A team of four researchers at the Max Planck Institute for the Physics of Complex Systems in Dresden, Germany, including Zuo Wang and Marios H. Michael, on October 21, 2026 posted a preprint to arXiv proposing a new state resembling a time crystal, which they call a "discrete time liquid." The team ran numerical calculations on classical spins arranged on a 100×100 square lattice and coupled to a resonator in contact with a thermal bath. Although no periodic drive was applied from outside, within a certain temperature range the spins showed "period doubling," oscillating at twice the period of the resonator's oscillation. However, the phase of this oscillation switched irregularly and never stayed the same over long times. The authors named this state a discrete time liquid, but what was confirmed is a metastable state that eventually collapses through heating, and no experimental demonstration has yet been made. How does this "liquid" differ from a conventional time crystal, and how far do the numerical results support the authors' view that it could exist even in thermal equilibrium?
How temporal order can arise from thermal fluctuations without a periodic external drive
In a conventional discrete time crystal, the periodic drive applied from outside serves as the reference for temporal order. A discrete time crystal is a state in which many particles or spins, subjected to a periodic stimulus, oscillate regularly with a period longer than that of the stimulus. Consider, for example, a case where an interaction makes neighboring spins try to align in the same direction while an external stimulus flips all the spins once per cycle. The spins alternate between up and down, so two cycles of the stimulus are needed to return to the original state. This is called "period doubling," meaning the response has twice the period of the external drive. Such a response is understood as the collective of spins spontaneously breaking the discrete time-translation symmetry set by the external drive.
What distinguishes the present study is that it uses no such periodic external drive. The authors placed spins on a 100×100 square lattice with periodic boundary conditions, connecting each edge to the opposite edge. To make long, large-scale calculations feasible, the spins are treated not as quantum objects but as classical vectors. This spin lattice is coupled to a single resonator mode, modeled as a damped harmonic oscillator in contact with a thermal bath. The only time-dependent influence on the whole system is the random thermal fluctuation originating from the bath.
At first glance, it seems odd that regular oscillation could emerge from random noise. The resonator, however, plays a key role here. White noise from the thermal bath contains components at many frequencies, but a weakly damped resonator responds strongly and selectively to components near its own resonant frequency. As a result, thermal fluctuations that were originally irregular become an "almost periodic signal" after passing through the resonator. This signal is not perfectly periodic, but over a limited time it behaves like a regular oscillation.
In the team's model, in step with the resonator's oscillation, the strength of the Ising interaction, which tries to align neighboring spins, and the strength of the transverse field, which rotates the spins, vary in opposite directions. When the Ising interaction becomes stronger, the transverse field becomes weaker, and conversely, when the transverse field becomes stronger, the interaction becomes weaker. This makes a process that aligns the spins and a process that rotates them appear alternately, approximately reproducing the periodic operations used in conventional discrete time crystals. Unlike a perfectly periodic external drive, however, the amplitude of the signal depends on the temperature of the bath, and the time over which the phase stays aligned is finite. This time is called the resonator's "coherence time," and is given by 2/γ, where γ is the damping rate.
In the numerical calculations, the behavior of the spins fell broadly into three regimes depending on temperature. At low temperature, thermal fluctuations are too weak to affect the spin motion sufficiently, and the spins continue free precession, similar to the wobble of a spinning top. At intermediate temperatures, the spins synchronize to half the resonator's oscillation frequency, and clear period doubling appears. When the temperature is raised further, thermal fluctuations become too strong and disrupt the spin motion, and period doubling is lost within a few cycles. In other words, in this model moderate thermal fluctuations create temporal order, and the order fails to form when they are either too weak or too strong.
For period doubling to appear, the thermal fluctuations must be large enough to switch the strengths of the Ising interaction and the transverse field sufficiently within one resonator cycle. The temperature at which order begins to form therefore has a certain threshold. Heat is normally thought to disrupt regular structure, but in this system, interestingly, raising the temperature conversely produces temporal order. Moreover, the resonator signal itself contains no half-frequency component, so the period doubling arises anew as a response of the spin lattice. According to the authors, the period doubling produced this way is stabilized by interactions among many spins and is maintained over a certain temperature range.
What does it mean for a time crystal to "melt"? The oscillation continues, but the phase is disrupted
The period doubling that appeared at intermediate temperatures had a feature different from that of an ordinary time crystal. The spins keep oscillating regularly, returning to their original state every two resonator cycles, yet the phase of that oscillation is not maintained over long times. Examining individual calculation runs in detail, the team found that during the period doubling, two equivalent oscillation patterns swapped irregularly. For example, spins that had been oscillating as "up, down, up, down" switch at some point to the pattern "down, up, down, up." Both are period-doubled, but their oscillation phases are shifted by half a cycle.
The research team called this phase switching a "time-soliton" and treated it as a topological defect in temporal order. Importantly, the occurrence of a time soliton does not make the period doubling itself disappear immediately. After the phase switches, the spins continue oscillating regularly in the new phase. But when such switches are repeated, oscillations at widely separated times no longer have aligned phases, and temporal coherence over long durations is lost. Coherence here refers to the property describing how long the phase relationship of an oscillation is preserved.
The name "discrete time liquid" arose by analogy with the melting of spatial crystals. In an ordinary crystal, atoms are arranged regularly over long distances, but as defects that disrupt the structure increase, long-range order is lost and the system moves to a liquid-like state. The paper cites the 1978 theory of Halperin and Nelson, which explained the melting of two-dimensional crystals through the proliferation of topological defects, and extends that idea to the time direction. Just as the arrangement of atoms is disordered in a spatial crystal, in a time crystal the oscillation phase switches irregularly. As a result, periodic oscillation can be confirmed over short times, while regularity of phase is lost over long times. The authors named precisely this kind of state a discrete time liquid.
To check whether this interpretation agrees with the numerical calculations, the team also built a simple probabilistic model focusing only on the occurrence of time solitons. In an ordinary time crystal, all spins are rotated by 180 degrees (π) each cycle, but in this model the rotation angle is replaced by 360 degrees (2π) with a certain probability p. A spin rotated by 360 degrees returns to its original orientation, so the flip that should have occurred in that cycle is skipped, and the phase of the subsequent oscillation pattern switches. In other words, the probability p controls how frequently time solitons occur.
The team compared this simple model with the numerical calculations of the spin lattice coupled to the resonator. As a result, the probabilistic model could also quantitatively reproduce the distribution of times over which phase correlations persist in individual calculations. The probabilistic model takes as input the thermalization time measured in the spin-lattice calculations, and the time-soliton density is obtained from correlation times averaged over multiple runs, so there are no free parameters adjusted to fit the results. This supports the idea that the behavior of the complex spin lattice can be explained by the occurrence of time solitons.
Specifically, under the condition that the resonator's coherence time is 20 times the oscillation period, the value corresponding to the average number of time solitons occurring within the thermalization time was about 37. Under this condition, the oscillation phase switches many times before the spin lattice loses its order through thermalization. Also, when the resonator's coherence time is sufficiently shorter than the thermalization time, the correlation time averaged over multiple runs nearly matched the resonator's coherence time. This corresponds to a relationship in which a time soliton occurs about once per twice the resonator's coherence time. Conversely, when the resonator's coherence time is sufficiently longer than the thermalization time, the spin lattice thermalizes before time solitons occur, so a single phase is maintained while the order lasts. This behavior corresponds to a "prethermal discrete time crystal," and how many time solitons occur during the lifetime is an important indicator distinguishing a discrete time crystal from a discrete time liquid.
A phase diagram separating four states: the time liquid persists even at short coherence times
The team repeated the calculations while varying the bath temperature and the resonator's coherence time, and summarized which states appear under which conditions in a phase diagram. In Fig. 1c of the paper, the spin behavior is classified into four categories: "free precession," "discrete time crystal," "discrete time liquid," and "disordered." The key to the classification is the "correlation time," which expresses how long phase correlations persist, and the "thermalization time," the time until the order of the spin lattice is lost through heat absorption. Comparing the two makes it possible to examine whether phase regularity is also maintained while the period-doubled oscillation continues.
| State | Input producing temporal structure | Local period doubling | Phase order over long times | Criterion / feature in the paper |
|---|---|---|---|---|
| Conventional Floquet discrete time crystal (for reference) | Nearly perfectly periodic drive applied externally | Yes | Yes | Stable subharmonic response to a periodic drive |
| Free precession | Resonator with weak thermal fluctuations | No | Not applicable | Low-temperature region identified by a precession indicator |
| Disordered | Resonator with overly strong thermal fluctuations | Lost within a few cycles | No | Averaged correlation time is under 10 resonator cycles |
| Discrete time crystal | Resonator at intermediate temperature | Yes | Maintained for a length close to the thermalization time | Ratio of correlation time to thermalization time exceeds 0.418 |
| Discrete time liquid | Resonator at intermediate temperature | Yes | Lost through time solitons | Correlation time is 10 cycles or more, and ratio to thermalization time is 0.418 or less |
The first row of the table is for comparison only and is not among the four regions classified in this phase diagram. Also, the 0.418 used for the boundary between discrete time crystal and discrete time liquid is not a universal constant of nature but a threshold set by the research team to classify the numerical results.
The classification proceeds in stages. First, an indicator of precession characteristics is used to identify the low-temperature free-precession region. For the remaining conditions, the correlation times obtained from individual calculations are averaged; if this is under 10 resonator cycles, the state is judged disordered. If correlation of 10 cycles or more remains, local temporal order is deemed present, and if the ratio of correlation time to thermalization time exceeds 0.418 it is classified as a discrete time crystal, otherwise as a discrete time liquid. In other words, even if the oscillation itself continues for a long time, when phase correlations are lost in a much shorter time, the state is judged to be a "liquid."
An important result readable from the phase diagram is that as the resonator's coherence time becomes shorter, the temperature range in which a discrete time crystal appears becomes narrower. Under the present calculation conditions, when the coherence time fell below about 130 times the resonator's oscillation period, the region judged to be a discrete time crystal disappeared. The discrete time liquid, on the other hand, was confirmed at all the coherence times examined in the calculations. Even if the resonator's phase is disrupted in a short time, the spin lattice can maintain period doubling itself, but it can no longer maintain a state in which the phase stays aligned for a long time. The authors interpret this result as a phenomenon in which a discrete time crystal turns into a discrete time liquid as the resonator's coherence time shortens.
However, the boundaries separating these states have not been confirmed as physically sharp phase transitions. According to the supplementary material, the distribution of the correlation-time ratio that separates discrete time crystals from discrete time liquids has a valley, but the valley is shallow, and the number of points classified as discrete time crystals changes depending on how the threshold is set. The boundary between the discrete time liquid and the disordered state is even more ambiguous. When the correlation-time threshold is varied from 10 cycles over a range of 3 to 30 cycles, the number of points classified as disordered changes from 256 to 816. The authors explain this as being because local temporal order is gradually lost as the temperature rises. The value of about 130 times is therefore also only a guide based on this model and judgment method, not a boundary common to all physical systems.
A history of time-crystal research: from the 2012 proposal to a new "liquid in time"
The concept of a time crystal was proposed in 2012 by physicist Frank Wilczek. The idea was that, just as an ordinary crystal has a repeating structure in space, there might exist a state with a spontaneous periodic structure in the time direction. Later research, however, showed that this idea faces theoretical constraints. In 2013 Bruno, and in 2015 Haruki Watanabe and Masaki Oshikawa, presented "no-go theorems" forbidding ordinary time crystals in thermal equilibrium. In particular, Watanabe and Oshikawa's work proves that for general Hamiltonians whose interactions are not too strongly long-ranged, time crystals do not exist in the ground state or the canonical ensemble.
This theorem, however, concerns systems in thermal equilibrium, and does not forbid non-equilibrium systems that continue to receive a periodic external drive. Researchers therefore turned to systems in which a periodic drive makes the time-translation symmetry discrete from the outset. In such systems, spins and the like can break the discrete time-translation symmetry by responding at integer multiples of the period of the external stimulus. A theory of discrete time crystals based on this mechanism developed in 2016, and experimental observations were reported in 2017.
| Year | Main result | Paper / journal |
|---|---|---|
| 2012 | Frank Wilczek proposes the quantum time crystal | Physical Review Letters 109, 160401 |
| 2015 | Haruki Watanabe and Masaki Oshikawa prove the absence of time crystals in the ground state and canonical ensemble. Submitted to arXiv in October 2014 | Physical Review Letters 114, 251603 |
| 2016 | Else, Bauer and Nayak develop the theory of discrete time crystals in periodically driven systems | Physical Review Letters 117, 090402 |
| 2017 | Zhang et al. observe discrete time crystals in a trapped-ion chain, and Choi et al. in a dipolar spin system in diamond | Nature (Choi et al.: 543, 221–225) |
| 2021 | Kyprianidis et al. observe a prethermal discrete time crystal in a trapped-ion quantum simulator | Science 372, 1192–1196 |
| 2022 | Kongkhambut et al. observe a continuous time crystal in a continuously pumped atom–cavity system | Science 377, 670–673 |
| 2026 | Wang et al. propose a discrete time liquid from calculations on a spin lattice driven by thermal fluctuations | arXiv:2610.30266 (preprint) |
This table excerpts representative results in time-crystal research and is not exhaustive of related work. Also, the 2017 work by Zhang et al. and Wilczek's 2012 paper are not included in the reference list of the present paper.
Against this background, the present discrete time liquid can be seen as a proposal that further broadens the mechanisms by which temporal order can arise. It is not, however, the first study to create temporal order without applying a perfectly periodic stimulus from outside. The continuous time crystal reported in 2022 was realized through continuous pumping rather than a periodic drive. What is new here is that thermal fluctuations lacking periodicity are selectively amplified by a resonator, and period doubling is produced in the spin lattice as a result. The study further proposes a state different from conventional time crystals, in which that period doubling is partially disrupted by time solitons.
The common ground with prethermal discrete time crystals is also important. Prethermalization refers to a phenomenon in which a relatively stable state persists for a long time before energy is eventually distributed throughout the whole system, for example under an external drive. In the prethermal discrete time crystal observed in 2021, heating was suppressed by a high-frequency periodic drive, and the lifetime of the order grew substantially with the drive frequency. In the present numerical calculations too, under a condition where the resonator's coherence time is extremely long at 200,000 times the oscillation period and time solitons hardly occur, the correlation time of the period doubling grew exponentially with the resonator frequency. This is behavior similar to a prethermal discrete time crystal. On the other hand, under the discrete-time-liquid condition of a coherence time 20 times the period, the thermalization time lengthened as frequency increased, but more slowly than a single exponential.
Note that the research by Shanghai Jiao Tong University on the melting of spacetime crystals, reported by XenoSpectrum on July 28, 2026, also deals with the breakdown of temporal order. However, that is an experiment using discs on a vibrating plate, and differs in both subject and method from the present study of numerical calculations on a resonator and spin lattice. The two are related in considering the formation and breakdown of temporal order, but they are not studies demonstrating the same physical phenomenon.
Can it exist in thermal equilibrium? Theoretical limits and challenges for experiments
What is particularly interesting in the paper is the authors' outlook that a discrete time liquid could exist even in thermal equilibrium. Conventional time crystals face the constraint of no-go theorems stating that ordinary time crystals cannot exist in thermal equilibrium. In a discrete time liquid, however, time solitons occur at irregular times, so even though period doubling remains in individual oscillations, the bias toward a particular phase disappears when the results of many trials are averaged. In other words, while the motion of individual spins has temporal regularity, continuous time-translation symmetry may be restored on average. The authors propose that this property could avoid conflict with the conventional no-go theorems while allowing for the existence of temporal order in thermal equilibrium.
It is important to note here that the present numerical calculations have not demonstrated a discrete time liquid in thermal equilibrium. What was confirmed in the calculations is a metastable state that eventually loses its order through thermalization. The spin lattice gradually absorbs energy under the influence of the resonator, and after a certain time loses both its spatial order and its period doubling. What the authors showed is that, until this thermalization occurs, local period doubling is maintained even though long-time phase correlations are lost through time solitons. The possibility of existence in thermal equilibrium is a theoretical hypothesis derived from this behavior, and does not directly show that the same order persists in an equilibrium state.
This difference also shows in the method used to evaluate correlations in the numerical calculations. The authors examined whether period doubling remains after the oscillation phase has switched irregularly, by averaging products of the magnetization at different times. They found that the signature of period doubling is maintained even at times far beyond the resonator's coherence time, and is lost only as the thermalization time is approached. This is because time solitons swap the oscillation phase but largely preserve the amplitude itself. The time over which phase correlations are lost must therefore be distinguished from the time over which period doubling itself disappears.
Furthermore, the model contains important approximations. First, because the spins are treated classically, it has not been directly confirmed that the same phenomenon occurs in quantum many-body systems. Second, the main numerical calculations ignore the "back-action" of the spins on the resonator. That is, the resonator signal is generated independently of the state of the spin lattice, and thermal fluctuations influence the spins in only one direction. The supplementary material also includes calculations with back-action, confirming that the spin motion changes the statistical properties of the resonator signal, but how this changes the phase diagram remains unclear.
If back-action is included, the resonator may be influenced by the past motion of the spins, and correlations may arise among time solitons. In the current model time solitons occur independently, but if correlations arise, they might line up at regular intervals, or conversely become more likely to occur. In that case, temporal coherence could become either stronger or weaker. The authors also point out that back-action and quasi-periodic fluctuations containing multiple frequencies could give rise to different kinds of discrete time liquids. All of these, however, are subjects for future research.
The paper is a non-peer-reviewed preprint submitted to arXiv on October 21, 2026, and no experimental confirmation has been made. The related data and code are registered on Zenodo, but automated access is restricted. According to the paper, the policy is to provide credentials for access to researchers who request it reasonably for academic research or verification purposes. The current results are based solely on the authors' theory and numerical calculations.
For experimental verification, the team lists two candidates. One is an ultracold Bose gas coupled to an optical cavity, a configuration close to the system in which a continuous time crystal was observed in 2022. Normally a laser directly excites the Bose–Einstein condensate, but one could instead drive the cavity with white noise. In this case the amplitude of the noise corresponds to the bath temperature in the present numerical calculations, so there is no need to actually heat the real system in the same way. The other is a method using "surface phonon polaritons," which arise when lattice vibrations at a material's surface couple with electromagnetic waves. It envisions using strong electric-field fluctuations near the surface to give a two-dimensional array of dipoles with ferroelectric-type interactions an action similar to that in the present model.
In either experiment, a major technical challenge is making the resonator's influence vary both the interaction between spins and the action that rotates them, in an appropriate relationship. The central observable is whether, as the strength of the fluctuations is varied, correlations between spins synchronize to half the resonator frequency. In particular, the authors believe that in cold atomic gases coupled to a cavity it may be possible to image individual time evolutions, allowing direct examination of the moments when time solitons occur and the distribution of intervals between them. If such phase switches are observed in an experiment and the same statistical behavior as in the numerical calculations is confirmed, the interpretation that time solitons form the discrete time liquid could be tested outside the calculations.
This study showed numerically that temporal order can form through thermal fluctuations even without a regular period supplied from outside. It also frames a state in which period doubling itself is maintained but only phase regularity is lost as a "discrete time liquid," and concretely explains how it differs from conventional time crystals. On the other hand, existence in thermal equilibrium and extension to quantum systems remain untested, and analysis of models including back-action and experimental reproduction are indispensable. The focus going forward will be whether time solitons can be observed in real physical systems, and whether this state can exist beyond a temporary metastable state.