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arXiv:2103.10818 · Physical review. B./Physical review. B

Critical properties of the prethermal Floquet time crystal

Muath Natsheh, Andrea Gambassi, Aditi Mitra

Published Jun 28, 2021• 18 citations
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Abstract

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.

DOI
10.1103/physrevb.103.224311

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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)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→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞, we determine the exponent \ensuremathν\ensuremath{\nu}\ensuremathν that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent \ensuremathβ\ensuremath{\beta}\ensuremathβ characterizing the growth of the amplitude of the order parameter, as well as the initial-slip exponent \ensuremathθ\ensuremath{\theta}\ensuremathθ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents \ensuremathν,\ensuremathβ,\ensuremathθ\ensuremath{\nu},\ensuremath{\beta},\ensuremath{\theta}\ensuremathν,\ensuremathβ,\ensuremathθ 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 \ensuremathω/(2v),v\ensuremath{\omega}/(2v), v\ensuremathω/(2v),v being the velocity of the quasiparticles, and \ensuremathω\ensuremath{\omega}\ensuremathω being the drive frequency.
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 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.
While a clean, driven system generically absorbs energy until it reaches "infinite temperature," it may do so very slowly exhibiting what is known as a prethermal regime. Here, we show that the emergence of an additional approximately conserved quantity in a periodically driven (Floquet) system can give rise to an analogous long-lived regime. This can allow for nontrivial dynamics, even from initial states that are at a high or infinite temperature with respect to an effective Hamiltonian governing the prethermal dynamics. We present concrete settings with such a prethernial regime, one with a period-doubled (time-crystalline) response. We also present a direct diagnostic to distinguish this prethermal phenomenon from its infinitely long-lived many-body localized cousin. We apply these insights to a model of the recent NMR experiments by Rovny et al. [Phys. Rev. Lett. 120, 180603 (2018)] which, intriguingly, detected signatures of a Floquet time crystal in a clean three-dimensional material. We show that a mild but subtle variation of their driving protocol can increase the lifetime of the time-crystalline signal by orders of magnitude.
Prethermalization, where quasisteady states are realized in the intermediate long-time regime (prethermal regime), in periodically driven (Floquet) systems is an important phenomenon since it provides a platform for nontrivial Floquet many-body physics. In this Rapid Communication, we consider Floquet systems with dual energy scales: The Hamiltonian consists of two different terms whose amplitude is either comparable to or much smaller than the frequency. As a result, when the larger-amplitude drive induces a ZN{\mathbb{Z}}_{N}ZN​ symmetry operation, we obtain the effective static Hamiltonian respecting the emergent ZN{\mathbb{Z}}_{N}ZN​ symmetry in high-frequency expansions, which describes the dynamics of such Floquet systems in the prethermal regime. As an application of our formulation, our formalism gives a general way to analyze time crystals in prethermal regimes in terms of the static effective Hamiltonian. We also provide an application to Floquet engineering, with which we can perform the simultaneous control of phases and symmetries of the systems. For example, this enables us to control symmetry protected topological phases even when the original system does not respect the symmetry.
The use of periodic driving for synthesizing many-body quantum states depends crucially on the existence of a prethermal regime, which exhibits drive-tunable properties while forestalling the effects of heating.This dependence motivates the search for direct experimental probes of the underlying localized nonergodic nature of the wave function in this metastable regime.We report experiments on a many-body Floquet system consisting of atoms in an optical lattice subjected to ultrastrong sign-changing amplitude modulation.Using a double-quench protocol, we measure an inverse participation ratio quantifying the degree of prethermal localization as a function of tunable drive parameters and interactions.We obtain a complete prethermal map of the drive-dependent properties of Floquet matter spanning four square decades of parameter space.Following the full time evolution, we observe sequential formation of two prethermal plateaux, interaction-driven ergodicity, and strongly frequency-dependent dynamics of long-time thermalization.The quantitative characterization of the prethermal Floquet matter realized in these experiments, along with the demonstration of control of its properties by variation of drive parameters and interactions, opens a new frontier for probing far-from-equilibrium quantum statistical mechanics and new possibilities for dynamical quantum engineering.
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.