Tunable colloidal swarmalators with hydrodynamic coupling

Domain: physics
Published: 2025-12-08
Editorial outcome: minor_revision

Abstract
Swarmalators—entities that combine swarming with synchronization—offer a powerful framework for understanding systems where spatial organization and internal degrees of freedom are bidirectionally coupled. Such interplay arises in diverse natural and engineered systems, from Japanese tree frogs and magnetic domain walls to robotics swarms. In contrast to an established theoretical framework, experimental realizations with tunable coupling between motion and phase remain elusive. Here, we present a controllable swarmalator system based on feedback-controlled self-propelling colloidal particles orbiting around reference points and interacting via hydrodynamic flows. We show that synchronization and spatial dynamics co-evolve, giving rise to collective states including synchronized clusters, rotating aggregates, and dispersive phases using a single control parameter. A rapid change of this parameter between regimes of attractive and repulsive phase-mediated interactions yields dynamic regimes inaccessible to systems with static interactions. Simulations incorporating squirmer and lubrication forces support our findings. We also find a new interaction channel through synchronization-dependent forces perpendicular to the connection axis between swarmalators. In general, our platform provides a versatile testbed for probing swarmalator physics but also offers novel strategies for the design of self-organizing active matter.

Introduction
Understanding the emergence of collective behavior from local interactions is a central question in the study of complex systems. In nature, large-scale spatio-temporal patterns frequently arise without centralized coordination as seen, e.g., from the coordinated motion in animal groups1–3 or the synchronized flashing of fireflies4,5, chirping of crickets6 or waving of crabs7. Two foundational models—the Vicsek model, which captures spatial organization via local alignment8, and the Kuramoto model, which describes phase synchronization among coupled oscillators9—have provided key insights into such emergent phenomena. However, many real-world systems feature an inseparable interplay between spatial and oscillatory dynamics, rendering these models individually insufficient. Examples include chorusing frogs10, swimming sperm cells11, interacting magnetic colloids12, or magnetic domain walls13, where agents not only exhibit translational motion but also internal oscillations which are mutually coupled. To address this type of behavior, the concept of swarmalators (swarming oscillators), i.e., agents that exhibit both collective motion and synchronization, has been proposed14. Bidirectional coupling between these modes, where spatial proximity alters phase dynamics and phase differences modify motion, yields rich behaviors such as synchronized clustering, rotating waves, and ring-like assemblies15,16.

Although several of the above-mentioned systems exhibit phenomenological signatures of swarmalator behavior, the underlying mechanisms linking synchronization to motion in these systems are usually not clear and, accordingly, cannot be controlled. Understanding the coupling mechanisms in more detail is not only crucial to describe experimental swarmalators within existing theoretical frameworks but also important for spatio-temporal control in swarm robotics17–19, self-assembled micromachines20–23, and neuromorphic or reservoir computing24,25.

In this work, we present an experimental realization of a swarmalator system using feedback-controlled self-propelled colloidal particles (denoted as active Brownian particles (ABPs)) with independently tunable spatial and oscillatory dynamics. Active particles are especially well suited for realizing swarmalators, since they pose a comparably easy way of manipulating high numbers of moving agents and because the interactions between active particles have yielded basic swarmalator characteristics before12,26. Hydrodynamic interactions at high particle density mediate strong coupling between motion and phase. By tuning a single control parameter, we observe transitions between tightly synchronized, dense clusters and loosely connected, weakly synchronized states, in good agreement with numerical simulations. Our explicit understanding of the coupling mechanism enables us, for the first time, to describe and understand our system within the theoretical framework of swarmalators. Crucially, our system exhibits not only synchronization-dependent attraction, but also lateral interaction forces that drive collective rotation dynamics reminiscent of rotating biological clusters, such as those formed by bacteria or starfish embryos27,28. Furthermore, because the timescales for synchronization and clustering of the swarmalators are clearly separated, we can dynamically modulate their interaction strength to access non-equilibrium swarmalator states inaccessible under static conditions. Our results demonstrate a controllable experimental platform for swarmalators and expand the theoretical framework to include previously unexplored physical interactions.

Results
Synchronization of oscillating ABPs
To experimentally realize oscillators, we employed ABPs whose propulsion directions are individually controlled by an incident laser beam29–31. The ABPs we used were made from carbon-coated silica spheres (radius a=3μm) suspended in a near-critical water-lutidine mixture. Illumination with a focused laser heats each particle slightly above the critical temperature of the solvent, inducing local demixing and self-propulsion. Crucially, when the laser is laterally offset from the particle center, propulsion occurs opposite to the offset direction (Fig.1a; for details, see Methods section “Experimental realization of oscillating ABPs”). To encode a phase into each ABP’s motion, we programmed them to perform quasi-circular oscillations around designated target positions. This was implemented by continuously steering each particle in the (memorized) direction of its reference position qi using a control signal delayed by one timestep δt (Fig.1b; Methods section “Experimental realization of oscillating ABPs”). As a consequence of this control scheme, each ABP performs an orbiting motion around its reference position qi32,33 with the oscillating phase defined by θ=tan−1 uy=ux and the spatial components ux and uy of the propulsion velocity u. In our experiments, the time delay was set to δt=0.4s, which is identical to the time interval between recordings of the particle positions. In combination with a propulsion speed of |u|≈2μm/s, this yields in an average oscillation radius of R≈1μm (Fig.1c). The initial direction of rotation is random for each particle and may spontaneously reverse due to thermal fluctuations.

When operating several such oscillators with small distances |qij| between their reference points, their oscillation becomes highly synchronized (see Supplementary video1). Figure1d shows experimental snapshots of synchronized oscillators with their trajectories colored according to time, where the synchronization is visible in the form of particles moving in the same directions at the same time. Such synchronization is due to hydrodynamic interactions as discussed in detail below. As a measure of synchronization σi of oscillator i with its neighbors j, we use

σi=1Nj∑jcosθi−θjT: (1)

(see Fig.1e). Here, 〈. . . 〉T denotes the average over time intervals T=10δt and Nj then the number of neighbors j of oscillator i. Accordingly, σi can vary between +1 and −1, corresponding to in- and out-of-phase oscillations, respectively. Figure1f shows an array of about 250 oscillators, with their reference positions arranged on a hexagonal lattice at a spacing of |qij|=12μm. The ABPs are color-coded according to their local synchronization σi. The experiments are complemented by numerical simulations incorporating the same geometry and hydrodynamic interactions between ABPs (see Methods section “Simulation model” and SI section 5). At short inter-oscillator distances, the system exhibits large synchronized domains interspersed with only narrow domain walls of weak coherence. As the spacing |qij| between neighboring reference positions increases, the mean synchronization across the array decreases (Fig.1g; Supplementary video1). This spatially dependent synchronization is a hallmark of swarmalator systems.

To probe the underlying hydrodynamic coupling mechanism, we measured the velocity of a passive colloid placed near a moving ABP as a function of its relative position (Fig.1h, i). The resulting strongly anisotropic velocity field and the associated force distribution indicates that hydrodynamic interactions mediate the coupling, consistent with prior observations in colloidal oscillators11,34,35. The measured flow field agrees well with a superposition of a squirmer model (with active stress parameter β=0.25) and short-range lubrication forces (Fig.1j; Methods section “Simulation model” and SI section 5). Incorporating this interaction model into simulations quantitatively reproduces the experimentally observed distance-dependent synchronization (Fig.1g, snapshots in Fig.S1, for more details about the hydrodynamic synchronization see SI section 4).

Motile oscillators
In addition to phase oscillations, a characteristic feature of swarmalators is a translational degree of freedom, with mutual coupling between position and phase. Experimentally, we implement this by allowing the ABP’s reference positions qi,t to evolve dynamically in time. At each timestep δt, they are updated according to

qi,t+δt=qi,t+ΓRi,t−qi,t: (2)

Here, di,t=Ri,t−qi,t is the displacement between the time-averaged particle position Ri,t=1T∫t−Tt ri,t and the reference position qi,t and Γ is a freely adjustable coupling parameter. As a result of this rule, the reference positions acquire a time-varying velocity that depends on the displacement di,t and is therefore sensitive to the orbiting dynamics of the ABPs. The averaging window T is chosen to slightly exceed a typical oscillation period (6–8δt) and is consistent with that used to compute the synchronization parameter σi in equation (1); moderate variations in T do not qualitatively affect the behavior of the system. To prevent phoretic clustering of the ABPs, we include a short-range repulsive interaction between neighboring reference positions as usual in swarmalator models (see Methods section “Reference position update”).

Figure2a,b illustrate the dynamics resulting from equation (2): an ABP circles around a motile reference point, which itself moves. The mean squared displacement (MSD) provides a quantitative measure of activity36, and we use it to characterize the mobility of the swarmalator population and its dependence on the coupling parameter Γ. The MSD of isolated oscillators for different Γ values is shown in Fig.2c. For Γ=0, reference positions remain fixed, and the MSD shows oscillations at short times and eventually saturates. For finite values of Γ, the reference positions move due to fluctuations in the mean position Ri,t of the ABPs. This leads to an increasing value of the MSDs at large times with a Γ-dependent exponent α. Figure2d shows that α, obtained from fits to simulated MSDs, increases from 0 to 1 with increasing |Γ| (for Γ>-0.2), inducing ballistic motion and quenching of oscillations. Because the ABPs cease to oscillate in this regime, we restrict our analysis to Γ≥−0.2 in the subsequent sections.

Synchronization-dependent swarmalator motion
So far, we have shown that ABP oscillations synchronize as a function of the inter-oscillator distance |qij|, with reference positions evolving dynamically according to equation (2). What fundamentally distinguishes swarmalators from simple motile oscillators, however, is the bidirectional coupling between phase synchronization and translational motion. In our system, this coupling arises naturally through hydrodynamic interactions, particularly pronounced at high particle densities. Figure3a,b show experimental and simulation snapshots for coupling values Γ=0.2 and Γ=−0.1, respectively. In both cases, systems were initialized with identical ABP densities and evolved for over 6000δt. For Γ=0.2, oscillators mutually attract, leading to the emergence of dense, highly synchronized clusters. In contrast, for Γ=−0.1, this cohesion breaks down: the system expands, the particle density decreases, and synchronization is lost (also see Supplementary video2).

Figure3c shows that the experimentally measured fraction fc of oscillators within clusters (defined as those with at least one neighbor within |qij|≤15μm distance) increases continuously with Γ, consistent with simulations. Notably, isolated oscillators display diffusive motion for both positive and negative Γ (Fig.2d), indicating that the observed structure formation at high densities must arise from interactions between oscillators. Considering the reference point update rule equation (2), this behavior results from the displacement di,t of oscillating ABPs from their reference positions.

A defining feature of swarmalators is that their motion depends jointly on their distance and phase synchronization. To probe this relationship, we analyzed di,t as a function of both the reference point separation |qij| and the pairwise synchronization σij. Figure3e presents the average longitudinal displacement component d∥,ij of di,t, projected along the direction q^ij to neighboring oscillators. The data reveals a clear dependence of d∥,ij on both distance |qij| and synchronization σij, confirming that motility in our system is jointly modulated by spatial proximity and synchronization—consistent with theoretical swarmalator models. FigureS2 shows a graph of d∥,ij against σij averaged over distance |qij| between 8 and 15μm. When two oscillators i and j are negatively synchronized, the longitudinal displacement d∥,ij is negative, indicating that each ABP is, on average, displaced away from its neighbor relative to its own reference position. Conversely, for positively synchronized pairs, ABPs are displaced toward each other (i.e., d∥,ij>0). The magnitude of this displacement decays with increasing separation between reference positions. This behavior provides a mechanistic explanation for the Γ-dependent clustering observed in Fig.3a,b. When two oscillators are synchronized, their ABPs tend to shift toward each other, yielding positive d∥,ij values (Fig.3d). For positive Γ, this displacement drives the reference positions closer together via equation (2), thereby promoting aggregation. In contrast, for negative Γ, the reference points move opposite to the displacement, causing synchronized oscillators to effectively repel. This interpretation is supported by direct measurements of the relative velocities between neighboring reference positions (Fig.3f), which vary systematically with Γ, confirming the predicted dependence of effective interactions on the sign of Γ. Simulations based on the hydrodynamic interaction model detailed in Section “Synchronization of oscillating ABPs” reproduce the experimentally observed trends in both ABP displacement and relative reference-point velocity (Fig.S3).

Moreover, the nature of the interaction between oscillators depends sensitively on the character of their hydrodynamic propulsion mechanism: whether they behave as pushers, pullers, or neutral squirmers (Figs.S9, S10). This underscores the critical role of the hydrodynamic flow fields generated by the orbiting ABPs34,37.

Our results show that the motile oscillators exhibit synchronization-dependent motion and a distance-dependent synchronization. Taken together, these features are hallmarks of swarmalator behavior, consistent with theoretical studies. Standard swarmalator models are characterized by two parameters, the synchronization strength K, and the spatial interaction parameter J, controlling the synchronization-dependent attraction (J>0) or repulsion (J<0)14,38,39. For constant, distance-independent attraction, theory predicts that swarmalators with K>0 and J>0 form a single, dense, synchronized cluster14. However, when attraction decays with distance—as in our system—smaller, isolated synchronized clusters are expected39,40, in agreement with our experimental observations for Γ>0 (Fig.3a). The regime J<0 corresponding to Γ<0 is rarely explored theoretically. In our experiments, it leads to dispersed, weakly synchronized swarmalators.

Swarmalator model
Beyond a simple qualitative comparison, our quantitative characterization of the interactions among swarmalators enables us to model their dynamics within the general swarmalator framework. Specifically, we represent the phase coupling between oscillating colloids as a distance-dependent, Kuramoto-like coupling term (for more details, see SI section 4). With the oscillation frequency ω0 of isolated swarmalators, the corresponding equation of motion for the phase can be written as

θ_i(t)=ω0+∑j≠i|qij|−3Ksin〈Δθij(t)〉+ξi,θ(t): (3)

Here, Δθij(t) is the phase difference between neighboring swarmalators and ξi,θ(t) Gaussian, δ-correlated noise. Opposed to frequently used swarmalator models with long-range coupling14, hydrodynamic interactions results in a shorter-ranged |qij|−3 decay of the synchronization41.

As demonstrated in the previous section, the swarmalators in our experiments exhibit attraction or repulsion depending jointly on their relative position and phase synchronization. This interaction is mediated by the displacement of the ABPs. We find that the displacement d∥,ij can be described by the empirical relation d∥,ij∝|qij|−2sin(πσij). With the assumption that the net displacement di,t is the superposition of displacements in the direction of neighbors, i.e., di,t=1Nj∑j d∥,ij q^ij, the motion of swarmalators can be described as

q_i(t)=1Nj∑j≠i|qij|−3Jsin(πcos(Δθij(t)))+Frep,ij(t)+ξi,q(t): (4)

ξi,q(t) is Gaussian, δ-correlated noise, and Frep,ij adds a repulsive force for small distances (see Methods section “Reference position update”). For more details on the motivation of equation (4), see SI section 7. By integrating equations (3) and (4) with K>0 and varying J, we recover both the synchronized clusters and the dispersed states observed in experiments and simulations (Fig.4; Fig.S4; Supplementary video3). The model also enables exploration of anti-synchronizing interactions (K<0), which are not accessible in our experiments due to the synchronizing interactions of the oscillators. Hydrodynamic anti-synchronization is, however, in principle possible and has been reported in other colloidal systems34,42–44. For K<0 and J>0, the system evolves into a dispersed, weakly synchronized state. In contrast, when both synchronization and spatial interaction are repulsive (K<0, J<0), swarmalators self-organize into chain-like aggregates in which adjacent particles oscillate in antiphase. We can rationalize these effectively one-dimensional structures considering that denser configurations lead to frustrated phase arrangements, destabilizing synchronization, and inducing repulsion between synchronized pairs. Similar alternating phases have also been observed in numerical simulations of swarmalators with entirely different coupling and interaction mechanisms45. This suggests that the states observed in Fig.4 are rather generic beyond the specific swarmalator realization in our study. By using synchronization, cluster size, and cluster asphericity as order parameters, the data for different J and K values separate into four distinct phases: synchronized clusters, dispersed-synchronized, chain-like anti-synchronized, and dispersed anti-synchronized states (see SI section 8).

Outlook
Previous numerical studies of swarmalator systems have predominantly focused on quasi-static pattern formation, where steady-state configurations emerge rapidly. In contrast, our experimental system evolves on markedly longer timescales. Notably, the two principal observables—phase synchronization (σ) and the fraction of swarmalators in clusters (fc)—exhibit different relaxation dynamics (Fig.5a). Following a change in the coupling parameter Γ, synchronization decays rapidly (within a few oscillation periods τosc), whereas structural reorganization, as measured by fc, proceeds over tens of τosc. This timescale separation gives rise to non-trivial state trajectories when Γ is modulated dynamically. In particular, switching Γ between positive and negative values generates a closed loop in the (σ, fc) state space (Fig.5b), where the system transiently explores configurations that are inaccessible under static coupling. These intermediate states depend on the switching direction, reflecting a form of hysteresis. At higher switching rates, the system no longer equilibrates between transitions, but remains in a transient intermediate state, displaying a form of dynamic memory (Fig.5c, d). Since this effect is a direct consequence of the asynchronous relaxation of spatial and phase degrees of freedom, we expect it to be a generic feature of swarmalators, which offers additional possibilities regarding the control of swarmalator systems46.

Besides gaining control over swarmalator systems, another important direction in swarmalator research is the search for new types of interaction that can expand the range of phenomena these models can explain47,48. The swarmalators in our experiments reveal a qualitatively new interaction channel in the form of synchronization-dependent forces, which are not limited to the axis connecting swarmalators to which theoretical models are usually restricted. We find that the hydrodynamic interactions also give rise to lateral forces, i.e., forces acting orthogonal to the connecting axis, arising from the angular component of the flow field. Similar to the parallel displacement, which gives rise to the synchronization-dependent attraction (see Fig.3), this lateral displacement results in a rotational motion of swarmalators around each other. This kind of rotational motion originating from hydrodynamics is observed in multiple biological systems, like groups of starfish embryos27, but has not been reproduced in swarmalator models before. In the phenomenological model in the section “Swarmalator model”, we have omitted these lateral forces for simplicity, but this effect can be readily incorporated by adding another term to equation (4). Figure5g,h shows a simulation snapshot of a rotating cluster of swarmalators and the size and Γ-dependent angular velocity. The rotation speed of the cluster can also be tuned by Γ, which couples the hydrodynamic interactions of the swarmalators to their motion. Importantly, the individual particles have no intrinsic chirality; instead, symmetry is broken through noise-assisted flips in rotation and synchronization mediated by hydrodynamic interactions, resulting in an emergent global rotation of the entire cluster with the same sense as its constituent swarmalators (see Fig.5e, SI section 9). Incorporating lateral forces into swarmalator models fundamentally enriches the emergent phenomenology, enabling the formation of dynamic rotational states reminiscent of biological systems, such as vortical structures in bacterial colonies and starfish embryo aggregates27,28. Our results, therefore, expand the physical scope of swarmalator dynamics and underscore the need to include anisotropic interaction channels in future theoretical frameworks.

Methods
Experimental realization of oscillating ABPs
The Janus colloids we used as ABPs were prepared by coating commercially available silica particles (radius 3μm) with 80nm of carbon on one hemisphere. We dispersed these particles in a binary mixture of water and 26.8wt% 2,6-Lutidine, which has a critical demixing temperature of 34°C. The dispersion was placed inside a quartz glass sample cell with a height of 200μm, which was kept at 28°C. Focusing a laser with a wavelength of 532nm on these particles heats up the carbon cap, which leads to anisotropic demixing of the critical mixture around them and self-phoretic motion. The propulsion speed can be controlled via the laser intensity. We controlled the motion direction by offsetting the center of the laser spot from the particle center by 2.4μm. Using an acousto-optical deflector, we scanned the laser beam over the particles at 100kHz, allowing us to move multiple hundreds of ABPs in parallel. To achieve continuous steering of the particles, we captured microscope images, detected the particle positions, and steered each particle towards its assigned reference position in a feedback loop at 2.5Hz. As the particles move after the image capture at time t, the laser spots, which we apply at t+δt (δt=0.4s), push them in directions misaligned with the reference position. This kind of time delay and direction mismatch in the steering signal leads to a continuous oscillating motion of the ABPs around their reference points32.

The ABPs used in this study do, in principle, show negative phototaxis, i.e., their uncapped side and therefore their swimming direction reorient to point down light intensity gradients with reorientation times in the order of 1s30. However, we find that in our experiments, where we change the driving direction of each ABP with 2.5Hz (i.e., a delay time of δt=0.4s), the colloids cannot reorient sufficiently fast and are most of the time oriented with their carbon cap pointing in the direction orthogonal to the substrate they are moving on. As a result, the Janus colloids employed as ABPs here essentially behave like isotropic active colloids.

The relaxation time of the propulsion mechanism of the ABPs is around 0.1s30. Since the time δt in between changes of the laser spot positions is larger than this propulsion relaxation time, the ABPs react quasi-instantaneously to changes in the driving direction.

To compensate for the mechanical instabilities of the optical setup that result in slight misalignments between the laser focus and the ABP positions, we carried out periodical recalibrations throughout all our measurements.

Simulation model
The motion of the swarmalators is described by an ABP model of self-propelled particles with hydrodynamic interactions. An ABP experiences a propulsion force fp acting along an orientation vector ei. The motion of such an ABP at position ri is governed by

m€ri=fpei−γr_i+√2Dtγζi+FHI(ri), (5)

where γ is the damping coefficient related to the fluid viscosity, Dt is the thermal translational diffusion coefficient of the particle, and ζi is a Gaussian random process with 〈ζi(t)〉=0 and 〈ζi(t)ζj(t0)〉=δijδ(t−t0). For a particle speed of u0, the required propulsion force is given by fp=γu0. The hydrodynamic force FHI(ri) is given by the fluid velocity generated by the other swarmalator. Here, the flow field UHI(ri) is given by

UHI(ri)=∑jUflow,j(ri−rj)+Ulub,j(ri−rj) (6)

where the sum is over all particles j within some cutoff distance RHI, i.e., |ri−rj|<RHI. The first term Uflow,j is the hydrodynamic force generated at the position of particle i due to the self-propulsion of the particle j, and is modeled using the squirmer flow field49,50

Uflow,j(r)=−p/r2[1−3(ej·r^)^2]r^−s/r3[ej−3(ej·r^)r^] (7)

with r=ri−rj, r=|r| and r^=r/r. The first term decaying as 1/r2 is due to a force dipole of strength p=−3βu0a2/4, and the second term corresponds to a source dipole of strength s=u0a3/2, where β is the squirmer parameter that measures the asymmetry in the flow field and a is the particle radius. For β>0, we have a puller, and for β<0, we have a pusher. For light-activated colloids, we expect no strong dipole contributions, i.e., |β|<1.

In addition to the squirmer (far-field) interactions, we also include (near-field) lubrication forces between swarmalators51, as squirmer interactions alone are insufficient to account for the strong hydrodynamic interactions observed in the experiment (see SI section 2; Fig.S6). The flow field related to lubrication forces is given by

Ulub,j(r)=−h(r_·r^)Θ(r)r^ (8)

where r_=r_i−r_j is the relative velocity between particle j and i, h=r−2a is the shortest distance between the surface of the two particles, and Θ(r)=exp(−r/R0) is a cutoff function whose decay radius R0 is determined by matching the data to the experiment (see SI section 2; Fig.S6).

The ABP orientation vector ei(t) at any time instant t is determined by the position of the laser spot li(t), so that ei(t)=(ri(t)−li(t))/|ri(t)−li(t)|. By choosing the laser spot position li(t) such that the particle moves towards the reference position qi,t, we are able to closely replicate the experimental conditions and obtain very good agreement for the single particle motion (see SI section 1; Fig.S5). The laser spot is updated every delay time δt, with the update rule

li,t+δt=ri,t+δt−l0(qi,t−ri,t)/|qi,t−ri,t| (9)

where l0 is the distance from the particle center at which the laser is applied, and qi,t−ri,t is the displacement between the particle's position and the reference position. The system lies in the regime of low Reynolds number, so that we simulate equation (5) in the over-damped regime, i.e., m/γ≪δt, so that effectively

r_i≈v0ei+√2Dtζi+UHI (10)

The equation of motion (5) is solved using the Velocity-Verlet integration scheme52. All the simulation parameters along with the corresponding experimental values are collected in Table 1.

Reference position update
To couple the motion of the swarmalator to its rotational oscillations, a reference-point update rule based on the swarmalator past trajectory is implemented, with reference position qi,t updated every delay time as

qi,t+δt=qi,t+Γ(Ri,t−qi,t) (11)

where Ri,t=1T∫t−Tt ri,t is the average position of the swarmalator measured over the last T timesteps, and Γ controls the strength of the displacement of the new reference position. Moreover, to prevent the reference positions from coming too close together—which could cause collision between particles—an exclusion algorithm between the reference positions is implemented following a Monte-Carlo procedure. Thus, every reference update qi,t is accepted with probability

Pi=exp(−ΔE/kbT) (12)

where

ΔE=∑i≠j VLJ(|qi,t−qj,t|−VLJ(|qi,t−δt−qj,t−δt|)) (13)

and VLJ is the truncated Lennard-Jones potential given by

VLJ(r)={4ε(Rmax/r)12 r≤Rmax 0, r>Rmax (14)

Here Rmax is the minimum desired reference-point separation. The particle distribution around the reference points indicates that particles can reach distances of up to 2.0μm from their reference positions (see Fig.S5a), thereby suggesting a minimum center-to-center separation of Rmax=10μm (particle radius 3μm) for the reference points to avoid overlapping ABPs. We therefore set ϵ/kbT=0.2, so that while particles typically maintain a separation of Rmax, strong hydrodynamic forces can bring the particles closer together. Given the initialization density is around 0.003μm−2, for weak or no clustering (Γ≃0.0), the clustering coefficient is expected to remain low, as there is, on average, only one particle per box of side length 18μm. Therefore, we choose a cutoff of 15μm to measure clustering in the system. Additionally, in simulations, particles closer than 6.25μm are excluded from the analysis, since lubrication forces become too strong and the behavior is uncontrolled and ill-defined.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors studied experimental realization of swarmalators using feedback-controlled self-propelling colloidal particles. By tuning a single parameter, they observed the co-evolution of spatial and phase dynamics, leading to a variety of collective states such as synchronized clusters, rotating aggregates, and dispersive phases. Importantly, they demonstrated that rapid changes of this parameter can generate dynamic regimes inaccessible to systems with static interactions. Simulations with squirmer and lubrication forces supported the experimental results, and a new synchronization-dependent interaction channel was also identified.\nThe manuscript sounds incomplete except the following improvements:\n1. The study utilizes active Brownian particles (ABPs), realized using silica spheres suspended in a near-critical water–lutidine mixture. However, the rationale for selecting this particular system is not clearly discussed in the manuscript and should be elaborated.\n2. The authors stated that operating several such oscillators with small distances q_{ij} between their reference points, their oscillation becomes highly synchronized (Fig. 1d). However, it is not immediately clear from Fig. 1d how this conclusion can be drawn. The authors should provide a brief explanation to clarify this point.\n3. As a measure of synchronization \\sigma_j authors defined in equation (1). But N_j is not defined anywhere in the manuscript. The authors should clarify.\n4. The authors presented the mean squared displacement (MSD) of isolated oscillators and its plot in Fig. 2(c). However, the definition of MSD and the justification for its use in the analysis are not clearly provided and should be clarified.\n5. In Fig. 3c, the fraction of oscillators within clusters is shown to increase continuously with Γ. However, in the region 0<Γ<0.10 the figure indicates a decreasing trend in the fraction of clustered oscillators. The authors should highlight and discuss this feature. The authors should also clarify whether the clustering behavior depends on the distance between neighboring oscillators.\n6. To clearly demonstrate the dependence of d_{||, ij} on both distance and synchronization it would be more effective if authors include a plot of d_{||, ij} vs \\sigma_{ij} in the manuscript.\n7. In the swarmalator model [Eqs. (3) and (4)], authors observed emerging states for different parameter values, It would strengthen the work if a parameter space varying K and J is included, showing the regions where different collective states appear.\n8. Mention the initial conditions and methods for simulations.\n9. In the caption of Fig. 2, the word ‘of’ appears twice and should be corrected. In addition, the figure captions throughout the manuscript could be enhanced to provide clearer explanations of the figures. The quality of the figures themselves may also be improved for better clarity.\n10. I would like to note recent reviews that would fit well in the introduction section on swarmalators https://doi.org/10.1098/rspa.2025.0315 and SIAM Journal on Applied Mathematics 85 (4), 1475-1499 (2025).", "reviewer_2": "In the submitted manuscript, the authors presents a controllable experimental swarmalator system using feedback-controlled self-propelled colloidal particles coupled by hydrodynamic flows. Combining experimental observation, simulation and theory, they show that the synchronization and spatial dynamics in the system together give rise to several collective states including synchronized clusters, rotating aggregates, and dispersive phases controlled by a coupling parameter. A new interaction synchronized from the direction perpendicular to the connection axis of particles are also revealed. Overall, the results are interesting and helpful for the understanding the swarmalator systems. However, there are a few comments for the authors to improve the manuscript.\n1. There is a big issue about the output of the figures in the main text. All color bars in figures (Fig. 1f, Fig. 3b, Fig. 3e, Fig. 5b, Fig. 5f, Fig. 6c) and heat maps (Fig. 3e, Fig. 5f) are mistakenly shifted somehow. They are confusing. However, those in the SI and videos seem all right.\n2. The particles used are Janus particles with one hemisphere covered by the carbon. While some reference work used uniform gold nanoparticles. Does the distribution of the heat-absorbing materials affect the particle dynamics? Will the orientation of carbon caps be considered and detected for each time interval?\n3. What is the relaxation time of the flow generated by the heating and particle movement? The time delay is 0.4 s. What happens if the time delay is tuned, especially around the flow relaxation time?\n4. The density or the average distance in each realization needs to be clarified since it is also a controlled parameter. The definition of clusters uses the criterium of 15 microns. Particles are not quite close considering the radius of the particles is 3 microns. Any reason for choosing this number?\n5. It seems that there is a small oscillation flow between left bottom to right up initially in the experiment part of Video 2. Is it true (and why?) or just a visual misperception?\n6. There is not experimental counterpart of the mini state loops shown as shown in Fig. 5d of the simulation results. How is the experimental one?", "reviewer_3": "The manuscript presents experimental work on the collective behavior of active Brownian particles that simultaneously exhibit synchronized rotation and translational motion in a coupled fashion, corroborated with numerical simulations and theoretical modelling. A several hundred colloidal particles perform orbiting motion around the reference position of each, steered by a scanning laser with a feedback loop. The reference positions are also moved to induce translational motion of the particles controlled by a coupling parameter (Γ); the reference position of each particle decays toward its average position in the short past for Γ>0, while it is moved away from the average position for Γ< 0.\nWhen the reference positions are fixed on a hexagonal lattice, the particles exhibit synchronized rotation with the average degree of synchronization negatively correlated with the lattice spacing. For motile reference positions, in-phase synchronized pairs tend to attract each other, while anti-phase pairs repel each other. Simulations based on the squirmer model with short-range lubrication interactions reproduce the experimental observation qualitatively, if the swimmers are of the puller-type. For pusher-type swimmers, the correlation between the displacement and degree of synchronization behaves differently.\nThe authors also introduce a phenomenological swarmalator model with a long-range Kuramoto-type phase coupling and forces that depend on the phase difference between particles. The coupling parameters K for the phase and J for the repulsion/attraction control the collective phase behavior, which is qualitatively consistent with the experimental results for positive (synchronizing) K.\nFurthermore, (i) time-varying interactions are realized in the experiment, and a fast switching between positive and negative Γ caused bistable collective states; (ii) lateral forces that are perpendicular to the axis connecting two particles causes global rotation of a cluster in the simulations.\nThe manuscript reports on a state-of-the-art experiment that provides the first controllable realization of a swarmalator system. The results are highly interesting and represent a significant and novel contribution to the emerging field at the intersection of nonlinear science and active matter physics. The qualitative agreement between the experimental results and the numerical simulations is impressive given the system's complexity. Therefore, I can recommend this manuscript for publication in Nature Communications provided the authors can satisfactorily address the following points.\nMajor issues:\n1. The derivation and justification of the phenomenological swarmalator model [Eqs. (3,4)] should be clarified. The authors should provide a more detailed explanation, perhaps in the Supplementary Material, that addresses the following points:\n(i) The 1/r^3 coupling should be associated with the no-slip boundary condition at the substrate, and the coupling parameter K should be related to the distance from the substrate and other parameters.\n(ii) There appears to be a contradiction between the squirmer model, which is based on the Oseen tensor for bulk fluid, and the 1/r^3 coupling in the swarmalator model, which is associated with hydrodynamic interactions near a no-slip boundary. The authors should elaborate on the connection between these two descriptions or justify the use of the 1/r^3 form in their phenomenological model.\n(iii) The approximation for the longitudinal distance d_\\parallel needs justification by experimental data. It is unclear how this quantity is related to the cosine term in Eq.(4).\n2. The phase diagram presented in Fig. 4 is currently depicted as a collection of representative snapshots. To make this a quantitative and more rigorous phase diagram, the authors should define and apply quantitative order parameters to clearly delineate the boundaries between the different collective states (e.g., static synchronous, rotating cluster, async).\n3. The emergence of global cluster rotation from lateral forces [Fig. 5g,h] is an intriguing result that requires further explanation. The authors should clarify: (i) What parameters control the magnitude and speed of this global rotation? (ii) How is the direction of the collective rotation (e.g., clockwise or counter-clockwise) determined by the properties or chirality of the individual particles.\nMinor issues:\n- To place their findings in a broader context, the authors might consider discussing the observed global cluster rotation in relation to similar phenomena in biological systems, such as the collective rotation of starfish embryos [26].\n- Optical feedback control to steer rotation of colloidal particles and switch synchronization patterns was studied in: A. Maestro et al, Commun. Physics 1, 28 (2018). The paper seems more relevant to the present work than the other quoted examples of anti-phase hydrodynamic synchronization.\n- In page 5, left column, “Figure 3d presents the average longitudinal displacement ...”, it should refer to Fig. 3e.\n- The emergence of global cluster rotation from lateral forces [Fig. 5g,h] is an intriguing result that requires further explanation. The authors should clarify: (i) What parameters control the speed of rotation? (ii) How is the direction of the collective rotation (e.g., clockwise or counter-clockwise) related to the rotating direction and other properties of the individual particles.\n- In Fig.S5, caption, \"the data is shows”might be a typo."}, "author_response": "Response to the referee reports for the manuscript\nNCOMMS-25-57131-T Tunable colloidal swarmalators with hydrody-namic coupling\nTheauthorsstudiedexperimentalrealizationofswarmalatorsusingfeedback-controlledself-propelling\ncolloidal particles. By tuning a single parameter, they observed the co-evolution of spatial and\nphase dynamics, leading to a variety of collective states such as synchronized clusters, rotat-ing aggregates, and dispersive phases. Importantly, they demonstrated that rapid changes of\nthis parameter can generate dynamic regimes inaccessible to systems with static interactions.\nSimulations with squirmer and lubrication forces supported the experimental results, and a new\nsynchronization-dependent interaction channel was also identified.\nR1.1 The study utilizes active Brownian particles (ABPs), realized using silica spheres suspended\ninanear-criticalwater–lutidinemixture. However, therationaleforselectingthisparticularsystem\nis not clearly discussed in the manuscript and should be elaborated.\nReply The investigation of a real-world system of swarmalators with physical interactions requires\nthe control of the motion of hundreds of moving agents which must physically interact with each\nother. Systems of self-propelled objects have been suggested as ideal candidates because they\nexactly fulfill such requirements and have shown basic swarmalator characteristics before (see e.g.\nYan et al., Nature 2012, 491, 578-581 and Zhang et al., Nat Commun 2020, 11, 4401). Compared\nto other active moving agents (e.g. macroscopic robotic devices or drone swarms), active colloidal\nparticles can be experimentally realized in table-top experiments under controlled lab conditions.\nA particular advantage of the system of light-activated active colloids, which is used in our study,\nis that the particle motion can be precisely controlled by a scanning laser beam, offering precise\nand individual control over the motion of the individual agents. In addition, due to the absence\nof a chemical fuel (the self-propulsion mechanism is created by light), this allows for experiments\nover several days – crucial to obtain sufficient statistics which makes this system very convenient\nfor swarmalator studies. We have added an explanation regarding our choice of this experimental\nplatform in section 2 of the manuscript.\nR1.2 The authors stated that operating several such oscillators with small distances q_{ij} between\ntheir reference points, their oscillation becomes highly synchronized (Fig. 1d). However, it is not\nimmediately clear from Fig. 1d how this conclusion can be drawn. The authors should provide a\nbrief explanation to clarify this point.\nReply Yes, we agree that the image alone does not immediately convey this point. The conclusion\ncan, however, be drawn quantitatively from the parameter σ_j. We have revised the text to clarify\nthis point and added a reference to an Supplementary video 1, in which the synchronization can\nbe perceived more clearly.\nR1.3 As a measure of synchronization σ authors defined in equation (1). But N_j is not defined\nanywhere in the manuscript. The authors should clarify.\nReply We have now added a line defining N_j in the manuscript.\nplot in Fig. 2(c). However, the definition of MSD and the justification for its use in the analysis\nare not clearly provided and should be clarified.\nReply The MSD quantifies the dynamical regime of motion (diffusive, subdiffusive, superdiffu-sive/ballistic, confined) and is widely used in soft matter, biophysics, and statistical physics to\nclassify particle dynamics. We agree that a proper mathematical definition is required which has\nnow be added to the SI. In particular in the context of swarmalators, the MSD is frequently used to\ncharacterize the swarmalators activity and in particular its dependence on the coupling parameter\nΓ. To facilitate the comparision of our findings with other swarmalator studies, we have also used\nthis measure in our work.\nR1.5 In Fig. 3c, the fraction of oscillators within clusters is shown to increase continuously with\nΓ. However, in the region 0 < Γ < 0.10 the figure indicates a decreasing trend in the fraction of\nclusteredoscillators. Theauthorsshouldhighlightanddiscussthisfeature. Theauthorsshouldalso\nclarify whether the clustering behavior depends on the distance between neighboring oscillators.\nReply The small dip in the cluster fraction f_c noted by the referee originates from our initial\nexclusion of pairs separated by less than 7 µm in the cluster analysis. This cutoff was introduced\nto exclude near-contact pairs (r ≈ 2a) which generate very large lubrication forces in simulations.\nHowever, following the comment of the referee, we have reexamined the data and found that only\npairs closer than about 6 µm actually exhibit such effects, while the 7 µm threshold unnecessarily\nexcluded physically valid pairs. At small Γ, where motility is low, this resulted in an artificial re-duction of f_c as particles tend to cluster near 7 µm. Thus, the dip is actually an artifact resulting\nfrom an improper choice of the cutoff distance. We have therefore reduced the cutoff (for exclusion\nin the clustering analysis) to 6.25 µm, which removes only near-contact pairs and avoids spurious\nexclusions. With this correction, f_c now increases smoothly and continuously in this region, as\nexpected. We have added a few lines in the method section to clarify this calculation.\nThe synchronization-dependent attraction that leads to clustering in our system does indeed de-pendonthedistancebetweenswarmalators. Theheat-mapinFig.3edisplaystheexactdependence\nof the clustering behaviour on both the neighbour distance |q_{ij}| and on their synchronization σ_{ij},\nand Γ couples to the displacement d_{||} to induce clustering in the system.\nR1.6 To clearly demonstrate the dependence of d_{||,ij} on both distance and synchronization it\nwould be more effective if authors include a plot of d_{||,ij} vs σ_{ij} in the manuscript.\nReply The main purpose of Fig. 3e was to highlight that the attractive and repulsive interactions\nthat emerge from the hydrodynamics between ABPs depend on both the distance and the syn-chronization between the swarmalators, as usually assumed in swarmalator models. However, we\nagree that a heat map is not the most effective way to show the dependency of d_{||,ij} on σ_{ij} and have\ntherefore added a plot of both the parallel and normal displacement against the synchronization\naveraged over the same distances as for the relative velocities in Fig. 3f (8-15 µm) in the new\nExtended Data Fig. 7. In this form, the dependency of the displacement on the synchronization\ncan be more easily perceived.\nparameter values, It would strengthen the work if a parameter space varying K and J is included,\nshowing the regions where different collective states appear.\nReply We thank the referee for this suggestion. In the Supplementary Information, we had\npreviously included only a qualitative extension of the phase space. In response, we have now\nconstructed an order-parameter space – defined by synchronization, cluster size, and cluster as-phericity – that allows us to identify and distinguish the various swarmalator phases. Our analysis\nshows that the data separate clearly into four phases, corresponding to the representative states\npresentedearlier. Usingtheseorderparameters, wearenowabletopresenttheswarmalatorphases\nin the form of a phase diagram. We have included this analysis in the Supplementary Information,\nas it is somewhat technical and does not add significantly beyond the qualitative presentation in\nFig. 4. The four phases occupy the four quadrants of the JK parameter space (corresponding to\nthe positive and negative values of J and K), which is captured qualitatively by the phase diagram\nin Fig. 4.\nR1.8 Mention the initial conditions and methods for simulations.\nReply The Methods section already provides some description of the simulation model and pa-rameters (Section 7.2,7.3, Table 1). We have further added some details on initialization for the\ndifferent cases and data acquisition in the figure captions and Supplementary Information.\nR1.9 In the caption of Fig. 2, the word ‘of’ appears twice and should be corrected. In addition,\nthe figure captions throughout the manuscript could be enhanced to provide clearer explanations\nof the figures. The quality of the figures themselves may also be improved for better clarity.\nReply We have corrected the typo. In addition, we have added further information to the figure\ncaptions. We agree that the figures were quite dense and have increased their size and added\nexplanatory schematics for better clarity.\nR1.10 I would like to note recent reviews that would fit well in the introduction section on\nswarmalators https://doi.org/10.1098/rspa.2025.0315 and SIAM Journal on Applied Mathematics\n85 (4), 1475-1499 (2025).\nReply Many thanks for pointing out these articles about non-Kuramoto and higher-order inter-actions between swarmalator. We have added them to outlook section of our manuscript.\nIn the submitted manuscript, the authors presents a controllable experimental swarmalator system\nusing feedback-controlled self-propelled colloidal particles coupled by hydrodynamic flows. Com-bining experimental observation, simulation and theory, they show that the synchronization and\nspatial dynamics in the system together give rise to several collective states including synchronized\nclusters, rotating aggregates, and dispersive phases controlled by a coupling parameter. A new in-teraction synchronized from the direction perpendicular to the connection axis of particles are also\nrevealed. Overall, the results are interesting and helpful for the understanding the swarmalator\nsystems.\nReply We are very pleased by the referee’s positive assessment of our manuscript.\nR2.1 There is a big issue about the output of the figures in the main text. All color bars in figures\n(Fig. 1f, Fig. 3b, Fig. 3e, Fig. 5b, Fig. 5f, Fig. 6c) and heat maps (Fig. 3e, Fig. 5f) are mistakenly\nshifted somehow. They are confusing. However, those in the SI and videos seem all right.\nReply During submission of the manuscript we did not encounter any issues. We have, however\nrealized now that on Mac devices, most heat maps and color bars are distorted and think the\nproblem was the format in which we exported these graphics from Matlab. We are very sorry for\nthis issue and have tried to adjust the figures of the revised version to solve this issue.\nR2.2 The particles used are Janus particles with one hemisphere covered by the carbon. While\nsome reference work used uniform gold nanoparticles. Does the distribution of the heat-absorbing\nmaterials affect the particle dynamics? Will the orientation of carbon caps be considered and\ndetected for each time interval?\nReply The Janus colloids we employ as ABPs in our study do in principle align their carbon cap\ntowards the laser, resulting in negative phototaxis, and therefore show different propulsion dynam-ics compared to isotropic active particles like, e.g., colloids evenly coated with gold nanoparticles.\nIn our experiments, we do, however, change the relative position of the laser spot with respect to\neach particle at a higher frequency than the Janus particles can follow given their finite reorienta-tion time (around 1 s, see Lozano et al., Nat Commun 2016, 7, 12828). As a result, the particles\nare always found with their carbon cap oriented vertically (i.e. parallel to the laser beam) and their\nbright and dark hemispheres can not be distinguished. For this reason, the Janus particles behave\nlike isotropic active colloids in the oscillating mode in which they are driven in this study. To\nreduce the image processing time and because the particle orientation did not play an important\nrole for the dynamics, we did not not track the orientations of the particles in our experiments.\nWe have added an additional paragraph for clarification in the Methods section.\nR2.3 What is the relaxation time of the flow generated by the heating and particle movement?\nThe time delay is 0.4 s. What happens if the time delay is tuned, especially around the flow\nrelaxation time?\nCommun 2016, 7, 12828). Tuning the delay time between the detection of the particle positions\nand the propulsion towards the reference positions changes the oscillation frequency, the amplitude\nand the persistence of the orbiting motion of the ABPs (with a small delay time, the ABPs switch\ntheir direction of rotation less often). For our experiments, we have optimized the delay time to\nachieve oscillations with high amplitude and long correlation times of the angular velocity (i.e.\nfew switches in rotation direction). The time it takes to process the microscope image and detect\nthe particle positions sets a lower limit for the delay time we can achieve of 0.3 s. Thus, we\nwere unfortunately not able to test delay times below the hydrodynamic relaxation time of the\npropulsion mechanism. We would however expect that effects similar to inertia start playing a\nrole when the delay time approaches the relaxation time of the propulsion mechanism, where the\npropulsion direction in the previous time step influences the particle velocity in the next time step\nand therefore alters the dynamics of the oscillation. We have added more information about the\nexperimental time scales in the Methods section 7.1.\nR2.4 The density or the average distance in each realization needs to be clarified since it is also\na controlled parameter. The definition of clusters uses the criterium of 15 microns. Particles are\nnot quite close considering the radius of the particles is 3 microns. Any reason for choosing this\nnumber?\nReplyWefullyagreethattheparticledensityisanimportantparameterandintherevisedversion,\nwe have now added this information to the corresponding figure captions. Regarding the definition\nof clusters, we first implemented a repulsion algorithm to prevent particles from approaching too\nclosely, asdirectcontactwouldcausethemtosticktogetherandbecomeunresponsive. Theparticle\ndistribution around the target demonstrates that they can approach up to about 2.0µm from the\ntarget, which imposes a minimum center-to-center separation of about 10µm (particle radius is\n3µm). This value was therefore used in the repulsion algorithm to prevent sticking. Given the\ninitialization density of 0.003µm−2, for weak or no clustering (Γ ≃ 0.0), the clustering coefficient\nis expected to remain low, as there is on average only one particle per box of side length 18µm.\nTherefore a choice of 15µm is appropriate to identify clusters. We have clarified this point in the\nfigure captions and the Supplementary Information.\nR2.5 It seems that there is a small oscillation flow between left bottom to right up initially in\nthe experiment part of Video 2. Is it true (and why?) or just a visual misperception?\nReply The small global oscillatory drift of ABPs originates from a slight misalignment between\nthe laser focus and the particle center. Because unavoidable mechanical instabilities occur on\ntimescales of hours, we carry out periodic calibrations during the experiments to minimize global\ndrifts. This procedure results in the minor oscillatory global motion correctly noted by the referee.\nImportantly, this effect does not influence the relative particle motion, and in particular not the\nswarmalator behavior reported in our study, as confirmed by the excellent agreement with the\nnumerical (drift-free) data. We have added a clarifying note to the Methods section.\nR2.6 There is not experimental counterpart of the mini state loops shown as shown in Fig. 5d\n6\nof the simulation results. How is the experimental one?\nReply We did indeed run experiments where we applied the same fast switching of the coupling\nparameter as in the simulations to reproduce the mini state loops. Each run of these experiments\ntook much longer than the experiments with static interaction (which already took around one\nhour for each run), so we could switch back and forth between the different coupling parameters\nmultiple times. It is very demanding to control the experimental conditions precisely over such\nlong times and mitigate the inevitable instabilities that lead to the drifts the reviewer has also\nmentioned in the previous question. Additionally, switching the coupling parameter during the\nmeasurement, which also determines the motility of the swarmalators, complicated the periodic\nre-calibrations during the measurements. For these reasons, we were not able to maintain stable\nexperimental conditions with the high precision necessary to observe the mini state loops.\nThe manuscript presents experimental work on the collective behavior of active Brownian particles\nthat simultaneously exhibit synchronized rotation and translational motion in a coupled fashion,\ncorroborated with numerical simulations and theoretical modelling. A several hundred colloidal\nparticles perform orbiting motion around the reference position of each, steered by a scanning laser\nwith a feedback loop. The reference positions are also moved to induce translational motion of\nthe particles controlled by a coupling parameter (Γ); the reference position of each particle decays\ntoward its average position in the short past for Γ > 0, while it is moved away from the average\nposition for Γ < 0.\nWhen the reference positions are fixed on a hexagonal lattice, the particles exhibit synchronized\nrotation with the average degree of synchronization negatively correlated with the lattice spac-ing. For motile reference positions, in-phase synchronized pairs tend to attract each other, while\nanti-phase pairs repel each other. Simulations based on the squirmer model with short-range lu-brication interactions reproduce the experimental observation qualitatively, if the swimmers are of\nthe puller-type. For pusher-type swimmers, the correlation between the displacement and degree\nof synchronization behaves differently.\nThe authors also introduce a phenomenological swarmalator model with a long-range Kuramoto-typephasecouplingandforcesthatdependonthephasedifferencebetweenparticles. Thecoupling\nparametersKforthephaseandJfortherepulsion/attractioncontrolthecollectivephasebehavior,\nwhich is qualitatively consistent with the experimental results for positive (synchronizing) K.\nFurthermore, (i) time-varying interactions are realized in the experiment, and a fast switching\nbetween positive and negative Γ caused bistable collective states; (ii) lateral forces that are per-pendiculartotheaxisconnectingtwoparticlescausesglobalrotationofaclusterinthesimulations.\nThe manuscript reports on a state-of-the-art experiment that provides the first controllable real-ization of a swarmalator system. The results are highly interesting and represent a significant and\nnovel contribution to the emerging field at the intersection of nonlinear science and active matter\nphysics. The qualitative agreement between the experimental results and the numerical simula-tions is impressive given the system’s complexity. Therefore, I can recommend this manuscript\nfor publication in Nature Communications provided the authors can satisfactorily address the\nfollowing points.\nWearegratefultotherefereeforthecarefulreadingofourmanuscript, andfortherecommendation\nof publication.\nMain comments:\nR3.1 The derivation and justification of the phenomenological swarmalator model [Eqs. (3,4)]\nshould be clarified. The authors should provide a more detailed explanation, perhaps in the Sup-plementary Material, that addresses the following points:\n(i) The 1/r3 coupling should be associated with the no-slip boundary condition at the substrate,\nand the coupling parameter K should be related to the distance from the substrate and other\nparameters.\n(ii) There appears to be a contradiction between the squirmer model, which is based on the Oseen\ntensor for bulk fluid, and the 1/r3 coupling in the swarmalator model, which is associated with\nhydrodynamic interactions near a no-slip boundary. The authors should elaborate on the connec-tion between these two descriptions or justify the use of the 1/r3 form in their phenomenological\n8\nmodel. (iii) The approximation for the longitudinal distance d_\\parallel needs justification by experimental\ndata. It is unclear how this quantity is related to the cosine term in Eq.(4).\nReply We agree that the justification for the phenomenological swarming model requires further\nclarification. Intherevisedmanuscript, wehaveaddedadetailedexplanationinthesupplementary\ninformation, complemented by simulations, to better motivate the form employed. We would also\nlike to take this opportunity to clarify several related aspects:\n(i) The long-range behavior of light-activated colloids can be modeled as that of a (nearly)\nneutral squirmer, exhibiting a characteristic 1/r3 decay in the bulk, arising from the source\ndipole. While this has the same distance dependence the hydrodynamic interactions of a\npulled bead (force monopole) near a no-slip substrate (Oseen tensor), the underlying physics\nis fundamentally different: the particles in our study are force-free swimmers, and the 1/r3\nlong-range coupling arises from the squirmer flow fields rather than from any net external\nforce. At short distances however, lubrication forces may effectively act as “pulling” or\n“pushing” forces, but their exact functional form is not known. To account for these in-teractions, we implemented in our model a power-law decay with an exponential cutoff at\nlonger distances, with the squirmer radius as the characteristic length scale. This ensures\nthat short-range effects are incorporated without altering the intrinsic long-range coupling,\nwhich remains dictated by the squirmer model. Consequently, the resulting long-range in-teractions in our phenomenological swarmalator model are consistent with the 1/r3 scaling,\neven though the mechanism differs from the 1/r3 interactions of externally actuated particles\nnear a no-slip boundary. There is therefore no contradiction between the squirmer-based de-scription and the form of the long-range coupling used in the phenomenological model. We\nhave now clarified this aspect.\n(ii) The coupling constant K depends on the strength of the underlying hydrodynamic interac-tions, such as lubrication forces and the source dipole strength s. In the phenomenological\nmodel, this dependence is approximated by treating K as an effective constant, which we\ncan vary to explore new phases that are not experimentally accessible – like anti-synchronous\nphases.\n(iii) The form of d_\\parallel is directly motivated by experimental data (Fig. 3e), where d_\\parallel ∼ sin(πσ_{ij})\nand σ_{ij} ∼ cos∆θ_{ij}. We have now added a section in the SI that explains the derivation of\nequation (4) in more detail to avoid any potential confusion.\nR3.2 The phase diagram presented in Fig. 4 is currently depicted as a collection of representative\nsnapshots. To make this a quantitative and more rigorous phase diagram, the authors should\ndefine and apply quantitative order parameters to clearly delineate the boundaries between the\ndifferent collective states (e.g., static synchronous, rotating cluster, async).\nReply We thank the referee for this valuable suggestion. We agree that our previous depiction was\nprimarily qualitative and did not employ quantitative order parameters to clearly delineate the\nboundaries between the different collective states. In response, we have now constructed an order-parameter space—based on synchronization, cluster size, and cluster asphericity—that enables us\nto identify and distinguish the various swarmalator phases. Our analysis shows that the data\nseparate clearly into four phases, corresponding to the representative states presented previously.\nHowever, since this analysis is somewhat technical and does not reveal any additional phases\nbeyond those already shown in Fig. 4, we have included it in the Supplementary Information.\nR3.3 The emergence of global cluster rotation from lateral forces [Fig. 5g,h] is an intriguing result\nthat requires further explanation. The authors should clarify: (i) What parameters control the\nmagnitude and speed of this global rotation? (ii) How is the direction of the collective rotation\n(e.g., clockwise or counter-clockwise) determined by the properties or chirality of the individual\nparticles.\nReply The rotation speed is found to depend on the cluster size (as shown in Fig. 5h), as well\nas on the strength of hydrodynamic interactions and cluster synchronization. Since hydrodynamic\ninteractions modify the trajectory, which couples to the swarmalator motion through Γ, we expect\nthe rotation speed to increase with this parameter. We have now performed additional simulations\nto verify this, and the corresponding results have been included in the main text.\nThe rotation direction is determined by the emergent final chirality of the individual swarmalators\nwithin the cluster – i.e. the adopted collective rotational state. The individual particles themselves\ndo not possess an intrinsic chirality; they are initially randomized between clockwise and counter-clockwise states. However, noise (which facilitates occasional flips in rotation direction) together\nwith hydrodynamic interactions leading to synchronization causes rotational symmetry breaking,\nso that all swarmalators adopt the same rotational sense (either clockwise or counterclockwise).\nThe cluster then rotates in the same sense, i.e clusters of CW/ACW moving particles rotates\nCW/ACW. We have now added this explanation regarding the cluster rotation and supported it\nwith additional data in the Supplementary Information.\nMinor issues:\nR3.4 To place their findings in a broader context, the authors might consider discussing the\nobserved global cluster rotation in relation to similar phenomena in biological systems, such as the\ncollective rotation of starfish embryos [26].\nReply We have added a remark to this system in the manuscript where we first mention the\nobservation of the rotating clusters.\nR3.5 Optical feedback control to steer rotation of colloidal particles and switch synchronization\npatterns was studied in: A. Maestro et al., Commun. Physics 1, 28 (2018). The paper seems\nmore relevant to the present work than the other quoted examples of anti-phase hydrodynamic\nsynchronization.\nReply We thank the referee for pointing out this publication. We have added it as a reference to\nour manuscript.\nshould refer to Fig. 3e.\nReply Corrected.\nR3.7 In Fig. S5, caption, “the data is shows” might be a typo.\nReply Corrected.\n11"}, {"round": 2, "reviewer_comments": {"reviewer_1": "Authors have revised the manuscript nicely.\nNow it can be accepted for publication.", "reviewer_2": "I am good with the response and recommend the draft for a publication.", "reviewer_3": "The revised manuscript presents a significantly improved and more comprehensive study of tunable colloidal swarmalators with hydrodynamic coupling. The authors have clarified the physical basis of the phenomenological swarmalator model, linking the 1/r^3 phase coupling to hydrodynamic interactions near a no-slip boundary and providing experimental validation for the longitudinal displacement relation. They have also introduced quantitative order parameters — cluster size, asphericity, and synchronization — to define the phase diagram, resolving the earlier concern that the classification was only qualitative. The explanation of global rotation and lateral forces has been expanded and now includes clear dependence on the coupling parameter Γ and cluster size. The simulation codes look correct and consistent with the models described in the manuscript.\nOverall, the revision successfully addresses all of my previous comments. The experimental realization, numerical simulation, and theoretical modeling are now coherently integrated, and the results are presented with improved clarity and rigor. The manuscript now represents a definitive and publishable demonstration of a controllable swarmalator system. I therefore recommend its publication in Nature Communications."}, "author_response": ""}]

Decision letter