Time crystalline solitons and their stochastic dynamics in a driven-dissipative φ^4 model

TL;DR

The work addresses nonequilibrium topological excitations in a periodically driven, dissipative one-dimensional phi-four theory and reports a time-crystalline soliton (TCS) that spontaneously breaks discrete time translation while maintaining a spatial soliton profile. The authors show that the TCS exhibits a period-doubled response with a localized, solitonic structure and that dissipation stabilizes this nontrivial temporal order. Under white noise, the TCS center diffuses and can drive transitions between two degenerate Z2-symmetry-breaking DTC phases, evidenced by a pi-phase shift in the field along certain paths. For two separated TCSs, the annihilation time follows a power-law with initial separation, indicating deconfinement in contrast to confinement observed in some 2D nonequilibrium spin-ice systems, highlighting a new class of nonequilibrium topological excitations with potential extensions to higher dimensions and quantum regimes.

Abstract

Periodically driven systems provide unique opportunities to investigate the dynamics of topological excitations far from equilibrium. In this paper, we report a time-crystalline soliton (TCS) state in a driven-dissipative $φ^4$ model. This state exhibits spontaneous breaking of discrete time-translational symmetry while simultaneously displaying spatial soliton behavior. During time evolution, the soliton pattern periodically oscillates between kink and anti-kink configurations. We further study TCS dynamics under noise, demonstrating that soliton random walk can induce a dynamical transition between two distinct $Z_2$ symmetry-breaking time-crystalline phases in time domain. Finally, we examine the annihilation of two spatially separated TCSs under noise. Importantly, in contrast to the confined behavior of time-crystalline monopoles reported in [Phys. Rev. Lett. 131, 056502 (2023)], the dynamics of time-crystalline solitons is deconfined despite the nonequilibrium nature of our model: the statistically averaged annihilation time scales as a power law with the solitons' initial separation.

Figures (6)

• Figure 1: Time-crystalline soliton (TCS) in the driven-dissipative $\phi^4$ model. (a) Spatiotemporal evolution of $\phi(x,t)$. The bright central line traces the soliton center. (b) Top: Period-doubled dynamics of $\phi(x_0,t)$ showing $2T$-periodicity ($T=2\pi/\omega_0$). Bottom: Fourier spectrum with dominant peak at $\omega_0/2$, confirming discrete time-translational symmetry breaking. (c) Snapshot of the TCS at time slice $t_0$ and $t_0+T$. (d) Two degenerate DTC phases related to each other by shifting $T$ along the temporal direction. $x_0=-40$, and $x_0^\prime=40$. Parameters in our simulation are chosen as $\eta=0.5$, $J=5$, $\omega_0=\pi$, $\mathcal{D}=0$.
• Figure 2: Stable multi-TCS configurations with kink and anti-kink. Parameters in our simulation are chosen as $\eta=0.5$, $J=5$, $\omega_0=\pi$, $\mathcal{D}=0$.
• Figure 3: Random walk of TCS under noise and the intanton excitations. (a) Dynamics of $\phi(x,t)$ under a single noise trajectory. (b) Mean square displacement of the center of TCS compared to its equilibrium counterpart. The ensemble average is taken over $2000$ noise trajectories. (c) Zoomed-in view of the region in (a). (d) Temporal evolution of $\phi(x,t)$ along the blue path (blue curve) and red path (red curve) indicated in (c). The yellow circle indicates an instanton excitation between two degenerate $Z_2$ symmetry-breaking DTC phases. Parameters in our simulation are chosen as $\eta=0.5$, $J=5$, $\omega_0=\pi$, $\mathcal{D}=0.5$.
• Figure 4: Annihilation of a kink-antikink pair. (a) Evolution of the field $\phi(x,t)$ under a single noise trajectory. A kink and anti-kink annihilate after a characteristic time scale $\tau$. (b) Average annihilation time $\langle \tau\rangle_\xi$ as a function of the initial separation between the solitons. The ensemble average is taken over 2000 noise trajectories. Other parameters are chosen the same as in Figure (\ref{['fig3']}).
• Figure 5: Comparison between dynamics with different $\Delta t$, $\Delta x$, $L$ and different initial states. Other parameters are chosen the same as Figure\ref{['fig1']} in the main text.