Time Glasses: Symmetry Broken Chaotic Phase with a Finite Gap

Abstract

We introduce the time glass, a non-periodic analogue of the discrete time crystal that arises in periodically driven dissipative quantum many-body systems. This phase is defined by two key features: (i) spatial long-range order arising from the spontaneous breaking of an internal symmetry, and (ii) temporally chaotic oscillations of the order parameter, whose lifetime diverges with system size. In other words, a time glass is a state of matter in which all components evolve in a synchronized yet chaotic manner. To characterize the time glass phase, we focus on the spectral gap of the one-cycle (Floquet) Liouvillian, which determines the decay rate of the slowest relaxation mode. Numerical studies of periodically driven dissipative Ising models show that, in the time glass phase, the Liouvillian gap remains finite in the thermodynamic limit, in contrast to time crystals where the gap closes exponentially with system size. We further demonstrate that the Liouvillian gap converges to the decay rate of the order-parameter autocorrelation derived from the classical (mean-field) dynamics in the thermodynamic limit. This result establishes a direct correspondence between microscopic spectral features and emergent macroscopic dynamics in driven dissipative quantum systems. At first glance, the existence of a nonzero Liouvillian gap appears incompatible with the presence of indefinitely persistent chaotic oscillations. We resolve this apparent paradox by showing that the quantum Rényi divergence between a localized coherent initial state and the highly delocalized steady state grows unboundedly with system size. This divergence allows long-lived transients to persist even in the presence of a finite Liouvillian gap.

Figures (24)

• Figure 1: Schematic illustrations of the time evolution of the order parameter $M$ in four phases of a spin system. (a) Disordered phase: Spins point in random directions, resulting in a vanishing order parameter. (b) Static ordered phase: Spins uniformly align along a specific direction, yielding a nonzero, time-independent order parameter. (c) Time crystal phase: The system exhibits spatial long-range order, and the order parameter remains nonzero while undergoing periodic oscillations that persist indefinitely in the thermodynamic limit. (d) Time glass phase: Although the system develops spatial long-range order similar to the time crystal, the order parameter displays irregular, chaotic oscillations that persist indefinitely in the thermodynamic limit.
• Figure 2: Schematic illustrations of the autocorrelation $C_M^\infty(t)$ in the thermodynamic limit. (a) Disordered phase: The autocorrelation is zero at all times. (b) Static ordered phase: The autocorrelation remains nonzero and constant. (c) Time crystal: The autocorrelation exhibits persistent periodic oscillations. (d) Time glass: Although $C_M^\infty(0)>0$, indicating a finite order parameter, the autocorrelation decays to zero for $t \to \infty$.
• Figure 3: Relationship between the dynamics of the order parameter and the spectrum of the time evolution map. The first row shows the time evolution of the order parameter $M$ for a period-2 time crystal, a period-4 time crystal, a period-8 time crystal, and a time glass. The second row shows the corresponding autocorrelations of $M$, and the third row shows the spectra of the time evolution map $\mathcal{U}$. In the time crystal, the autocorrelation of $M$ shares the same period as its time-series oscillations. By contrast, in the time glass, the autocorrelation decays exponentially with a rate $g$. From a spectral perspective, a $p$-period time crystal arises when there are non-decaying eigenmodes with eigenvalues $\lambda_\alpha = e^{2 n \pi / p} \: (n = 0, 1, \ldots, p-1)$. For a time glass, although one might naively expect infinitely many eigenvalues to cluster on the unit circle, our results show this is not the case. Instead, a finite gap $\Delta$ appears, matching the decay rate $g$ of the classical autocorrelation.
• Figure 4: Bifurcation diagrams for the classical dynamics of the kicked collective spin model. Panels (a) and (b) show the long-time values of $m^x$ and $m^z$, respectively, plotted as functions of the kick strength $\omega_{xx}$, after discarding transient dynamics over the first $100$ cycles. The parameters are fixed at $\omega_z=\pi/2$ and $\kappa=1$. As $\omega_{xx}$ increases, the system undergoes a sequence of transitions from a stable fixed point, to a limit cycle, and eventually to chaotic behavior.
• Figure 5: Bifurcation diagrams for the classical dynamics of the kicked spin chain model with all-to-all coupling. Panels (a) and (b) show the long-time values of $m^x$ and $m^z$, respectively, plotted as functions of the kick strength $J$, after discarding transient dynamics over the first $100$ cycles. The parameters are fixed at $\omega_z=\pi/2$ and $\kappa=1$. As $J$ increases, the system undergoes a sequence of transitions from a stable fixed point, to a limit cycle, and eventually to chaotic behavior.