Transverse Ising chain under periodic instantaneous quenches: Dynamical many-body freezing and emergence of slow solitary oscillations

We study the real-time dynamics of a quantum Ising chain driven periodically by instantaneous quenches of the transverse field between $+{\Gamma }_0$ and ${\Gamma }_0$ back and forth in equal intervals of time. Two interesting phenomena are reported and analyzed. (i) We observe dynamical many-body freezing (DMF), i.e., strongly nonmonotonic freezing of the response with respect to the driving parameters (pulse width and height) resulting from coherent suppression of dynamics of all quasiparticle modes. For certain combinations of the pulse height and the period, maximal freezing (DMF peaks) is observed, where a massive collapse of the entire Floquet spectrum occurs and the many-body system remains frozen extremely close to the initial state for all time. (ii) Second, away from the freezing peak, we observe the emergence of a distinct oscillation with a single nontrivial frequency, which can be much lower than the driving frequency. This remarkable slow oscillation involving many high-energy modes dominates the response in the limit of long observation time. We identify this slow oscillation as the unique survivor of destructive quantum interference between the many-body modes. The oscillation tends to decay algebraically with time to a constant value. All the key features are demonstrated analytically with numerical evaluations for specific results.

Figure 1

The DMF behavior of the resulting response. (a) Variation of $M_z$ with $t$ for different $p$($p=\frac{{\Gamma }_{0}T}{\pi }$) for ${\Gamma }_0=20$. (b) Variation of $Q$ with $2\pi /T$ for ${\Gamma }_0=20$. (c) Variation of $Q$ with $p$ for different ${\Gamma }_0$. Maximal freezing is seen for integer $p$. (d) Magnetization after the 100th and the 1000th cycle at different ${\Gamma }_0$ for $T=0.1$ ($\hslash =1$).

Figure 2

Features of magnetization obtained by numerical integration of Eqs. (25) and (19). (a) Oscillations of two distinct time scales are observed in $M_z\left(t\right)$. In addition to the expected one with period $T$ (matching to the driving), there is an additional longer time scale prominently visible. (b) Fourier transform of $M_z\left(t\right)$ showing two time scales visible in frame (a): two distinct peaks at angular frequencies ${\omega }_Q$ and ${\omega }_0$ are observed. The peak at ${\omega }_Q=1.822$ in the figure matches quite well with the estimation (1.815) from Eq. (35), while ${\omega }_0\approx 2\pi /T$, where $T=0.1$ is the driving period. (c) Long-time behavior of $M_z\left(t\right)$ obtained numerically (points) and that given by fitting Eq. (34) (continuous curve) is shown. The envelop corresponding to the $1/\sqrt{n}$ decay (Eq. (34)) is visible. (d) Variation of $T_Q=2\pi /{\omega }_Q$ with $p$ for $T=0.1$ obtained by Fourier transform. $T_Q$ tends to blow up (i.e., ${\omega }_Q$ vanishing up to order $1/{\Gamma }_0$) at integer values $p$—consistent with the observed maximal freezing at integer $p$. The points correspond to the values obtained from Fourier transform and the continuous line is obtained from Eq. (35) ($\hslash =1$).