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Critical properties of the Floquet Time Crystal

Muath Natsheh, Andrea Gambassi, Aditi Mitra

Published Mar 19, 2021• 0 citations
OpenAlex

Abstract

The critical properties characterizing the formation of the Floquet time crystal are investigated analytically in the periodically driven $O(N)$ model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and absence of long-range spatial order from the non-trivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for $N\to\infty$, we determine the exponent $\nu$ that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent $\beta$ characterizing the growth of the amplitude of the order-parameter, as well as the initial-slip exponent $\theta$ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents $\nu, \beta, \theta$ are found to be identical to those in the absence of a drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period-doubled, are also found to oscillate in space at the wave-vector $\omega/(2 v)$, $v$ being the velocity of the quasiparticles, and $\omega$ being the drive frequency.

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The critical properties characterizing the formation of the Floquet time crystal in the prethermal phase are investigated analytically in the periodically driven $O(N)$ model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and the absence of long-range spatial order from the nontrivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for $N\ensuremath{\rightarrow}\ensuremath{\infty}$, we determine the exponent $\ensuremath{\nu}$ that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent $\ensuremath{\beta}$ characterizing the growth of the amplitude of the order parameter, as well as the initial-slip exponent $\ensuremath{\theta}$ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents $\ensuremath{\nu},\ensuremath{\beta},\ensuremath{\theta}$ are found to be identical to those in the absence of the drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period doubled are also found to oscillate in space at the wave vector $\ensuremath{\omega}/(2v), v$ being the velocity of the quasiparticles, and $\ensuremath{\omega}$ being the drive frequency.

arXiv:2103.10818•Physical review. B./Physical review. B•Jun 28, 2021•18 citations

Critical properties of the Floquet time crystal within the Gaussian approximation

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arXiv:2101.04140•Journal of High Energy Physics•May 1, 2021•27 citations
The critical properties characterizing the formation of the Floquet time crystal are investigated analytically in the periodically driven O(N)O(N)O(N) model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and absence of long-range spatial order from the non-trivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for N→∞N\to\inftyN→∞, we determine the exponent ν\nuν that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent β\betaβ characterizing the growth of the amplitude of the order-parameter, as well as the initial-slip exponent θ\thetaθ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents ν,β,θ\nu, \beta, \thetaν,β,θ are found to be identical to those in the absence of a drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period-doubled, are also found to oscillate in space at the wave-vector ω/(2v)\omega/(2 v)ω/(2v), vvv being the velocity of the quasiparticles, and ω\omegaω being the drive frequency.
The critical properties characterizing the formation of the Floquet time crystal in the prethermal phase are investigated analytically in the periodically driven O(N)O(N)O(N) model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and the absence of long-range spatial order from the nontrivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for $N\ensuremath{\rightarrow}\ensuremath{\infty}$, we determine the exponent $\ensuremath{\nu}$ that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent $\ensuremath{\beta}$ characterizing the growth of the amplitude of the order parameter, as well as the initial-slip exponent $\ensuremath{\theta}$ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents $\ensuremath{\nu},\ensuremath{\beta},\ensuremath{\theta}$ are found to be identical to those in the absence of the drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period doubled are also found to oscillate in space at the wave vector $\ensuremath{\omega}/(2v), v$ being the velocity of the quasiparticles, and $\ensuremath{\omega}$ being the drive frequency.
The periodically driven O(N)O(N)O(N) model is studied near the critical line separating a disordered paramagnetic phase from a period doubled phase, the latter being an example of a Floquet time crystal. The time evolution of one-point and two-point correlation functions are obtained within the Gaussian approximation and perturbatively in the drive amplitude. The correlations are found to show not only period doubling, but also power-law decays at large spatial distances. These features are compared with the undriven O(N)O(N)O(N) model, within the Gaussian approximation, in the vicinity of the paramagnetic-ferromagnetic critical point. The algebraic decays in space are found to be qualitatively different in the driven and the undriven cases. In particular, the spatiotemporal order of the Floquet time crystal leads to position-momentum and momentum-momentum correlation functions which are more long-ranged in the driven than in the undriven model. The light-cone dynamics associated with the correlation functions is also qualitatively different as the critical line of the Floquet time crystal shows a light cone with two distinct velocities, with the ratio of these two velocities scaling as the square-root of the dimensionless drive amplitude. The Floquet unitary, which describes the time evolution due to a complete cycle of the drive, is constructed for modes with small momenta compared to the drive frequency, but having a generic relationship with the square-root of the drive amplitude. At intermediate momenta, which are large compared to the square-root of the drive amplitude, the Floquet unitary is found to simply rotate the modes. On the other hand, at momenta which are small compared to the square-root of the drive amplitude, the Floquet unitary is found to primarily squeeze the modes, to an extent which increases upon increasing the wavelength of the modes, with a power-law dependence on it.
We show that the correlation functions of a class of periodically driven integrable closed quantum systems approach their steady-state value as ${n}^{\ensuremath{-}(\ensuremath{\alpha}+1)/\ensuremath{\beta}}$, where nnn is the number of drive cycles and $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ denote positive integers. We find that, generically, $\ensuremath{\beta}=2$ within a dynamical phase characterized by a fixed $\ensuremath{\alpha}$; however, its value can change to $\ensuremath{\beta}=3$ or $\ensuremath{\beta}=4$ either at critical drive frequencies separating two dynamical phases or at special points within a phase. We show that such decays are realized in both driven Su-Schrieffer-Heeger (SSH) and one-dimensional (1D) transverse field Ising models, discuss the role of symmetries of the Floquet spectrum in determining $\ensuremath{\beta}$, and chart out the values of $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ realized in these models. We analyze the SSH model for a continuous drive protocol using a Floquet perturbation theory which provides analytical insight into the behavior of the correlation functions in terms of its Floquet Hamiltonian. This is supplemented by an exact numerical study of a similar behavior for the 1D Ising model driven by a square pulse protocol. For both models, we find a crossover timescale nc{n}_{c}nc​ which diverges at the transition. We also unravel a long-time oscillatory behavior of the correlators when the critical drive frequency, ${\ensuremath{\omega}}_{c}$, is approached from below $(\ensuremath{\omega}<{\ensuremath{\omega}}_{c})$. We tie such behavior to the presence of multiple stationary points in the Floquet spectrum of these models and provide an analytic expression for the time period of these oscillations.
Despite being forbidden in equilibrium, spontaneous breaking of time translation symmetry can occur in periodically driven, Floquet systems with discrete time-translation symmetry. The period of the resulting discrete time crystal is quantized to an integer multiple of the drive period, arising from a combination of collective synchronization and many body localization. Here, we consider a simple model for a one-dimensional discrete time crystal which explicitly reveals the rigidity of the emergent oscillations as the drive is varied. We numerically map out its phase diagram and compute the properties of the dynamical phase transition where the time crystal melts into a trivial Floquet insulator. Moreover, we demonstrate that the model can be realized with current experimental technologies and propose a blueprint based upon a one dimensional chain of trapped ions. Using experimental parameters (featuring long-range interactions), we identify the phase boundaries of the ion-time-crystal and propose a measurable signature of the symmetry breaking phase transition.
We propose a Floquet period-doubling time-crystal model based on a disordered interacting long-range spin chain where the periodic swapping of nearby spin couples is applied. This protocol can be applied to systems with any local spin magnitude sss and in principle also to systems with nonspin (fermionic or bosonic) local Hilbert space. We explicitly consider the cases s=1/2s=1/2s=1/2 and s=1s=1s=1, using analytical and numerical methods to show that the time-crystal behavior appears in a range of parameters. In particular, we study the persistence of period-doubling oscillations in time and the time-crystal properties of the Floquet spectrum (quasienergy $\ensuremath{\pi}$-spectral pairing and long-range correlations of the Floquet states), and introduce a quantity (the local imbalance) to assess what initial states give rise to a period-doubling dynamics. We also consider the quasienergy average level-spacing ratio, and find that the interval of parameters where the system dose not thermalize and persistent period doubling is possible corresponds to the one where the Floquet spectrum shows time-crystal properties.
Abstract We study the properties of a conformal field theory (CFT) driven periodically with a continuous protocol characterized by a frequency ωD. Such a drive, in contrast to its discrete counterparts (such as square pulses or periodic kicks), does not admit exact analytical solution for the evolution operator U. In this work, we develop a Floquet perturbation theory which provides an analytic, albeit perturbative, result for U that matches exact numerics in the large drive amplitude limit. We find that the drive yields the well-known heating (hyperbolic) and non-heating (elliptic) phases separated by transition lines (parabolic phase boundary). Using this and starting from a primary state of the CFT, we compute the return probability (Pn), equal (Cn) and unequal (Gn) time two-point primary correlators, energy density(En), and the mth Renyi entropy ( Snm {S}_n^m Snm​ S n m ) after n drive cycles. Our results show that below a crossover stroboscopic time scale nc, Pn, En and Gn exhibits universal power law behavior as the transition is approached either from the heating or the non-heating phase; this crossover scale diverges at the transition. We also study the emergent spatial structure of Cn, Gn and En for the continuous protocol and find emergence of spatial divergences of Cn and Gn in both the heating and non-heating phases. We express our results for Snm {S}_n^m Snm​ S n m and Cn in terms of conformal blocks and provide analytic expressions for these quantities in several limiting cases. Finally we relate our results to those obtained from exact numerics of a driven lattice model.