Discrete time liquids

Time is usually thought of as a steady river, flowing forward at a constant rate, indifferent to the events happening along its banks. In the world of physics, this steady flow is known as time-translation symmetry: the laws of nature do not change from one moment to the next. For decades, scientists believed that if a system was left alone in a state of thermal equilibrium—where heat and energy are evenly distributed—it would eventually settle into a static, unchanging pattern. It was thought impossible for such a system to spontaneously start ticking or oscillating on its own, as if it had its own internal clock. This belief was so strong that it was codified into "no-go theorems," mathematical rules that seemed to forbid any form of time-based order in a quiet, balanced system. However, the discovery of "discrete time crystals" challenged this view. These are special states of matter that, when pushed by a regular, repeating external force, respond by moving at a slower, rhythmic pace. They break the symmetry of the drive, locking into a rhythm that is a fraction of the push, creating a kind of temporal structure that persists without fading away. But a lingering question remained: could such a rhythmic order exist without a perfect, external metronome? Could order emerge from pure noise?

A team of researchers at the Max Planck Institute for the Physics of Complex Systems and South China Normal University has now answered this question by identifying a new state of matter they call a "discrete time liquid." In a series of detailed computer simulations, they modeled a two-dimensional grid of tiny magnetic spins, similar to the atoms in a magnet, coupled to a cavity—a box that traps light or electromagnetic waves. Unlike previous experiments that used a perfectly timed, external laser to drive the system, this setup relied entirely on the natural, random thermal fluctuations of the cavity itself. The cavity was in contact with a heat bath, meaning its internal vibrations were driven by random thermal noise rather than a precise clock. The researchers found that when the temperature of this heat bath was set to a specific intermediate level, the spins did not just jitter chaotically. Instead, they spontaneously locked into a rhythm that was exactly half the frequency of the cavity's natural oscillation. This period-doubled response is the hallmark of a time crystal, but with a crucial difference: the rhythm was not perfect.

In a standard time crystal, the rhythm is rigid and unchanging, like a metronome that never skips a beat. In this new "time liquid" state, the rhythm is robust but fragile in a specific way. While the spins maintained their period-doubled motion for a long time, individual paths of the system would occasionally and randomly switch between two opposite phases of the rhythm. Imagine a clock that keeps perfect time for hours, but every now and then, the hands suddenly jump forward or backward by a full cycle before settling back into the correct beat. The researchers identified these sudden jumps as "time-solitons," which are topological defects in the flow of time. These defects act like glitches in the temporal order; they preserve the local rhythm but destroy the long-range synchronization across the entire system. Because these glitches can happen at any random moment, if you were to watch a large collection of these systems all at once, the random jumps would cancel each other out, and the average behavior would look like a smooth, continuous flow of time. This effectively restores the time-translation symmetry that was broken in the individual trajectories, suggesting that this new form of order can exist even in thermal equilibrium, a realm where such order was previously thought impossible.

The study reveals that the emergence of this order is counterintuitive. Usually, heat and noise are seen as enemies of order, blurring patterns and destroying structure. Here, however, the researchers found that the temporal order only appeared when the thermal noise was strong enough to be significant, yet not so strong that it overwhelmed the system entirely. If the temperature was too low, the thermal fluctuations were too weak to trigger the rhythmic response, and the spins simply precessed freely. If the temperature was too high, the noise was too chaotic, and the system became disordered. It was only in a narrow window of intermediate temperature that the spins could harness the noise to create a stable, period-doubled rhythm. This phenomenon was observed in simulations where the cavity's coherence time—the duration over which its oscillations remained predictable—was varied. When the coherence time was long, the system behaved like a classic time crystal with a fixed rhythm. As the coherence time shortened, the system transitioned into the "time liquid" phase, where the rhythm persisted but was constantly interrupted by the time-solitons.

The researchers confirmed that this behavior was not an artifact of their specific setup but a fundamental property of the system. They developed a simple mathematical model that treated the system as a series of random switches, where the probability of a "glitch" occurring was determined by the density of time-solitons. This minimal model successfully reproduced the full distribution of time correlations observed in the complex simulations, suggesting that the behavior of the many-body system could be understood through the simple statistics of these temporal defects. The findings imply that the "time liquid" is a distinct phase of matter, different from both a disordered gas and a rigid time crystal. It represents a state where temporal order is maintained locally but is fluid globally, allowing the system to exist in a state of equilibrium while still exhibiting complex, rhythmic dynamics.

This work opens a new avenue for understanding how order can arise from disorder. It suggests that the landscape of possible states of matter is richer than previously thought, with phases that can exist in thermal equilibrium and still display the signatures of time-crystalline behavior. The researchers propose that this phenomenon could be observed in real-world experiments, such as with ultracold atoms trapped in optical cavities or in condensed matter systems involving surface phonon polaritons, which are waves of electric charge and vibration on a material's surface. In these setups, the natural fluctuations of the environment could serve as the noisy drive, potentially allowing scientists to create and study these time liquids in the laboratory. The discovery challenges the long-held view that thermal equilibrium must be a state of static stillness, showing instead that it can harbor a subtle, liquid-like form of time that flows, stutters, and recovers, driven by the very noise that was once thought to destroy it.