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Dynamics of Magnus Dominated Particle Clusters, Collisions, Pinning and Ratchets

Motivated by the recent work in skyrmions and active chiral matter systems, we examine pairs and small clusters of repulsively interacting point particles in the limit where the dynamics is dominated by the Magnus force. We find that particles with the same Magnus force can form stable pairs, triples and higher ordered clusters or exhibit chaotic motion. For mixtures of particles with opposite Magnus force, particle pairs can combine to form translating dipoles. Under an applied drive, particles with the same Magnus force translate; however, particles with different or opposite Magnus force exhibit a drive-dependent decoupling transition. When the particles interact with a repulsive obstacle, they can form localized orbits with depinning or unwinding transitions under an applied drive. We examine the interaction of these particles with clusters or lines of obstacles, and find that the particles can become trapped in orbits that encircle multiple obstacles. Under an ac drive, we observe a series of ratchet effects, including ratchet reversals, for particles interacting with a line of obstacles due to the formation of commensurate orbits. Finally, in assemblies of particles with mixed Magnus forces of the same sign, we find that the particles with the largest Magnus force become localized in the center of the cluster, while for mixtures with opposite Magnus forces, the motion is dominated by transient local pairs or clusters, where the translating pairs can be regarded as a form of active matter.


I. INTRODUCTION
There are a variety of systems that can be described as local clusters of interacting particles, including colloids 1-3 , Coulomb clusters 4,5 , vortices in type II superconductors 6,7 , dusty plasmas 8 , Wigner crystals 9 , vortices in superfluids 10,11 , skyrmions 12,13 , granular matter 14 , and active matter assemblies 15 . In many of these systems, the cluster formation arises when the particles experience a local confinement or selftrapping due to the nature of the pairwise particleparticle interactions 16,17 . Under various types of driving, these systems can exhibit interesting dynamical effects including self assembly 18,19 , rotating gear behavior 20,21 , and depinning phenomena 22 . In most of these systems, the dynamics is overdamped; however, some systems also include nondissipative effects such as inertia or Magnus forces. In particular, Magnus forces produce a velocity component that is perpendicular to the net force experienced by a particle, and such forces arise for vortices in fluids [23][24][25][26] , active spinners [27][28][29][30][31] , chiral active matter 32 , charged particles in magnetic fields 33 , and skyrmions in chiral magnets [34][35][36] . One consequence of this is that pairs or clusters of particles can undergo rotations or spiraling motion when they enter a confining potential [37][38][39][40][41] or are subjected to a quench 42 . If damping is present, these spiraling motions are transient unless there is some form of external driving. Less is known about how Magnusdominated particles would interact with obstacles or pinning sites; however, there are some studies which indicate that the Magnus force strongly modifies the dynamics compared to overdamped systems [40][41][42][43][44][45][46][47][48][49] .
Motived by our previous work on point particle models of skyrmions interacting with each other and with random 41,50 or periodic pinning 42 , where the particles have both a Magnus and a damping force, we consider the limits of zero damping or very low damping and study the Magnus-dominated dynamics of pairs and small clusters of particles interacting with each other and with pinning sites. We consider mixtures with identical Magnus forces, dispersion in the Magnus force, and assemblies with opposite Magnus forces. In the case of a pair of particles with the same sign and magnitude of the Magnus force, we find that a bound rotating pair forms despite the repulsive particle-particle interactions, and that under an external drive the pair remains coupled and translates at 90 • with respect to the driving direction. If the magnitude of the Magnus force is not the same for both particles, a drive dependent decoupling transition occurs. For higher numbers of particles we find various types of stable rotating states, including rotating pairs that rotate around each either. For larger clusters we observe chaotic dynamics in which the system breaks up into smaller clusters with some particles jumping from one cluster to another. When the Magnus forces of a pair of particles are of the same magnitude but different sign, the particles form a dipole which translates in a direction determined by the initial orientation of the pair, with dipoles of smaller size translating more rapidly. If the Magnus forces are of different sign and magnitude, the particles form a translating pair that can break apart and reform if a collision with an obstacle or other particles occurs. We argue that assemblies of particles with mixed Magnus force sign represent a new example of an active matter system.
When repulsive obstacles are added to the system, we find that the particles can form stable bound circulating orbits around the obstacles and exhibit a depinning transition under an applied drive. We show that it is possible for pairs and clusters of particles to collide with and become localized by a obstacle. If damping is present, these pinned states are transient and the particle gradually winds away from the obstacle. A particle interacting with a cluster of defects can enter an orbit that encircles all of the defects. In the overdamped limit, an asymmetric cluster of defects produces a diode-like effect for driving in different directions, but in the Magnus-dominated limit this diode effect disappears and the particles circle around the entire cluster. A particle driven toward a line of obstacles experiences a Magnus force-induced deviation in its direction of motion as it approaches the line until it breaks through the line, and this deviation is reduced for increased driving force. We also find that it is possible to produce a ratchet effect for a particle that is placed by a line of obstacles when a biharmonic ac drive is applied. Here the particle can form circular orbits that create a gear-like motion when combined with the periodicity of the line of obstacles. Reversals in the ratchet current occur as a function of ac amplitude and Magnus force. Finally, we examine the chaotic dynamics of smaller clusters and show that if there is dispersion in the Magnus force, the particles with the largest Magnus forces become localized in the center of the cluster.
Our results should be relevant for skyrmions in the absence of damping or in the low damping limit in the presence of a drive, for certain models of point vortex dynamics in superfluids or Bose-Einstein condensates with fluid flows, and for active spinners and active chiral colloidal systems.

II. SIMULATION
We consider a two-dimensional system with periodic boundary conditions in the x and y-directions containing N particles that are initially placed at fixed distances from each other. Typically we use initial conditions in which the particles are in one-dimensional lines. The dynamics of particle i are governed by the following undamped equation of motion: where v i is the velocity of particle i and α i m is the coefficient of the Magnus term, which creates a velocity component perpendicular to the net applied forces. Each particle can be assigned a different amplitude or sign of α i m . The particle-particle interaction force is given by F pp i = N j=1 K 1 (r ij )r ij , where r ij = |r i − r j | is the distance between particles i and j,r ij = (r i − r j )/r ij , and the modified Bessel function K 1 (r) falls off exponentially for large r. This form of the interaction was previously used in particle-based models of skyrmions in two-dimensional systems [40][41][42]50 . The driving force F D = F Dx is applied uniformly to all particles. An individual particle in the Magnus force-dominated limit moves at 90 • with respect to the driving force, so that when the drive is applied in the x direction, the particle moves in the y direction. The term F obs = Np k=1 FIG. 1. The particle locations (dots) and trajectories (lines) for pairs of interacting particles. (a) When α 1 m = α 2 m = 1.0, the particles form a clockwise rotating bound pair. (b) For α 1 m = 2.0 and α 2 m = 1.0, the particles form nested orbits where the particle with the higher Magnus force is closer to the center. (c) A system with α 1 m = α 2 m = 1.0 in which a finite damping term α d = 0.1 has also been added, causing the particles to spiral out gradually. (d) The α 1 m = α 2 m = 1.0 system from (a) with an additional drift force FD = 0.075 applied in the x-direction, causing the pair to translate in the negative y-direction. represents the force from N p obstacles, which take the form of particles that are permanently fixed in place. In some cases, we add a damping term α i d v i to the equation of motion which aligns the velocities in the direction of the external forces. Under a drive, a particle experiencing both Magnus and damping forces moves at an angle θ = arctan(α m /α d ). We measure the particle velocities both parallel,

III. DYNAMICS OF COUPLED PARTICLES
We first consider particles with the same sign and magnitude of the Magnus force. In Fig. 1(a) we show an image of the trajectories of two particles with α 1 m = α 2 m = 1.0 initialized a distance R apart. In an overdamped system, the particles would move away from each other, but here they form a pair and rotate around each other in a clockwise manner. The particles remain confined to the pair due to the Magnus force which generates velocities that are perpendicular to the net forces on each parti- cle. When α 1 m = α 2 m , the particles form a nested pair as illustrated in Fig. 1(b) for α 1 m = 1.0 and α 2 m = 2.0, with the larger Magnus force particle orbiting closer to the center. If we add a finite damping term of α d = 0.1 to the α 1 m = α 2 m = 1.0 system in Fig. 1(a), the particles gradually spiral away from each other as shown in Fig. 1(c), and in the long time limit, the presence of damping eventually causes the particles to come to a standstill. If only one particle has damping, the overall motion still damps away since the damped particle couples to the undamped particle and dissipates its energy, so as long as there is some damping in the system both particles will eventually come to rest unless an external drive is applied. In the zero damping limit, when there is an applied drive the rotating pair remains coupled and its center of mass translates, as shown in Fig. 1(c) for the α 1 m = α 2 m = 1.0 system under a drive of F D = 0.075. The x direction drive causes the pair to translate in the negative y-direction, giving a skyrmion Hall angle of 90 • . Here the intrinsic skyrmion Hall angle is defined as θ int sk = arctan(α m /α d ) 41,43,46 . In the presence of damping, the driven pair in Fig. 1(d) gradually spiral away from each other and translate separately at a Hall angle less than 90 • .

A. Systems with opposite Magnus force
When two particles that have Magnus forces which are equal in magnitude but opposite in sign are brought together, they form a bound pair that translates in a fixed direction even in the absence of an applied drive. The repulsive interaction between the two particles produces an outwardly directed force on each particle, and the Magnus term rotates this force by 90 • for one particle and by −90 • for the other, producing a net translation instead of a rotation. In Fig. 2 is determined by the initial placement of the particles. The speed of the dipole pair increases as the initial distance R between the particles decreases, since the pairwise interaction force increases at smaller distances, while the dipole drift velocity V d is given by Fig. 3(a) we plot the measured velocity V d versus α m for the system in Fig. 2(a) at a fixed initial separation distance of R = 2.0. The solid line is a fit to 1/α m . In Fig. 3(b) we show V d versus R for fixed α m = 1.0 in the same system. The dipole velocity decreases approximately exponentially with increasing distance at large R, as expected for the function K 1 (R). In Fig. 2(b) we illustrate the dipole trajectory for a system with Magnus forces of opposite sign but unequal magnitude, α 1 m = 1.65 and α 2 = −2.0, where the dipole curves into a localized circular orbit. As the difference in magnitude of the Magnus forces increases, the circular orbit becomes tighter.

IV. DYNAMICS UNDER A DRIVE
We next consider the effect of applying a driving force in the positive x-direction, which causes isolated particles with a positive Magnus force to move in the negative y direction. For a pair of particles with Magnus forces of equal sign and magnitude, the pair remains coupled when the drive is applied and translates perpendicular to the drive, as shown in Fig. 1(d). If the magnitude of the Magnus forces are unequal, there is a critical driving force above which the pair decouples. In Fig. 4 we plot the velocities V 1 and V 2 of a pair of particles versus driving force F D for a system with α 1 m = 1.6 and α 2 m = 2.0. For F D ≤ 0.15, V 1 = V 2 and the particles are coupled into a dipole, while for F D > 0.15, the pair decouples as indicated by the change in the velocities. The critical driving force F c at which the decoupling occurs decreases as the difference |α 1 m − α 2 m | increases, while F c increases as the separation R decreases.
A cluster containing more than two particles that all have the same α m remains coupled under an applied drive, but when some of the particles have different values of α m , multiple decoupling transitions can occur.

A. Dynamics with Obstacles and Depinning
We next study the effects of driven particles interacting with a repulsive obstacle. To begin, we consider a single particle under an applied drive interacting with an obstacle which is modeled as another particle that is fixed permanently in place, giving a repulsive force between the particle and the obstacle. In the overdamped limit, there is no pinning effect and the particle simply moves away from the obstacle due to the pairwise repulsion. In Fig. 5(a), a particle with α m = 2.0 under a driving force of F D = 0.005 initialized at a distance of R = 1.5 from the obstacle forms a localized pinned orbit around the obstacle. At F D = 0.015 in the same system, Fig. 5(b) indicates that the particle is still localized but the orbit becomes distorted by the drive. In Fig. 5(c) at F D = 0.165, the particle has depinned and translates in the y direction, interacting with the obstacle during each pass through the periodic boundary conditions. At F D = 0.025 in Fig. 5(d), the interaction with the obstacle is diminished and the pinch point in the trajectory near the obstacle has disappeared. In Fig. 6(a) we plot the FIG. 5. The particle position (red dot) and trajectory (line) with the obstacle location (blue dot) for a single particle interacting with a stationary obstacle in the form of a permanently fixed particle. The particle has αm = 2.0 and is initialized at a distance R = 1.5 from the obstacle, and the applied drive is absolute value of the average particle velocity in the ydirection, | V y |, versus F D . We find a clear region where the particle is pinned, as indicated by | V y | = 0, along with a critical depinning force at F c = 0.016. In most systems where depinning occurs, there must be an attractive interaction between the particle and a defect so that the particle can settle into a potential energy minimum and stop moving. It is possible in some overdamped systems for the particle to become trapped behind a repulsive barrier, but even in that case the particle comes to rest and can be described as jammed 22 . Here we find a depinning transition in which the particle is always moving but remains localized below depinning. If the sign of the Magnus force is reversed, the same dynamics occurs but the particle depins in the opposite direction. The depinning threshold depends on the magnitude of α m and the initial distance R at which the particle is placed from the obstacle. If we add some damping to the particle dynamics, the particle does not remain localized but always escapes via an unwinding transition. This process is illustrated in Fig. 7(a) for the system from Fig. 5 with an added damping of α d = 0.01 at a fixed drive of F D = 0.01, which is below the threshold depinning force found in Fig. 6 for the undamped system. When damping is present, after each orbit the particle gradually moves away from the obstacle until eventually it depins and then translates at an angle θ sk = arctan(α m /α d ). If the interaction between the particle and the obstacle is attractive, when damping is present the particle gradually spirals into the obstacle, while if a drive is also applied, the particle spirals inward until it reaches an equilibrium point at which the driving force balances the attractive force from the obstacle.
A single repulsive obstacle can also capture multiple particles. An example of this process appears in Fig. 8(a) for a sample with two particles where α 1 m = α 2 m = 2.0, R = 1.5, and F D = 0.005, where the two particles form a pair that rotates around the obstacle. Due to the applied drive, the trajectories are denser on the left side of the obstacle. When the drive is increased, a depinning transition occurs in which one particle depins while the other remains localized, as shown in Fig. 8(b) for the same system at F D = 0.01. Due to the periodic boundary conditions, the depinned particle returns and interacts with the obstacle again, passing through a spiraling orbit before escaping. At a higher drive of F D > 0.015, the second particle also depins. If the two particles are initially in a pair away from the obstacle, then when they collide with the obstacle under a driving force, the obstacle can trap the pair, only one particle, or neither particle. In Fig. 8(c) we show the collision of a pair with the obstacle at F D = 0.01, where one particle becomes trapped and the other escapes. For F D > 0.015, the pair stays together after encountering the obstacle, while for F D < 0.05, both particles become trapped. If the Magnus force is different in a pair of trapped particles, two orbits form with two different average distances from the obstacle. Even if the two particles have Magnus forces of

B. Interaction with Multiple Obstacles and Ratchet Effects
When multiple obstacles are present, a single particle can move around or encircle a cluster of obstacles to create an edge current effect. In an overdamped system, when particles interact with an asymmetric array of defects, it is possible to create a diode effect in which the depinning threshold is higher in one direction than the other. In Fig. 9(a) we show seven obstacles that have been arranged into a funnel shape. When a mobile particle is initially placed near one of the obstacles, it can encircle a single obstacle or it can encircle all of the obstacles, as shown in Fig. 9(a) for an α m = 1.0 particle placed at a distance of R = 1.5 from the funnel, where F D = 0.0. This ability to encircle multiple obstacles indicates that the Magnus dominated particle exhibits an  Fig. 9(b,c) for motion in the negative y-direction (blue) and positive y-direction (pink). For either direction of drive, in Region I, the particle moves around the obstacles, and in Region II, the particle breaks through the funnel between the outer two obstacles. The dashed line at FD = 0.95 indicates a transition for the positive y direction motion to the flow illustrated in Fig. 9(d). Changes in the breakthrough location are associated with small cusps in the velocity-force curve, and additional breakthrough cusps occur at higher drives (not shown). (b) | Vy | vs FD for the same system in the overdamped limit of αm = 0.0 and α d = 1.0. There is a finite depinning threshold for motion in the negative y-direction (blue) but not for motion in the positive y direction (pink), creating a diode effect.
edge current behavior of the type observed in chiral active matter systems 51,52 . Under application of a drive in the negative x-direction with F D = 0.01, Fig. 9(a) indicates that the particle moves in the positive y direction and curves around the array of obstacles. The same drive of F D = 0.01 applied in the positive x direction causes the particle to move in the negative y-direction, and as shown in Fig. 9(c), the particle skirts around the funnel tip without getting trapped. Under varied parameters, we have not found a case in which the funnel tip is able to trap the particle for driving in any direction. At higher F D , the particle breaks through the funnel array rather than moving around it, as illustrated in Fig. 9(d) for the system from Fig. 9(c) at F D = 0.25. In Fig. 10(a) we plot the absolute y-direction velocity | V y | versus F D for the system in Fig. 9 for driving in both the positive and negative x directions. There is no pinned regime and the velocities are almost identical for both directions of driving. The label I indicates the regime in which the particle moves around the outer edge of the obstacles as shown in Fig. 9(b,c), while the label II denotes the regime in which the particle passes between the outer two obstacles. For motion in the positive y direction, the next breakthrough point occurs at F D = 0.1, which appears as a cusp in the velocity, and is associated with a transition to the motion illustrated in Fig. 9(d). This breakthrough transition occurs at a drive higher than the range shown for motion in the negative y-direction.
If finite damping is present, we can observe a diode effect which is the most pronounced in the fully overdamped limit. In Fig. 10(b) we plot | V y | versus F D for the system from Fig. 10(a) but with α m = 0.0 and α d = 1.0 under both positive and negative y direction driving. Since the Magnus force is zero, the particle motion is aligned with the driving force direction. There is a finite depinning threshold for motion in the negative y-direction, but no threshold for driving in the positive y-direction. In Fig. 11(a) we plot the particle trajectory in the overdamped limit of the system in Fig. 10 for a drive of F D = 0.04 in the negative y-direction, where the particle becomes trapped by the funnel tip, while in Fig. 11(b) the same system under driving in the positive y direction has continuous flow of the particle around the obstacles.
The appearance of a diode effect in the overdamped system with a funnel array geometry also implies that if an ac drive is applied, a ratchet effect will appear in which the particle translates along the easy flow direction of the funnel during one portion of the ac cycle. This type of ratchet is known as a rocking ratchet 53 and it has been observed in overdamped superconducting vortices interacting with asymmetric pinning [54][55][56][57][58] and in skyrmion systems where there is a combination of damping and a Magnus effect 59,60 . In the skyrmion system there are even cases where a ratchet effect only occurs when the Magnus force is present 59  out damping, the ratchet effect is absent, indicating that some damping is necessary for ratcheting to occur; however, we next show that it is still possible to achieve a ratchet effect in the Magnus-dominated regime if the symmetry is broken by a combination of ac driving and the chirality of the Magnus force.
In Fig. 12(a) we plot the trajectory of a particle moving in the negative y-direction interacting with a line of obstacles with a period of a = 1.0 for a system with F D = 0.007 and α m = 2.0. In the absence of obstacles, the particle moves in a straight line at a constant velocity; however, as the particle approaches the line of obstacles, it begins to bend away from the line due to the repulsive force from the particles in the positive y direction. The Magnus force changes this repulsive force into a positive x direction velocity component of the moving particle. The particle accelerates as it comes closer to the obstacles, and eventually it passes through the barrier. As F D increases, the particle experiences a smaller x direction deviation of its motion when it approaches the obstacle line, as shown in Fig. 12(b) for F D = 0.07, while for even higher values of F D , the deviation in the x-direction nearly disappears. If a particle is placed near the line of obstacles in the absence of a driving force, the particle moves at a constant velocity parallel to the line of obstacles due to the Magnus force.
If we place the particle near the line of obstacles and subject it to an ac driving force given by F AC = A cos(ωt)x + B sin(ωt)ŷ, we observe not only directed motion but a reversal in the direction of motion as a function of the ac drive amplitude, Magnus force, and dissipation. This occurs due to the fact that the ac drive induces a rotation of the particle that interacts like a gear mechanism with the periodicity of the line of obstacles. In Fig. 13(a) we plot the trajectory of a particle with α m = 2.0, A = B = 0.05, and ω = 0.00005 which is placed at distance of R = 12a from the line of obstacles. This is sufficiently far away that there is no inter- action between the particle and the obstacles, and the particle executes a circular counterclockwise orbit with no directed motion. In Fig. 13(b), we keep everything the same but place the particle a distance R = 2.0a from the line of obstacles. The particle now translates in the positive x-direction and passes an integer number of obstacles during each ac drive cycle. In Fig. 13(c), the same system with A = B = 0.1 has a larger particle orbit and the particle translates in the negative x direction, indicating a reversal of the current. The effectiveness of the reversed ratchet effect is much lower, with the particle translating at 1/4 the speed of its motion in the positive x direction in Fig. 13(b). In Fig. 13(d), we show the system from Fig. 13(b) with a much larger value of α m = 10. The particle translates in the positive x-direction but at a much smaller velocity.
In Fig. 14(a) we plot V x versus A for the system in Fig. 13(b,c) with B = A. There is a reversal in the current from positive to negative at A = 0.7, while at higher A, V x goes to zero. As A approaches zero, the particle moves in a straight line along the x direction at fixed V x = 0.056 due to the Magnus force created by the repulsion from the line of obstacles. Figure 14 shows V x versus α m for the system in Fig. 13(b) at fixed A = 0.05. For α m < 1.5, the particle moves in the negative x direction, while the motion is in the positive x direction when α m ≥ 1.5. The efficiency of the ratchet as measured by the magnitude of V x reaches a maximum near α m = 3.0 and then gradually decreases with increasing α m . The step near α m = 2.5 is produced by a change in the nature of the translating orbit. In Fig. 15(a) we plot the trajectory of a particle moving in the negative x direction for the system in Fig. 14(b) at α m = 1.0. For smaller α m , the orbit increases in extent and the particle encircles up to three obstacles per ac drive cycle. The magnitude and direction of the rectified current depends on the starting position of the particle relative to the line of obstacles, and there can also be translating orbits that do not encircle any obstacles in which the particle skips along the edge of the line of obstacles, as shown in Fig. 15(b,c) for a particle with A = B = 0.025 initially placed either above or below the line of obstacles, respectively. The ratchet can also occur as function of only a single ac drive. When the ac driving force is applied only along the x-direction, we find a series of ratchet effects as illustrated in Fig. 15(d) for the same system as in Fig. 13(b) but with A = 0.05 and B = 0.0.
Here the particle is ratcheting in the positive direction with V x = 0.009, which is about half the velocity found for a ratchet effect with simultaneous x and y ac driving, A = B = 0.05. The ratchet effect is strongly affected by the damping. A finite damping term causes a particle placed near a line of obstacles to move away from the obstacles gradually; however, the ac driving can maintain the ratcheting motion. In Fig. 16(a) we plot V x versus time in simulation time steps for a particle with α m = 2.0 and R = 4a at two different values of the damping, α d = 0.005 and α d = 0.001. For the larger damping, the particle gradually moves in the positive y-direction away from the line of defects since the damping term aligns the particle velocity with the direction of the repulsive force from the defect line. In this case, as the particle moves further away from the obstacles, the ratcheting effect is reduced. For the smaller damping, the particle oscillates across the line of obstacles until it ends up below the line of obstacles and then gradually gets pushed further away from the obstacles in the negative y-direction, causing a reduction in the ratchet effect. In this case, there is also a window of time during which the particle becomes localized on an obstacle, giving a ratchet velocity of zero, while when the particle begins to spend most of its time below the line of obstacles, it begins to ratchet in the negative x-direction. There are also several points at which discrete jumps occur in the velocity due to the jumping of the particle between different orbits that are commen- surate with the periodicity of the obstacle line.
In Fig. 16(b), we plot V x versus time in simulation time steps for the system in Fig. 16(a) with α m = 2.0 and α d = 0.005 for a particle placed above the line of obstacles at a distance of R = 1a, 2a, 4a, and 6a. In this case, a particle initially placed at R = a ends up below the line of obstacles and is gradually pushed further in the negative y direction while V x approaches zero. For R = 2a, the particle gradually moves away in the positive y-direction but the system passes through a series of different types of orbits that ratchet in the positive x direction, as indicated by the oscillations in V x , and there is even a peak in the velocity before it dies away to zero. For R = 4a, the particle enters a single orbit and gradually moves away from the line of obstacles. If we place the particle even further away, we observe the same behavior as for the R = 4a sample but with even lower values of V x , as shown for R = 6a.
We note that ratchet effects with biharmonic drives have been studied for skyrmions, where a Magnus effect can come into play; however, in these studies there was still a damping term, and the internal modes of the skyrmion were also important 61,62 . The ratchet effect we observe here is more closely related to ratchet effects found in colloids undergoing circular orbits while interacting with a magnetic bubble lattice, where the asymmetry necessary to produce the ratchet arises from the ac drive and the transport occurs due to a commensuration effect with the underlying substrate 61

C. Dynamics of Clusters
We next consider the case of three or more particles. In Fig. 17 we show some representative examples of possible multiparticle orbits. For N = 3 particles with α 1 m = α 2 m = α 3 m = 1.0 that are initially placed in a row along the x direction spaced 2a apart, Fig. 17(a) shows the formation of a spiraling pattern, which rotates due to precession of the orbits. The particular type of orbit that appears for N = 3 equivalent particles depends on the initial particle placement, but in general we find non-chaotic stable orbits. In Fig. 17(b) we plot the trajectories for N = 3 with α 1 m = α 3 m = 2.0 and α 2 m = 1.0, where the two α m = 2 particles form a pair that orbits in the center of the cluster while the α m = 1.0 particle follows an orbit with a larger radius. An N = 3 sample in which all of the particles are different, with α 1 m = 1.0, α 2 m = 3.0, and α 3 m = 2.0, appears in Fig. 17(c). Here a layering effect occurs in which particles with larger Magnus force spend more time closer to the center of the cluster. Figure 17(d) shows the same system with α 1 m = 1.0, α 2 m = 7.0, and α 3 m = 2.0, where three clear spatial layers appear and the α m = 7.0 particle is nearest to the center. This system has some similarities the ordering of small clusters of colloids in a trap; however, in this case, the particles are continuously undergoing motion and there is no external confining trap. In Fig. 17(e) we plot the trajectories for an N = 4 system with varied Magnus forces of α 1 m = 1.0, α 2 m = 2.0, α 3 m = 3.0, and α 4 m = 4.0, which forms a chaotic cluster. We note that if the variations in the Magnus forces are larger, ordered states can appear with ring like structures, which we describe in the next subsection. In Fig. 17(f) we show an N = 4 sample with α 1 m = α 2 m = α 3 m = α 4 m = 2.0. In this case, the particles form two rotating pairs which rotate around each other. For N > 3, most orbits are chaotic, but for special initial placement conditions, it is possible to stabilize different types of rotating states. In larger clusters where the particles all have the same Magnus force, the chaotic states typically involve a transient state of two or three particle subclusters that break up and reform over time.

D. Clusters with Strong Variation in Magnus Force Magnitude and Ring Formation
For particles with Magnus forces that are of the same sign but that have sufficiently different magnitudes, clusters appear that have well defined spacings between the particle orbits, with the particles that have the highest Magnus force localized at the center of the cluster. In Fig. 18(a) we plot the trajectories in an N = 4 system with α 1 m = 7.0, α 2 m = α 3 m = 2.0, and α 4 m = 1.0. The α m = 7.0 particle becomes localized at the center of the cluster and is surrounded by a ring containing the α m = 2.0 particles, while the α m = 1.0 particle jumps between the α m = 2.0 ring and a partially formed outer ring. A similar structure appears in Fig. 18 Fig. 18(c) where the two inner particles with α m = 10.0 are orbited by the α m = 3.0 particles to form a dumbbell shape. Strongly segregated ring structures can also occur when N = 4, as illustrated in Fig. 18(d) for a sample with α 1 m = α 2 m = α 3 m = 7 and α 4 m = 2, where the inner particles have the higher Magnus force. If the difference between the Magnus forces of the particles is reduced, the ring structures are lost.

E. Clusters and Collisions for Particles with Opposite Magnus Forces
As noted earlier, if two particles with equal and opposite Magnus forces come together, they can form a dipole that translates in a straight line. If the magnitude of the Magnus forces are different, an arching orbit appears instead. In Fig. 19(a,b) we show the trajectories of two particles with α 1 m = 2.0 and α 2 m = −2.0 under an external driving force of F D = 0.0075. The particles are initially placed at the same x position but are widely separated in y. Under the influence of the drive, the particles initially move in opposite directions, but as they approach one another, they form a pair that translates in the positive x direction, as shown in Fig. 19(a). The driving force causes the particles to move closer together and eventually pass each other as shown in Fig. 19(b). Figure 19(c) shows two particles with α 1 m = 2.0 and α 2 m = −1.5 that form a dipole which moves in an arch shape before the particles decouple again. In Fig. 19(d), a system with α 1 m = 2.0 and α 2 m = −1.0 undergoes multiple collisions due to the periodic boundary conditions, and the orbit performed during each collision has a small radius due to the large difference in the magnitude of the Magnus forces. If the particles are separated in y but also have a small offset in x, they do not collide head on, which creates spiraling orbits similar to that shown in Fig. 19(b) but with asymmetric loops. For a system of three particles in which the sign of the Magnus term of one particle is opposite from that of the other two particles, we generally observe closed periodic orbits; however, depending on the initial placement of the particles, it is also possible to have a pair of particles with opposite signs of Magnus force break off and move away as a dipole. In a system with mixed Magnus force amplitudes where one particle has a positive Magnus force and the other two have negative Magnus forces, a translating dipole can form that then rotates around the third particle. For example, in Fig. 20(a), a system with α 1 m = 1.0, α 2 m = −1.1, and α 3 m = 0.85 has a translating dipole moving in an orbit that gradually precesses counterclockwise while the third particle follows a tighter precessing orbit. For five or more particles with mixed Magnus force signs, in general we do not observe long-lived localized structures but instead find that pairs of particles with opposite sign form a gas of translating dipoles that are either broken up or deflected when a collision with another particle or dipole occurs. In Fig. 20 Fig. 20(c) showing jumps between zero velocity when the particle is not part of a dipole and finite velocity when the particle is part of a translating dipole.
site sign that translate. For this combination of Magnus forces, we always observe translating clusters, but the direction and velocity of the translation depends on the initial placement of the particles. If the Magnus forces are unequal, as in Fig. 21(b) where α 1 m = 1, α 2 m = α 4 m = −2, and α 3 = 2, similar dynamics occur but the cluster moves in a circle.
A collection of particles with opposite Magnus force signs can be considered an example of an active matter system. In active matter, the particles are self-propelled and can be described as undergoing driven Brownian diffusion or run and tumble dynamics. Typically, active particles show short time ballistic behavior and long time diffusive behavior due to collisions 65,66 . In the case of the Magnus dominated system, mixtures of opposite Magnus force signs form translating dipoles that act like active Brownian particles in the limit of zero orientational diffusion or like run and tumble particles with an infinite run time. When there are other particles in the system, collisions can cause the dipoles to change directions or to break up before reforming again. To highlight this effect, in Fig. 22 we plot a time series of the x-direction velocity V x for a single particle from the system in Fig. 20(c), where regions of constant velocity are interspersed with regions of zero velocity. The constant velocity regions correspond to periods in which the particle forms half of a translating dipole, while the zero velocity regions are periods in which the particle is no longer paired into a dipole and is therefore stationary. There can also be intervals in which the particle is part of a rotating pair composed of two particles with the same Magnus force sign. In future studies, it would be interesting to examine the velocity distributions in large collections of mixtures of Magnus dominated particles to see whether this system exhibits further similarities to active matter.

V. DISCUSSION
A number of the results we observe are similar to behavior found in point vortex models. In these models, vortices in fluids are represented as non-dissipative point particles with a logarithmic long range interaction and nondissipative dynamics that are controlled by a Coriolis or Magnus term [23][24][25][26] . A pair of point vortices with the same vorticity rotate around one another, while a pair with opposite vorticity translates. Additionally, the point vortex literature shows that clusters of four or more particles generally form chaotic states. Other work has shown that point vortex particles can effectively be trapped in orbits around defects such as a fixed point vortex 67,68 , or they can scatter off defects. In our case, the interactions are shorter range than the point vortex interactions; however, the smooth behavior of the Bessel function potential causes much of the dynamics of the Magnus-dominated particles to be fairly similar to the point vortices. In our work we considered scattering off multiple objects, ratchet effects, and particles with mixed Magnus force values. In most of the point vortex literature, the Magnus force is of the same magnitude, and in general, there is no driving force and dissipative effects are neglected.
Due to the dynamical nature of the states we observe, it is possible to imagine that for periodic obstacle geometries or arrangements of a large assembly of particles localized around an obstacle, some sort of dynamical but repeatable crystal could form which would be an example of a classical time crystal [69][70][71][72] . In a real system, some form of dissipation would likely arise that would eventually destroy the crystal, but it may be possible to create long lived transient Magnus time crystals. Experimentally, our system most closely resembles skyrmion or meron motion with no or low dissipation where the Magnus force dominates the dynamics, which should be achievable under certain conditions. Skyrmions can also be set into motion readily under a drive, so it should be possible to maintain the transient Magnus force dynamics in a low dissipation system indefinitely by applying ac or dc driving. It is also possible to have dispersion in the Magnus force component of a skyrmion system as well as skyrmions with opposite signs. Additional internal modes can arise in skyrmions that are not taken into account in our model; however, there have already been some studies of skyrmion dynamics in the zero dissipation limit with both Thiele equation and continuum modeling approaches 49 .
Our results could also be relevant to spinning charged colloids levitated acoustically or dusty plasmas in magnetic fields, where Magnus effects arise and dissipation effects are weak. Many active spinner systems include strong dissipation or have only short range contact interaction forces, so that two particles or a particle and an obstacle would only interact when they touch. Finally, our results for decoupling and depinning should also be applicable to vortices and point vortex models in the presence of some form of flow field.

VI. SUMMARY
We have examined the dynamics of individual pairs and small clusters of repulsive pairwise interacting particles in which the dynamics is dominated by a Magnus term. In the overdamped limit, clusters of such particles exhibit transient motion and settle into a stationary state. For particles without dissipation, when the Magnus terms have the same magnitude and sign, a pair of repulsively interacting particles rotate around each other at fixed distance. Similar rotating clusters appear up to sizes of N = 4, but for larger clusters the dynamics become chaotic. A pair of particles with opposite Magnus force sign forms a translating dipole. Under an applied drive, an individual particle moves at 90 • with respect to the drive direction, a rotating pair with the same Magnus force translates, and a pair with different Magnus force magnitudes has a decoupling driving force threshold. A particle interacting with repulsive obstacles forms a bound state with a critical driving threshold for the decoupling of the particle from the obstacle, while if the particle dynamics include damping, the particle gradually spirals away from the obstacle. A single obstacle can bind multiple particles simultaneously. When a rotating pair encounters a obstacle, one or both particles in the pair can become trapped. For particles interacting with clusters of obstacles, we find that it is possible for a particle to become bound to the cluster and form a circulating current around the outside of the cluster. In the overdamped limit, a particle interacting with obstacles arranged in a funnel shape exhibits a diode effect, but when there is only a Magnus force and no damping, the diode effect disappears. A line of obstacles causes a deviation in the direction of the trajectory of the driven particle, which eventually passes through the obstacle line. Under an ac drive, we show that it is possible to observe a ratchet effect for a particle placed near a line of obstacles due to a gear-like mechanism in which the particle orbit becomes commensurate with the periodicity of the obstacle line. The ratchet effect shows a reversal as a function of ac drive, Magnus force, and distance from the obstacle line. For large clusters of particles, we find that if the dispersion in the Magnus force is sufficiently large, the particles with the largest Magnus force become localized in the center of the cluster. In mixtures of particles with opposite signs, we find the intermittent formation of dipoles that can translate over some distance before breaking up or deflecting upon encountering other particles, and we show that these dipoles have certain similarities to active matter systems. Our results could applied to skyrmion systems in the absence of dissipation or in the low dissipation limit, or to chiral active matter in which there is low damping or continuous driving. Our results could also be useful for understanding transient dynamics in systems with Magnus dominated dynamics and weak damping.