Three-stage melting of a macroscopic continuous spacetime crystal

Abstract

A spacetime crystal is a phase of matter that spontaneously develops periodic order in both space and time. Spacetime crystals have been experimentally observed in microscopic quantum many-body systems and, very recently, in a mesoscopic nematic liquid crystal. However, the melting process of a spacetime crystal and its underlying physical mechanisms have not yet been experimentally reported. Here, we present a direct observation of a classical continuous spacetime crystal melting in a table-top experiment with macroscopic active granular disks in 2+1 spacetime dimensions. The spacetime crystal is characterized by the spontaneous formation of a coherent, rigid-body rotation of a 2D triangular lattice that persists for almost a day and remains remarkably robust to noise. By tuning the disk packing fraction, we observe a complex three-stage melting process involving a spatially hexatic phase and multiple coexistence regions. Importantly, we show that spatial and temporal crystalline orders melt separately through distinct mechanisms: spatial order is destroyed by the proliferation of topological defects, while temporal order is lost through the decay of directional persistence caused by the progressive weakening of many-body interactions. Our results demonstrate that the spontaneous breaking of spatial and temporal translational symmetries can be decoupled, leading to the emergence of exotic out-of-equilibrium classical phases of matter.

Figures (22)

• Figure 1: Macroscopic continuous spacetime crystal. A. Experimental setup: Particles are placed on an aluminum-alloy plate and confined within a circular boundary. Vertical vibrations are applied normal to the plate using an electromagnetic shaker. Inset: schematic of the ratcheted particles. B. Single-particle translational motion: Particles initially placed near the center exhibit random, disordered trajectories, consistent with 2D diffusive Brownian motion. C. Single-particle rotational motion: External vibrations apply torque, although most spin angular momenta remain near zero. Theoretically, a particle has zero mean spin due to the symmetric inclination of its six legs, whereas a disk acquires finite spin from manufacturing imperfections. D. Photograph of the macroscopic time crystal. Trajectories of three representative particles are shown. The color gradient from blue to red indicates particle positions over a time interval from $0$ to $2400$ s. The particles move collectively, forming a global spacetime crystalline state, as shown in Supplementary Movie \ref{['vid:835']}. E. The $X$ and $Y$ position of a benchmark particle in panel D as a function of time at $\varphi = 0.835$. F. Power spectrum of the phase signal. A pronounced peak appears at $f = 5.5 \times 10^{-5}\,\mathrm{Hz}$ corresponding to a rotation period of $5.05$ h. Inset: time-correlation function of the normalized projection of particles' motion $\tilde{y}$ in log--log scale, defined as $G(t)=\langle \tilde{y}(t)\tilde{y}(0)\rangle - \langle\tilde{y}(t)\rangle\langle\tilde{y}(0)\rangle$, where the ensemble average is taken over particles of radius $8.8\text{cm}<R<20.6\text{cm}$ to avoid center and boundary region. The black dashed line shows a fit to the envelope of |$G(t)$|, yielding a slope of $-0.09\pm0.05$. G. Magnified photograph of a portion of the spacetime crystal. Particles arrange into a triangular lattice. H. Longitudinal and transverse dynamical structure factors $S(q,\omega)$ after subtraction of the global rotation. Here $D$ denotes the pitch diameter ($D=8.8$ mm). The white dashed lines show fits to the maxima of $S(q,\omega)$, given by $\omega=26.34\sin(qD/2)$ for the longitudinal mode and $\omega=16.27\sin(qD/2)$ for the transverse mode, where both frequencies are measured in Hz (1/s).
• Figure 2: Three-stage melting of spacetime order. A. Phase diagram showing, from right to left, the spacetime crystal, time-coexistence, space-coexistence, and fluid phases. The temporal and spatial crystalline fractions decrease independently across the coexistence regimes. The critical packing fractions separating the phases are $\varphi_1 = 0.687$, $\varphi_2 = 0.709$, and $\varphi_3 = 0.734$. One representative packing fraction is selected for each phase to illustrate its characteristic behavior; in ascending order, these are $\varphi = 0.662$, $0.699$, $0.719$, and $0.835$. The corresponding locations are indicated by enlarged red diamonds. B. Particle trajectories at three selected packing fractions, as shown in Supplementary Movies \ref{['vid:662']}, \ref{['vid:699']}, and \ref{['vid:719']}. The color scale, ranging from blue to red, represents particle positions over the time interval $0$ to $2400$ s. Left: random trajectories. Middle: weak dynamical coherence. Right: coexistence of clustered and disordered particle motion. C. Normalized mean-square displacements (MSD) at the four selected packing fractions. At $\varphi = 0.662$, the MSD displays diffusive scaling (slope $=1$), whereas at $\varphi = 0.835$ it exhibits ballistic scaling associated with rigid-body rotation (slope $=2$). For clarity, the same color code is used throughout the figure to denote these four packing fractions. D. Spatial patterns of the static structure factor $S(q_xD,q_yD)$. As the packing fraction decreases from right to left, the system evolves from a crystalline state to a hexatic phase, then melts, and ultimately transitions into a fluid.
• Figure 3: Dynamical melting of time-crystalline order. A. Dynamical coexistence. The dynamical parameter $\log_{10}\!\left(D_{\min}^2/D^2\right)$ quantifies local non-affine displacements. In the spacetime crystalline phase, this quantity remains small, indicating highly ordered and coherent dynamics. In the fluid phase, no ordered domains are present. In the time-coexistence (T-coexistence) region, dynamically ordered and disordered domains coexist. B. Schematic of the coexistence between a time crystal and a fluid. Arrows indicate particle displacements over $200\,\mathrm{s}$ after rescaling. Color encodes the magnitude of the non-affine displacement (blue: small; red: large). C. Percolation of dynamical order. The spatial extent of the time-crystalline domain decreases rapidly across the T-coexistence regime. D. Directional persistence, which quantifies long-time correlations of velocity directions in the co-rotating frame of the time crystal. From right to left: Spacetime Crystal, where each particle maintains alignment with its initial direction of motion; T-coexistence, where the ability to sustain coherent tangential motion rapidly weakens as the packing fraction decreases; S-coexistence, where particles largely lose directional persistence, with only sporadic tangential motion; Fluid, where long-time velocity-direction correlations vanish completely. Inset: spatial map of the local directional persistence, where colors from blue to red denote long-time velocity-direction correlations from fully aligned to completely random. The red region near the center in the spacetime crystalline phase reflects deviations from perfect rigid-body rotation.
• Figure 4: Structural correlations and topological defects. A. Spatial distribution of the local hexatic order parameter $|\psi_6|$ in the S-coexistence phase at $\varphi=0.662$, where fragmented ordered domains coexist with a disordered fluid. Here, $|\psi_6|$ represents the modulus of $\psi_6$. B. Distribution of $\psi_6$ in the complex plane for the configuration shown in panel A. The central peak corresponds to disordered fluid particles, while the peripheral peaks indicate locally ordered domains. C. Percolation of structural order. The spatial extent of the space-crystalline domains decreases rapidly upon entering the S-coexistence regime. D. Spatial hexatic correlation function. Three distinct decay regimes are observed: constant (Spacetime Crystal), power-law (T-coexistence and S-coexistence), and exponential (Fluid). The black dashed line with slope $-1/4$ marks the critical decay separating the hexatic-like and fluid phases. E. Temporal hexatic correlation function. The black dashed line with slope $-1/8$ again indicates the boundary between the hexatic-like and fluid phases. F. Defect ratio, defined as the ratio between the number of defects and the total number of particles in each frame, shown as a function of $\varphi$. Free dislocations proliferate at $\varphi=0.734$, signaling the loss of spatial long-range order and the transition from the crystalline to the hexatic-like phase. Free disclinations proliferate at $\varphi=0.714$, indicating the loss of angular order and the onset of the fluid phase. Defect clusters proliferate throughout the hexatic-like and coexistence phases.
• Figure 5: Spatiotemporal phase diagram of spatial and temporal order. A summary of the different phases in terms of their spatial and temporal crystalline order is shown. Remarkably, spatial and temporal symmetries are either both preserved or both broken in the low-packing-fraction fluid phase and in the high-packing-fraction spacetime crystalline state. In contrast, in the two intermediate coexistence regions, spatial and temporal order decouple: they melt at distinct critical points and through fundamentally different physical mechanisms, producing an intricate three-stage melting scenario including a hexatic state.