Programmable time crystals from higher-order packing fields

TL;DR

This work presents a hydrodynamic protocol for programmable continuous time crystals in driven diffusive fluids by introducing a packing-field mechanism that excites the $m$-th density Fourier mode through the Kuramoto-Daido order parameter $z_m$, producing $m$ traveling condensates with a scalable relation to the base $m=1$ case. It derives an explicit instability threshold $\lambda_c^{(m)} = 4\pi m \frac{D(\rho_0)\rho_0}{\sigma(\rho_0)}$ and a traveling-wave velocity $\omega_m = 2\pi m \sigma'(\rho_0)\epsilon$, along with a current shift linking transport coefficients to the ordered phase. The authors demonstrate the framework across four paradigmatic models (RW, WASEP, KMP, KLS), revealing both continuous and first-order transitions and enabling decorated time-crystal phases via multi-mode driving. This approach offers a versatile, experimentally accessible route to tailor time-crystal phases with controllable number, shape, and speed of condensates in 1D driven diffusion, connecting to Kuramoto synchronization and opening avenues to study explosive transitions in driven systems.

Abstract

Time crystals are many-body systems that spontaneously break time-translation symmetry, and thus exhibit long-range spatiotemporal order and robust periodic motion. Recent results have demonstrated how to build time-crystal phases in driven diffusive fluids using an external packing field coupled to density fluctuations. Here we exploit this mechanism to engineer and control on-demand custom continuous time crystals characterized by an arbitrary number of rotating condensates, which can be further enhanced with higher-order modes. We elucidate the underlying critical point, as well as general properties of the condensates density profiles and velocities, demonstrating a scaling property of higher-order traveling condensates in terms of first-order ones. We illustrate our findings by solving the hydrodynamic equations for various paradigmatic driven diffusive systems, obtaining along the way a number of remarkable results, e.g. the possibility of explosive time crystal phases characterized by an abrupt, first-order-type transition. Overall, these results demonstrate the versatility and broad possibilities of this promising route to time crystals.

Figures (4)

• Figure 1: (a) Sketch of a $1d$ driven particle system sustaining a net current with an homogeneous density structure on average. By switching on a $m$th-order packing field ${\cal E}^{(m)}(\theta)$ with strength $\lambda$ beyond a critical value [see (c),(e) and shaded curves in (b),(d)], an instability is triggered to a time-crystal phase characterized by the emergence of $m$ rotating particle condensates. The magnitude $\lvert z_{m} \rvert$ of the complex packing order parameter indicates the packing of particles around $m$ emergent localization centers, placed at the argument of $(\space\sqrt[m]{z_{m}})_j$, with $j\in[0,m-1]$, and represented by the red arrows inside the ring in (b),(d). The convergent blue arrows around the ring signal the local direction of the packing field around the $m$ localization centers.
• Figure 2: (a) Mobility $\sigma(\rho)$ for the different models studied. Inset: diffusivity $D(\rho)$ in the KLS model. For $m=1$, $\epsilon=10$ and different $\rho_0$, the other panels display (b) the magnitude of the packing order parameter $|z_1|$, (c) the condensate velocity $\omega_1$, and (d) the average relative current $J/J_0$, as a function of $\lambda/\lambda_\mathrm{c}$ for the different models.
• Figure 3: Condensates density profiles for different models, order $m$ and couplings $\lambda$. (a) WASEP with $m=3$, (b) RW fluid with $m=4$, (c) KMP model with $m=2$, and (d) KLS lattice gas with $m=1$. In all cases $\rho_0 = 1/3$, $\epsilon=10~m$, and $\lambda/\lambda_{\mathrm{c}}^{(m)} = 1.01,~1.2,~1.5,~2$.
• Figure 4: Raster plots of the spatiotemporal evolution of the density field in the WASEP subject to different time-modulated generalized external fields $E_{x,t}[\rho]$ for $\rho_0=1/3$. In (a) we swap for $\epsilon=10$ between different number of condensates in time by switching on and off different orders $m=2,~3,~5$ modulating $\lambda_m(t)$ as shown in panel (b). In (c) a decorated time-crystal phase emerges for $\epsilon=0.5$ by modulating in time a higher-order $m=4$ mode using $\lambda_{4}(t)$ as in panel (d), in a constant background $m=2$ matter wave obtained by setting $\lambda_2>\lambda_c^{(2)}$.