A colloidal time crystal and its tempomechanical properties

arXiv:2112.04498 · cond-mat.soft, cond-mat.stat-mech, nlin.AO, physics.flu-dyn, quant-ph · Submitted 2021-12-08 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "A colloidal time crystal and its tempomechanical properties".

Mira: Here's the gist of the paper. The gist: Active colloidal particles can form classical continuous time crystals by spontaneously breaking time-translational symmetry, exhibiting complex nonequilibrium properties and tempomechanical behavior.

Kai: Introduction to Time Crystals and Active Matter A time crystal is characterized by spontaneously breaking timetranslational symmetry and forms an analog to spatial crystals that break translational symmetry in space.

Mira: Similar to spatial crystals, time crystals form a state of matter.

Kai: Time crystals cannot exist in equilibrium because they are a nonequilibrium phenomenon and never reach thermal equilibrium.

Title and authors: Kai: So, diving into the specifics of "A colloidal time crystal and its tempomechanical properties," we see that Marina Evers and Raphael Wittkowski are the authors.

Mira: I think it’s important to remember they are bridging two fields, active matter physics and condensed matter theory, which is where this paper lives.

Lev: From an engineering standpoint, I'm interested in how complex these self-organized states can be when you try to build them with real hardware.

Kai: The title itself tells you the scope: it’s not just about the movement, but also the physical properties of this organized state of matter.

Mira: It sets up a lot of expectation for what we might see in these systems, linking their dynamic behavior to macroscopic material science concepts.

Lev: For us, it means we need to consider how robust these time-crystalline structures are when we introduce real noise and imperfections into the system.

The paper's summary: Kai: So what they actually found is that seven spherical particles, or a crystallite, can self-organize into this periodic motion if they are arranged in a specific hexagonal pattern and set at a distance of d = 2 point 2R <ref:2112.04498#pg2>.

Mira: They model these particles as squirmers—which is the standard way we describe active swimmers—and when they are pushers with a squirming parameter beta of negative three, they get this periodic rotational motion.

Lev: So, the specific configuration they use involves pushers and that distance d = 2 point 2R, and it results in a central particle rotating clockwise at a constant speed <ref:2112.04498#pg2>.

Kai: That rotation is the hallmark of the time crystal state they observe after some initial transition phase, and it’s described by a simple two-dimensional spherical motion in twenty-one-dimensional phase space <ref:2112.04498#pg3>.

Mira: The paper confirms that this behavior fulfills all four criteria for what we define as a time crystal, which is pretty significant because it shows this isn't just an artifact of the setup.

Lev: It’s interesting that they specify the distance d = 2 point 2R because that seems to be the critical parameter for observing this ordered state, and they mention that for a larger distance of d = 2 point 4R, you still see motion but it gets more fluctuating <ref:2112.04498#pg2>.

The paper's improvements: Kai: The authors suggest a few ways to improve the understanding of this system, and one big thing is focusing on how the interaction between particles dictates the state of matter.

Mira: They point out that self-organization and time crystallinity are only observed for pushers, which narrows down the types of particles we need to consider for this specific phenomenon.

Lev: That helps a lot with simulation; you don't have to test every single type of swimmer; you can focus on the ones that actually show the desired behavior.

Kai: They also discuss how they could refine their state diagram by looking at different particle distances, showing that d = 2 point 2R gives them a clean periodic state, but d = 2 point 4R introduces clear fluctuations instead of perfect order <ref:2112.04498#pg2>.

Mira: And they mention that the interactions are crucial; specifically, the torque exerted on a second nearest neighbor is only about one point five percent of what you apply to a first nearest neighbor, which shows how important those local forces are.

Lev: That one point five percent figure is telling for real hardware; it means the interactions have to be tuned very precisely for this crystal to stay stable against noise, even if the Brownian motion is small compared to the propulsion speed.

Conclusion: Kai: So, to wrap up on "A colloidal time crystal and its tempomechanical properties," the main implication is that a self-assembling assembly of squirmers can create a classical continuous time crystal when you keep the particle distance and the squirming parameter small enough.

Mira: The paper really shows how these non-equilibrium states are not just theoretical curiosities; they are actual physical configurations that exhibit measurable, predictable mechanical properties.

Lev: For real systems, this means we need to focus on tuning those local interactions—like that torque function Tz(alpha, d, beta) = one/(ca zero + ca 1d) (alpha) + beta / (cb zero + cb 1d) (two alpha) —to make sure the structure is robust.

Kai: Right, so the next step they suggest is to calculate the state diagram with higher resolution and look at even larger ranges of parameters to see where this crystal state persists.

Mira: And for future work, they propose making this experimental by arranging microswimmers on a hexagonal lattice using optical or acoustical tweezers to see if we can realize this colloidal time crystal in a lab.

Lev: I agree, experimentally realizing it with tweezers would be the ultimate test of whether these theoretical predictions hold up when you move from simulation to a physical setup.

Kai: It’s definitely something worth watching as researchers try to build these active assemblies with actual microswimmers.

Marina Evers, *Raphael Wittkowski

Institut für Theoretische Physik, Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster

cond-mat.soft, cond-mat.stat-mech, nlin.AO, physics.flu-dyn, quant-ph

Submitted: 2021-12-08

Updated: 2021-12-08

Comments: 16 pages, 12 figures, 2 tables

Journal ref: Physica Scripta 98, 125240 (2023)

DOI: 10.1088/1402-4896/ad05ab

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 76/100

The gist: The gist: Active colloidal particles can form classical continuous time crystals by spontaneously breaking time-translational symmetry, exhibiting complex nonequilibrium properties and

Key concepts

Time Crystal: A state of matter that spontaneously breaks time-translational symmetry. Unlike normal crystals that break space symmetry, this state exhibits periodic behavior in time, meaning its properties repeat at regular intervals over time. It is a nonequilibrium phenomenon that cannot reach thermal equilibrium.
Active Colloidal Particles (Squirmers): These are microscopic particles capable of consuming environmental energy to move and self-propel. They are modeled as squirmers, which can be 'pushers' or 'pullers' based on their propulsion mechanism ($eta$ parameter). Their active nature is key to inducing the exotic time-crystalline behavior.
Temporal Stress-Strain Diagram: This diagram maps how the material responds mechanically under time-dependent stress. By measuring how the period length ($ au_f$) changes relative to a reference period ($ au_0$) when subjected to external torques, researchers can characterize the material's mechanical properties, showing distinct behaviors under compression and tension.

Summary

The gist: Active colloidal particles can form classical continuous time crystals by spontaneously breaking time-translational symmetry, exhibiting complex nonequilibrium properties and tempomechanical behavior.

Introduction to Time Crystals and Active Matter

A time crystal is characterized by spontaneously breaking timetranslational symmetry and forms an analog to spatial crystals that break translational symmetry in space. Similar to spatial crystals, time crystals form a state of matter. Time crystals cannot exist in equilibrium because they are a nonequilibrium phenomenon and never reach thermal equilibrium. Active colloidal particles are nano- or microparticles characterized by the capability to consume energy from their environment and to use it for self-propulsion. These active particles have been found to exhibit a fascinating nonequilibrium behavior where various exotic properties can arise. The authors establish a concrete connection between active colloidal crystals and time crystals.

Formation of the Time Crystal State

The paper investigates a small active crystallite consisting of seven spherical particles with radius R placed in form of a hexagon with one central particle and surrounded by a liquid. The particles are modelled as squirmers, which today is a standard model for active swimmers. For an active crystallite formed by pushers with squirming parameter β = −3, the system observed a periodic rotational motion after an initial transition phase. This periodic motion is mainly characterized by the rotation of the central particle which rotates clockwise. The system can be described by a simple two-dimensional spherical motion in the 21-dimensional phase space. This periodic motion spontaneously breaks the continuous time-translational symmetry in a way that is similar to the breaking of translational symmetry occurring in an ordinary crystal in space. All four criteria for a time crystal are fulfilled by this active crystallite.

State Diagram and Parameter Dependence

The state diagram shows that time crystallinity forms a state of matter and should not require fine-tuning the system’s initial conditions or parameters. The occurrence of this periodic state does not rely on particular initial conditions. The type of interaction between the particles is crucial for forming a time crystal, as self-organization and time crystallinity are observed only for pushers. For pushers, different states occurring for different particle distances can be distinguished. Time crystallinity with periodic motion is observed for the smallest considered particle distance d = 2.2R. For a larger particle distance of d = 2.4R, the motion is almost periodic but with clearly visible fluctuations. When comparing this state to classical systems, it can be interpreted as a dynamical coexistence between ordered and unordered (temporal) regions.

Stability Against Fluctuations and Interactions

The system is stable against fluctuations because for the periodic state, no such behavior was observed in simulations. The overall motion of the system is robust against Brownian motion if it is small compared to the propulsion of the particles. The torque exerted on a second nearest neighbor is, on average, only 1.5% of one applied on a first nearest neighbor. This shows that the interactions between the particles are crucial for forming a time crystal.

Temporal Stress-Strain Diagram

The authors investigate tempomechanical properties by determining a temporal stress-strain diagram. The change in period length τf −τ0 is the answer of the system and corresponds to the strain in the classical case. For negative and decreasing Text, τf declines and asymptotically approaches τf = 0, which corresponds to the mechanic behavior of an ordinary material under compression. For approaching intermediate positive values of Text from above, the curve runs towards strongly negative values of τf <ref:11pg2, when approaching the intermediate values from below, the curve runs towards large positive values of τf <ref:2112.04498#pg3>.

Conclusion and Future Work

The investigation reveals that an assembly of squirmers can self-assemble a classical continuous time crystal when particle distance and squirming parameter are sufficiently small. Future work should continue by calculating the state diagram with higher resolution and for larger ranges of parameter values. Experimental realization is proposed by arranging microswimmers on a hexagonal lattice with optical or acoustical tweezers. The spatial configuration of the microswimmers studied could also be achieved by self-assembly when the particles are equipped with suitable interactions.

Methods Overview

The active particles are modeled as spherical squirmers using a model proposed by Lighthill and Blake. The distinction between pullers, neutral squirmers, and pushers is possible via the squirming parameter β = B1/B2. Hydrodynamic interactions are approximated by the Stokes equation because the Reynolds number Re is much smaller than one.

Improvements for AI systems

The improved AI system can perform predictive modeling of non-equilibrium matter by identifying conditions for classical continuous time crystals. This allows the system to predict when active colloidal particles will self-organize into a state exhibiting spontaneous symmetry breaking in time, based on parameters like particle distance and squirming parameter.
The AI can develop tools for analyzing complex phase transitions in active systems by calculating order parameters such as the time average of the minimum distance of the 21-dimensional system trajectory u(t) to itself, which is defined as "¯δ = 0 applies to periodic states and ¯δ > 0 to non-periodic states."
The system can simulate and predict the temporal stress-strain behavior of these novel materials by calculating a temporal stress-strain diagram based on external torques, allowing it to characterize the material's mechanical response, specifically observing how the period length τf changes relative to τ0 for various external torques.
The AI can optimize particle design for specific active states by using the derived interaction torque function, Tz(α, d, β) = 1/(ca0 + ca1d) sin(α) + β / (cb0 + cb1d) sin(2α), to tune the system's behavior across different parameter regimes.
The improved AI can generalize its understanding of time-crystalline stability by incorporating larger systems, confirming that the time-crystallinity must also hold for larger systems, thus ensuring the predicted state is robust against boundary effects in extended colloidal assemblies.