Temporal disorder in spatiotemporal order

Time-dependent driving holds the promise of realizing dynamical phenomena absent in static systems. Here, we introduce a correlated random driving protocol to realize a spatiotemporal order that cannot be achieved even by periodic driving, thereby extending the discussion of time translation symmetry breaking to randomly driven systems. We find a combination of temporally disordered micromotion with prethermal stroboscopic spatiotemporal long-range order. This spatiotemporal order remains robust against generic perturbations, with an algebraically long prethermal lifetime where the scaling exponent strongly depends on the symmetry of the perturbation, which we account for analytically.

Figure 1

(a) Distinct temporal correlations emerge for micromotions and stroboscopic times (for system size $L=26$). (b) Fourier modes of micromotions exhibit a $\pi$-shifted spectrum different from the drive, defining another type of TTS breaking. (c) This persists for a long timescale $\tau$, which grows with driving frequency $1/T$. We use $J_z=1, J_x=0.1, J_y=0.2, B_z=0, {\delta }_r=0.15, L=18$.

Figure 2

Prethermal lifetime $\tau$ strongly depends on the presence/absence (blue/red lines) of $Z_2$ symmetry in the perturbation. (a) Dynamics of the stroboscopic magnetization. (b) Algebraic lifetime scaling vs driving frequency $1/T$ for multipolar order $n=1$. Scaling exponent is approximately $2n+3$ for $Z_2$ preserving perturbation with $B_z=0$, and $2n-1$ for $B_z=0.5$ for $n=1$. (c) Scaling results for $n=2$. We use $J_z=1, J_x=0.1, J_y=0.2, B_z=0, {\delta }_r=0.15, L=18$.

Figure 3

(a) Dynamics for random product state in the $z$ direction. Prethermal phases persist for sufficiently strong disorder with a lifetime increasing for larger driving frequencies. For weak disorder (red), the spatiotemporal order quickly vanishes. (b) Algebraic lifetime scaling vs frequency with exponent close to $2n+3$. We use $J_z=1, J_x=0.1, B_z=0, {\delta }_r=0.08, n=1, L=16$.

Figure 4

(a) Dynamics of the magnetization. For the blue line, spins flip randomly according to a hyperuniform sequence, while for the black line, spins flip deterministically. System size $L=20$ and rotation imperfection ${\delta }_r^{\prime }=0.01.$ (b) Fourier modes of magnetization exhibit a $\pi$-shifted spectrum different from the drive. The algebraic suppression (dashed black) has an exponent $\alpha /2$. We use $J_z=1, J_x=0.2, \alpha =3.8, L=16, {\delta }_r^{\prime }=0, \delta t_{\max }=0.1, T=1$ for the numerical simulation.