recurrences take a substantially larger number of kicks. We focused on periodic time-driving via femtosecond half-cycle laser pulses. We studied the quantum metric, the distance between the initial and evolved wave functions, in terms of participation rates of different bands and commensurate (non-commensurate) properties of Floquet eigenvalues. In weak driving, we observed open Lissajous curves, indicating non-periodic dynamics in the system. We proved that in Floquet systems with complex band structure, one can clearly formulate the necessary and sufficient conditions for subharmonic Floquet resonances that resemble single-particle Floquet time-crystal–like behavior. Namely, we proved that the necessary criterion for subharmonic Floquet resonances is the quantum Poincaré recurrence in the system, while the sufficient criterion is the Lissajous resonance. Acknowledgments A S acknowledges funding by the Grant JSF-22-10-0001A of the Julian Schwinger Foundation for Physics Research. Data availability statement All data that support the findings of this study are included within the article (and any supplementary files). Appendix A: Energy of effective Hamiltonian For the Hamiltonian defined in equation (35) with we can find the eigenenergy in following form and if we take and then we can find For a nonzero magnetic field case , energy leaves read: In case of the strong half-cycle pulses V m and , we deduce: Appendix B: Basic DOS structures in 2d Here we evaluate analytically the main DOS structures listed in section 4 in two spatial dimensions. In the case of a linear spectrum (here, the symbol is used as shorthand for ) we get In the case of a parabolic spectrum the corresponding DOS structure comes out as Quite surprisingly, the DOS of a 2d saddle-like spectral structure can be understood as the Fourier transform of a Coulomb potential in 1d. To see this we write the delta function in its Fourier representation where we switched into the polar coordinates. The momentum integral can be carried out directly where is the first-kind Bessel function, which leaves us with the aforementioned Fourier transform of 1d Coulomb potential which reproduces the standard Fourier-transform relation for the one-dimensional Coulomb kernel, resulting in the logarithmic Van Hove singularity The effective Hamiltonian for deserves a special consideration. It reads for and has two pairs of eigenstates Inverting the matrix from equation (54) and taking trace gives where in the second line a partial fraction decomposition has been performed and . Separating the imaginary part we further get We evaluate the first term for the band Analogously, for the fourth term, we get Using the property of the —function we then conclude that Second pair gives the same result hence the total DOS gives a constant value Thus, the density of states is a constant in the whole range of energies. Appendix C: Effective Hamiltonian of graphene in Dirac point using the notation of the Hamiltonian (35).