Strange Ripples That Grow and Merge Give Rise to a Time Crystal in a Machine-Like Material

When a piece of iron is magnetized, tiny magnetic domains twist and merge until the whole material points one way. When an emulsion of oil and water separates, droplets swallow their neighbors in an inexorable march toward larger and larger sizes. Physicists have long understood this behavior, known as coarsening, as a signature of systems relaxing toward equilibrium: domains compete, boundaries move, and structure grows coarser over time through slow, diffusive transport. Now a team reporting in Nature has discovered a dramatic vibrational counterpart to this familiar process, one that unfolds not through diffusion but through waves, and that ends in one of the most exotic states of modern physics, a time crystal.

The research, led by Jonas Veenstra and Jack Binysh of the Institute of Physics at the University of Amsterdam, together with colleagues in Delft, Leiden, Birmingham and Cambridge, including senior authors Anton Souslov and Corentin Coulais, introduces what the authors call wave coarsening. In their engineered materials, vibrations do not simply die away or stay put. Instead, oscillations emerge spontaneously, then progressively grow in wavelength, period and amplitude, until the entire system beats in unison as a single, self-sustained oscillation. That globally synchronized state is precisely what physicists mean by a time crystal, a phase of matter that exhibits periodic motion in time in much the same way an ordinary crystal exhibits periodic order in space.

The key ingredient is non-reciprocity. In ordinary elastic materials, Newton’s third law guarantees that if one part pushes a neighbor, the neighbor pushes back equally. The mechatronic metamaterials built by the Amsterdam group deliberately violate this rule. Each unit cell contains a compliant flexure housing a voice coil actuator and a Hall effect sensor, monitored by a microcontroller. When one hexagonal cell is deformed, the sensors detect the strain and the actuators exert forces on neighboring cells according to a programmed rule that is deliberately asymmetric: compression of one bond generates unequal, antisymmetric tensions in adjacent bonds. The result is a lattice in which cause and effect are decoupled, a mechanical analogue of the non-reciprocal interactions that theorists have identified as a wellspring of novel pattern formation far from equilibrium.

When driven past a critical threshold of this asymmetric activity, the metamaterial undergoes an instability reminiscent of a Hopf bifurcation, beyond which small perturbations grow into sustained limit-cycle oscillations. In a two-dimensional lattice of 37 such active cells, the team observed shear waves appear at short wavelength and then undergo a remarkable evolution: ripples merged into longer waves, which merged again, each stage carrying larger amplitude and longer period, until the deformations spanned the entire sample. The researchers characterized this process through spatial correlation functions, finding that the characteristic length scale of the pattern grows as a power law in time, the hallmark of self-similar, dynamical scaling similar to that seen in conventional coarsening and even in the growth of rough interfaces described by the Kardar-Parisi-Zhang equation.

To demonstrate that the phenomenon is not an artifact of a particular architecture, the team built a one-dimensional elastic ring of 24 torsional unit cells with the same non-reciprocal force rules. There, too, traveling waves appeared at high wave number and progressively combined into fewer, larger, slower oscillations, finally settling into a stable low-frequency time-crystalline state. Numerical simulations of a minimal field theory, in which a complex order parameter evolves under the combined influence of elasticity, nonlinearity and an antisymmetric, non-reciprocal coupling whose imaginary part is odd in wave number, reproduced the entire cascade. The simulations tracked the amplitude spectrum over five orders of magnitude in time, showing the peak of the spectrum marching steadily toward lower wave numbers as the lowest mode ultimately wins the competition.

What makes this coarsening genuinely new is its mechanism. Classical coarsening is driven by diffusion: material or magnetization slowly migrates across domain walls, which then recede. Wave coarsening involves no such transport. Instead, the nonlinear dynamics of the active elastic medium channels energy from short-wavelength vibrational modes into longer ones, with the amplitude of each mode growing and shifting as the total elastic energy in the system changes. The researchers showed analytically and numerically that the effective growth rate of each mode decreases and moves to lower wave numbers as the oscillations amplify, providing a quantitative picture of how the cascade proceeds. In three-dimensional simulations of an active solid, the inverse correlation length likewise fell as a power law in time, though with distinct rates along different directions, revealing anisotropic scaling in the coarsening dynamics.

Perhaps the most striking result concerns what happens when coarsening is deliberately stopped. The team placed the two-dimensional metamaterial on an unlubricated surface, introducing friction that breaks the conservation of momentum that ordinarily underpins the wave dynamics. Arrested mid-cascade, the waves no longer spanned the system, leaving behind a stable strain defect in the bulk. Yet the material did not fall silent. Instead, large-scale simulations revealed a surprise: edge waves came to dominate the dynamics, settling into steady, counter-rotating oscillations localized at the boundary. The researchers identify these persistent edge states as a far-from-equilibrium, nonlinear analogue of chiral topological edge modes, the unidirectional boundary waves familiar from quantum Hall systems and topological acoustics. That a dissipative, frictionally arrested, strongly nonlinear machine could exhibit behavior echoing topological edge physics came as an unexpected bridge between two previously separate domains.

The wider significance of the work lies in its breadth of application. Active elastic media, solids whose constituents generate their own motion or forces, appear throughout nature and technology: epithelial cell sheets that oscillate autonomously, bacterial and cellular colonies, muscle fibers with odd elasticity, dense human crowds that spontaneously synchronize, and swarms of fireflies flashing in unison. The Amsterdam-led team suggests that wave coarsening may serve as an organizing principle for nonlinear waves across acoustics, optomechanics, living matter and soft robotics. In robotics, in particular, the ability to program a metamaterial that spontaneously coordinates its motion from rapid, local jitter into slow, global, coherent deformation points toward machines that self-organize their gait without centralized control, a prospect the same group has explored in earlier work on adaptive locomotion in active solids.

The study also deepens the still-young physics of time crystals. Since the first discrete time crystal was observed in a chain of trapped ions in 2017, and continuous time crystals followed in atomic, photonic and Rydberg systems, most realizations have lived in quantum or driven-dissipative microscopic settings. Here the time crystal emerges from centimeter-scale mechanical components governed by classical, engineered forces, and, crucially, the route by which it forms has been traced step by step. The authors frame their results as opening new questions about the transient physics of systems with non-potential interactions, those whose forces cannot be derived from a single energy landscape, where intuition built on equilibrium thermodynamics fails. Theoreticians have recently predicted non-reciprocal phase transitions, order-by-disorder phenomena and turbulent cascades in such systems; wave coarsening now provides a concrete, measurable laboratory instantiation of how order in time crystallizes out of such frustration.

For a field that has long treated pattern formation and coarsening as the province of diffusion and equilibrium relaxation, the message of this work is that waves, when fed by non-reciprocal activity, can perform the same coarse-graining trick and carry matter all the way into temporal order. The data and code behind the study have been released on the public repository Zenodo, allowing other groups to reproduce the cascade and to test whether the same power-law scaling and frictionally arrested edge waves appear in their own active lattices. As mechatronic metamaterials grow more sophisticated, the boundary between designed machines and spontaneously ordered matter continues to blur, and this experiment suggests that the next organized states of matter may not sit still at all, but tick with a rhythm all their own.

Subject of Research: Wave coarsening and time crystallization in non-reciprocal mechatronic metamaterials

Article Title: Wave coarsening drives time crystallization in mechatronic metamaterials

Article References: Veenstra, J., Binysh, J., Seinen, V., Solé Borràs, R., Naber, R., Robledo-Poisson, D., Hunt, A., van Saarloos, W., Souslov, A., & Coulais, C. (2026). Wave coarsening drives time crystallization in mechatronic metamaterials. Nature. https://doi.org/10.1038/s41586-026-11163-6

Image Credits: AI Generated

DOI: 10.1038/s41586-026-11163-6

Keywords: time crystals, metamaterials, wave coarsening, non-reciprocal interactions, active elastic media, nonlinear waves, self-organization, topological edge modes, soft robotics, pattern formation, mechatronics, far-from-equilibrium physics