{'fraction_non_alphanumeric': 0.04063736498770185, 'fraction_numerical': 0.019655651801946315, 'mean_word_length': 4.07059971803492, 'pattern_counts': {'":': 0, '<': 18, '<?xml version=': 0, '>': 20, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 10, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Dynamics and Nonmonotonic Drag for Individually Driven Skyrmions

We examine the motion of an individual skyrmion driven through an assembly of other skyrmions by a constant or increasing force in the absence of quenched disorder. The skyrmion behavior is determined by the ratio of the damping and Magnus terms, as expressed in terms of the intrinsic skyrmion Hall angle. For a fixed driving force in the damping dominated regime, the effective viscosity decreases monotonically with increasing skyrmion density, similar to what is observed in overdamped systems where it becomes difficult for the driven particle to traverse the surrounding medium at high densities. In contrast, in the Magnus dominated regime the velocity dependence on the density is nonmonotonic, and there is a regime in which the skyrmion moves faster with increasing density, as well as a pronounced speed-up effect in which a skyrmion traveling through a dense medium moves more rapidly than it would at low densities or in the single particle limit. At higher densities, the effective damping increases and the velocity decreases. The velocity-force curves in the Magnus-dominated regime show marked differences from those in the damping-dominated regimes. Under an increasing drive we find that there is a threshold force for skyrmion motion which increases with density. Additionally, the skyrmion Hall angle is drive dependent, starting near zero at the threshold for motion and increasing with increasing drive before reaching a saturation value, similar to the behavior found for skyrmions driven over quenched disorder. We map dynamic phase diagrams showing the threshold for motion, nonlinear flow, speed-up, and saturation regimes. We also find that in some cases, increasing the density can reduce the skyrmion Hall angle while producing a velocity boost, which could be valuable for applications.


I. INTRODUCTION
Skyrmions are particle-like magnetic textures 1-5 that have been identified in a growing number of materials, including numerous systems in which they are stable at room temperature [6][7][8] . Skyrmions can be set into motion with applied currents 3,4,6,[8][9][10][11] , magnetic field gradients 12 , and thermal gradients 13 . Due to their size scale, room temperature stability, and mobility, skyrmions are promising candidates for various applications including memory [14][15][16] and novel computing architectures [17][18][19] . In terms of basic science, skyrmions represent a new class of systems that exhibit collective dynamics under an applied drive 20,21 , similar to vortices in superconductors 22,23 , colloids 24 , Wigner crystals 25 frictional systems 26 , and other soft matter systems 27 . A unique feature which distinguishes skyrmions from these other systems is the strong gyroscopic or Magnus force component of their dynamics, which generates velocities perpendicular to the net force experienced by the skyrmion 3,4,9,11,28 . One consequence of this is that a skyrmion moves at an angle, known as the skyrmion Hall angle, with respect to an applied driving force 3,28-32 . The Magnus term also produces many other effects, including accelerations or speed-ups of skyrmions interacting with barriers or walls [33][34][35][36][37] , ratchet effects [38][39][40][41] , and spiraling motion in a confined potential 5,42,43 or after a quench 44 . Such effects can be strongly modified by pinning or skyrmion-skyrmion interactions 5 .
Beyond moving skyrmions with currents, other methods that can be used to manipulate individual skyrmions include scanning tunneling microscope tips 45 , local magnetic fields 46,47 , and optical traps 48 . Extensive experi-mental and theoretical studies have been performed in systems such as colloids [49][50][51][52][53] , granular matter [54][55][56][57][58] , active matter 59 , superconducting vortices [60][61][62][63][64][65] , various soft matter systems [66][67][68][69] , and metallic glasses 70 examining the dynamics of individually driven particles moving through an assembly of other particles under either a constant force or an increasing force. Measurements of the velocity-force relations or the drag on the driven particle provide information about how the effective viscosity of the system changes across density-induced transitions such as solid to liquid 71,72 , liquid to glass 50,52 , or unjammed to jammed 54,55,58 . For example, the viscosity can increase strongly at the onset of a glass or solid phase produced by an increase in the density. When the probe particle is driven with an increasing force, there can be a force threshold below which the particle is unable to move 49,50,73,74 , as well as distinct features in the velocity-force curves 75,76 which change as the density increases.
In soft matter systems, driving a single probe particle through a background of other particles is known as active rheology 52,58,59,68,74 . Similar probes have been applied in type-II superconductors using individually dragged vortices [60][61][62][63][64][65] . Existing work on active rheology has involved overdamped systems, where the viscosity generally increases with increasing density. It is not clear how the addition of a Magnus term would impact active rheology. For example, it could change the threshold force for motion, increase or decrease the drag, modify the Hall angle, or alter the way in which the probe particle moves with respect to the background particles. Beyond skyrmions, there are many other chiral systems in which Magnus or gyroscopic effects can be important for active rheology, including chiral soft matter fluids and solids, active chiral systems [77][78][79][80][81] , charged particles in a magnetic field 82,83 , magnetic colloids in oscillating fields 84 , spinning particles in fluids 85,86 , fluid vortices 87,88 , and fracton systems 89 .
In this work we study active rheology for skyrmion systems in the absence of quenched disorder using a particlebased model. We apply an external driving force to one skyrmion and measure the velocity components and drag both parallel and perpendicular to the driving direction. For a constant drive, the velocity decreases monotonically with increasing skyrmion density in the dampingdominated regime. This is similar to what has been found for the active rheology of overdamped systems, where it becomes more difficult for the particle to pass through a denser medium due to the increased interactions with other particles. When the Magnus force is dominant, the driven skyrmion velocity has a strongly non-monotonic density dependence and increases with increased density. In this regime, the velocity can be boosted or accelerated beyond the expected velocity for an isolated driven skyrmion in the absence of surrounding skyrmions. As the density increases, the skyrmion Hall angle decreases and the velocity reaches a peak value before decreasing at higher densities. The velocity boost effect arises due to the creation of a locally asymmetric density profile around the driven skyrmion oriented perpendicular to the driving direction. This creates an additional repulsive force on the driven skyrmion which does not cancel out due to the lack of symmetry and is converted by the Magnus term into a velocity component aligned with the driving direction. The behavior is similar to the acceleration effect found for skyrmions interacting with walls and barriers [33][34][35][36][37] .
For the case of a driving force that is increased from zero, the driven skyrmion exhibits a critical threshold for the onset of motion. The skyrmion Hall angle is zero at the threshold and increases with increasing velocity until it saturates to a constant value at high drives, similar to the behavior found for skyrmions driven over quenched disorder 5,[30][31][32][90][91][92][93][94][95] . In the Magnus dominated regime, the boost effect is nonmonotonic, with reduced boost at low and high drives and maximum boost at intermediate drives. This drive dependence originates from the reaction of the surrounding skyrmions to the motion of the driven skyrmion. Local density fluctuations of the background skyrmions have time to relax when the driven skyrmion velocity is low, but do not have time to form when the velocity is high. In each case the boost effect is reduced. In contrast, for intermediate driven skyrmion velocities, density fluctuations in the surrounding skyrmions form and relax on the same time scale as the motion of the driven skyrmion.
In the strongly damped regime, the velocity-force curves are similar to those observed for the active rheology of overdamped systems, while in the Magnusdominated regime, negative differential conductivity can appear when part of the velocity in the driving direc-tion gets transferred into the direction perpendicular to the drive. We also find that under constant driving, there are regimes in which an increase in density produces a velocity boost with a reduction of the skyrmion Hall angle. This indicates that harnessing skyrmionskyrmion interactions may be a viable method for reducing the skyrmion Hall angle without reducing the skyrmion speed, which could be important for applications. We map dynamic phase diagrams as a function of skyrmion density, driving force, and the ratio of the damping term to the Magnus force. We discuss possible experimental realizations and the application of our results to the broader class of systems with gyroscopic forces.

II. SIMULATION
We consider an assembly of N skyrmions in a twodimensional system with periodic boundary conditions in the x and y-directions. A single skyrmion is coupled to an external drive and is driven through the other skyrmions in the absence of quenched disorder. The skyrmion dynamics are obtained using a particle-based model from the modified Thiele equation 5,29,30,44,96 . The equation of motion for skyrmion i is given by where v i = dr i /dt is the skyrmion velocity, α d is the damping coefficient, and α m is the coefficient for the Magnus force. The damping term aligns the velocities in the direction of the net external forces, while the Magnus term creates a perpendicular velocity component. The skyrmion-skyrmion interaction force is F ss i = N j=1 K 1 (r ij )r ij , where r ij = |r i − r j | is the distance between skyrmions i and j and K 1 is the modified Bessel function. The driving force F D i = F Dx is applied only to the driven skyrmion and is zero for all other skyrmions.
In the absence of other skyrmions, the driven skyrmion would move in the direction of the intrinsic Hall angle, θ int sk = − arctan(α m /α d ). In the overdamped limit where α m = 0, θ int sk = 0 • . When skyrmion-skyrmion interactions occur, the skyrmion Hall angle is reduced below its intrinsic value and is defined as θ sk = arctan( V ⊥ / V || ), where V || and V ⊥ are the average velocities of the driven skyrmion parallel and perpendicular to the direction of the drive, respectively. In this work, each individual simulation is performed at a constant value of F D and lasts 3 × 10 5 to 1 × 10 6 simulation time steps to avoid transient effects. For drives very near the critical threshold for motion, much larger time intervals are needed to reach a steady state 97 . The density of the system is given by ρ = N/L 2 , where the system size L = 36.

III. RESULTS
In Fig. 1(a) we show an example of our system for a collection of skyrmions at a density of ρ = 0.1. The red particle is being driven at a constant force F D = 0.5, while the blue particles are the non-driven or bath skyrmions. For this system, θ int sk = −44.4 • , so the ratio between the damping and Magnus forces is close to one. The driven skyrmion moves on average at an angle of θ sk = −29 • . This is smaller in magnitude than the intrinsic skyrmion Hall angle, indicating that collisions with the bath skyrmions are reducing θ sk . The magnitude of θ sk increases for higher values of F D but decreases with increasing ρ. Figure 1(b) shows the same system at a higher density of ρ = 0.5, where the driven skyrmion moves at an even smaller θ sk . The skyrmion Hall angle and net velocity depend strongly on the skyrmion density, the driving force, and the ratio of the Magnus force to the damping term.
If the driven skyrmion did not collide with any other skyrmions, it would move at the intrinsic skyrmion Hall angle of θ int sk = − arctan(α m /α d ) with an average absolute velocity of |V | = F D / α 2 d + α 2 m . We first consider systems in which we constrain α 2 d + α 2 m = 1.0. Under this condition, we can define a velocity V 0 = |V | N =1 = F D to be the velocity in the single particle limit. Once skyrmion-skyrmion interactions are introduced, in overdamped systems under fixed driving we expect to find |V | < V 0 . The velocity decreases with increasing ρ and there is a critical density ρ c at which |V | = 0.
In Fig. 2(a) we plot V || and V ⊥ versus ρ for a system in the damping dominated regime with α m /α d = 0.1, θ int sk = −5.739 • , and F D = 0.5. As ρ increases, V ⊥ and θ sk both decrease in magnitude and approach zero for ρ > 0.5. In Fig. 3 we plot |V | = V || 2 + V ⊥ 2 for the same system, where the dashed line indicates that |V | ≤ V 0 for all values of ρ. Here |V | monotonically deceases with increasing ρ, similar to the behavior found in fully overdamped systems. When α m = 0, V ⊥ = 0, θ sk = 0, and V || has the same shape shown in Fig. 2 but its magnitude is slightly reduced.
In Fig. 4 we plot V || , V ⊥ , and θ sk versus ρ for the same system from Fig. 2 in the Magnus dominated regime with α m /α d = 9.95 and θ int sk = −84.26 • . Here the velocities are highly nonmonotonic. V || initially increases with increasing ρ, reaches a maximum of V || = 0.78 near ρ = 1.25, and then decreases again, while V ⊥ has a small initial increase in magnitude followed by a decrease in magnitude to a value of zero near ρ = 1.6. The Hall angle has the value θ sk = −84.26 • at low ρ, decreases linearly in magnitude with increasing ρ, and reaches zero for ρ > 1. 25. The maximum value of V || = 0.78 is higher than V 0 = 0.5, indicating that the velocity in the driving direction is being boosted as the skyrmion density increases. There is also a small boost in | V ⊥ | near Fig. 3, the plot of |V | versus ρ for the system in Fig. 4 indicates that |V | reaches a maximum value close to 0.9. The dashed line represents V 0 = 0.5, so that a velocity boost is occurring whenever |V | > V 0 . We observe velocity boosting up to ρ = 2.0, while for ρ > 2.0, the net velocity decreases, indicating an increase in the effective damping.
In  no overshoot in |V | and we always find |V | < V 0 . When α m /α d > 1.0, an overshoot emerges with a peak velocity which increases and shifts to larger ρ with increasing Magnus force. Figure 5(b) shows the corresponding θ sk versus ρ. In every case, θ sk starts at its intrinsic value for small ρ, decreases in magnitude with increasing ρ, and reaches a saturation at high ρ.
In Fig. 6 we map the regions where a velocity boost is present and absent as a function of ρ versus θ int sk for the system in Fig. 5. As θ int sk increases, the upper edge of the velocity boost window shifts to higher values of ρ. The red squares indicate the values of ρ at which the boost of |V | takes its maximum value for each choice of θ int sk . The boost disappears for all ρ when θ int sk ≤ 44 • .

IV. DRIVE DEPENDENCE
The size of the velocity overshoot and the value of the skyrmion Hall angle at a given skyrmion density is also a function of the magnitude of the driving force. In Fig. 7(a) we plot |V | versus F D for the system in  Here |V | is always lower than the single particle limit. (b) ∆V = |V |/V0 versus FD, where V0 is the expected velocity in the single particle limit at a given drive. Here ∆V < 1.0, indicating increased damping compared to the single particle limit. At higher drives ∆V approaches the single particle limit. cle limit. For any given value of F D , Fig. 7(a) shows that |V | decreases monotonically with increasing ρ. In Fig. 7(b) we plot ∆V = |V |/V 0 versus F D , where V 0 is the velocity in the single particle limit at each value of F D . When ∆V = 1.0, the driven skyrmion is moving with the same velocity it would have in the single particle limit. ∆V < 1.0 indicates increased damping, while ∆V > 1.0 is a signature of a boosted velocity. Here we find ∆V < 1.0 over the entire range of ρ, indicating that the driven skyrmion is moving slower than the free particle limit. The damping effect is strongest at low drives, and the overall damping increases with increasing ρ. There is also a threshold F c below which the driven skyrmion does not move, as shown in Fig. 8(a) where we plot a blow up of the low drive region from Fig. 7(a). When F D < F c , |V | = 0.0. F c starts from zero in the single particle limit and increases with increasing ρ. The plot of ∆V versus F D in Fig. 8(b) indicates that ∆V is zero below F c and rises to a saturation value which approaches the single particle limit as ρ decreases. In general, ∆V increases towards ∆V = 1.0 with increasing F D since the surrounding skyrmions have less time to respond to a rapidly moving driven skyrmion. For overdamped systems with quenched disorder, the velocity of the probe particle has the power law form V = (F d −F c ) α , so the particle moves the most slowly just above the threshold depinning force 5,20,21 , while at high drives the velocity generally approaches the pin-free limit 20,21 . In |V | is generally above the dashed line, which is the expected behavior in the single particle limit. (b) The corresponding ∆V = |V |/V0 versus FD, where V0 is the expected velocity for the single particle limit at a given drive. Here ∆V > 1.0 indicates the occurrence of velocity boosting, with ∆V approaching the single particle limit at higher drives.
our system, there is no quenched disorder but the background bath of skyrmions effectively serves as a type of deformable quenched disorder. At high velocities, where the background skyrmions do not have time to respond before the driven skyrmion has passed beyond them, the system looks more like a single skyrmion moving through an array of fixed defects or scattering sites.
In Fig. 9 we plot |V | and ∆V versus F D for varied ρ in the Magnus dominated regime with α m /α d = 9.95 and α 2 d + α 2 m = 1.0 for ρ = 0.05, 0.1, 0.25, 0.5, 1.0, 1.5, and 2.0. The dashed line indicates the expected behavior in the single particle limit. Here we find that |V | is generally larger than the single particle limit, indicating a velocity boost. In Fig. 9(b), ∆V increases with increasing ρ and has a peak at F max which shifts to higher values of F D as ρ becomes larger. For large F D , ∆V decreases back to the single particle limit. When ρ = 2.0, the peak value of ∆V is almost 2.5 times larger than the free particle limit. We still observe a threshold force F c for motion which is almost the same as that found for the overdamped case. In general we do not observe a significant difference in the threshold for motion as a function of the ratio of the Magnus term to the damping. In systems with quenched disorder, it has been shown that the depinning threshold often decreases with increasing Magnus force since the motion of the skyrmions requires a longer time to relax when the Magnus force is larger 5,44 . This increases the time required in the presence of disorder for the spiraling motion of the skyrmions to settle into equilibrium after a finite applied drive is increased 5,29,30,98 . In our work we apply a constant drive and wait for the system to settle into a steady state. The transient time to reach that state increases near the threshold, suggesting that the observation of a reduced pinning threshold in other studies with quenched disorder could be the result of the increased relaxation time for larger Magnus forces. In studies where the drive is continuously increased from zero to a finite value, the sweeping rate above which transient effects become important should depend on both the damping and Magnus force. Such rate effects will be the subject of a future work.
In Fig. 9(b), we find that ∆V decreases with increasing drive for F D > F max and approaches ∆V = 1.0 at high drives. The decrease follows the form ∆V = [A/(F D − F max )] + 1.0. In Fig. 10 we plot ∆V versus F D for the specific case of ρ = 2.0 from the system in Fig. 9(b) along with a fit using A = 0.65 and F max = 0.85. A similar fit can be performed for the other ∆V versus F D curves. This type of 1/F D dependence on fluctuations or dynamic disorder is also often observed in driven systems with quenched disorder 5,29,30 .
The speed up effect as a function of ρ and F D arises due to the Magnus force. In systems where a single driven skyrmion interacts with a wall or barrier which is parallel to the x direction, an applied drive parallel to the x direction generates a skyrmion Hall component which deflects the driven particle in the y direction. This increases the skyrmion-wall interaction force in the y direction, which results in a perpendicular skyrmion velocity component along the x direction. The wallinduced velocity contribution adds to the x direction velocity from the driving, generating a velocity boost ef- fect. The magnitude of the boost velocity depends on the nature of the pairwise interaction between the wall and the skyrmion. If the interaction is of the power law form V (r) ≈ 1/r α , then as the skyrmion approaches the wall more closely, r decreases and the resulting velocity in the x direction increases [33][34][35][36][37] . The boost effect can also occur for skyrmions moving over a two-dimensional periodic substrate 29 or over random pinning 99 , since the effect arises whenever the component of the substrate force which is perpendicular to the direction of drive is not balanced by a compensating substrate force in the other direction. This is due to the skyrmion Hall angle which pushes the skyrmion to one side of the pinning sites. For our single driven skyrmion, there is no pinning or barrier wall; however, a velocity boost effect can still occur due to the creation of a local density gradient in the surrounding medium. The mechanism of this effect is outlined in the schematic of Fig. 11(a), where a driven skyrmion is moving under a finite driving force F D . For intermediate drives, the skyrmion tries to translate along θ int sk , which is at a negative angle with respect to the x axis. In the process, it displaces some of the surrounding skyrmions in the negative y direction, creating a local density imbalance ∆ρ such that the density is higher below the driven skyrmion and lower above it. Due to the repulsive pairwise interactions with the surrounding skyrmions, there is an unbalanced force in the positive y direction on the driven skyrmion, as indicated by the black arrow in the schematic. The Magnus term transforms this repulsive force into a velocity in the positive x- direction, parallel to the drive, as indicated by the green arrow. Thus, the boost velocity V boost = |V | − V 0 ∝ ∆ρ.
Here the x-direction velocity from the damping term is aligned with but opposite to the velocity induced by the Magnus force. In contrast, when F D is larger, the surrounding skyrmions do not have time to respond to the rapidly moving driven skyrmion, producing a small ∆ρ and reducing V boost , as shown schematically in Fig. 11(b). For drives that are only slightly above the threshold for motion, the driven skyrmion is moving sufficiently slowly that the surrounding skyrmions have enough time to completely relax any density perturbations, making ∆ρ small. The maximum boost velocity changes as a function of F D and ρ due to the different relaxation times. In general, at a fixed drive |V | has a damping dependence of |V damp | ∝ 1/ρ and a boost velocity contribution of |V boost | ∝ (α m /α d )∆ρ, where ∆ρ is a function of both the unperturbed density and the drive. For increasing ρ, larger values of ∆ρ can appear but the threshold driving force for the formation of density inhomogeneities also increases. For small ρ, ∆ρ is reduced and the boosting effect is diminished. The boosting effect occurs for any finite value of the Magnus term, but in general only when α m| /α d > 1.0 do we find a regime in which the net velocity is higher than the single particle limit.
Another effect of the emergence of a density inhomogeneity ∆ρ is a reduction in the skyrmion Hall angle at lower drives, since the skyrmions that accumulate below the driven particle partially block the motion in the −y direction. In Fig. 12 we plot θ sk versus F D for the system in Fig. 9 with α m /α d = 9.95 for ρ = 0.05, 0.1, 0.25, 0.5, 1.0, 1.5, and 2.0. The dashed line indicates the single particle limit, where θ sk = − arctan(α m /α d ) for all values of F D . We find a finite interval of F D over which the magnitude of θ sk increases with increasing F D , followed by a saturation close to the intrinsic value at higher drives. As the driven skyrmion approaches the saturation regime, it is moving fast enough that that surrounding skyrmions cannot respond to its presence, giving a small ∆ρ and a reduced boost effect, as shown in Fig. 11(b). The increase of θ sk with increasing F D has also been observed for skyrmion assemblies driven over random disorder, where at higher drives the skyrmions are only weakly perturbed by the pinning [29][30][31][32][90][91][92][93][94][95] .
From the features in V ⊥ , V || , |V |, and θ sk , we can construct a dynamic phase diagram as a function of F D versus ρ for the strongly damped and Magnus dominated limits. In Fig. 13 we show the phase diagram for the damping dominated system with α m /α d = 0.1 in Figs. 7 and 8. Red indicates the pinned regime where F D < F c . In the green region, the driven skyrmion is moving but its velocity is strictly along the driving direction so that θ sk = 0.0 • . In the blue region, there is a finite but growing θ sk , while in the orange region, the skyrmion Hall angle has saturated. The thresholds for motion parallel and perpendicular to the driving direction both increase with increasing ρ. Previous work in systems with quenched disorder also showed that there can be separate thresholds for the onset of motion parallel and perpendicular to the drive, with a region above the first depinning threshold where the skyrmions can flow at θ sk = 0 • for small intrinsic skyrmion Hall angles 31,100,101 . As the ratio of the Magnus force to the damping term increases, the threshold for transverse motion shifts to smaller F D . In the saturation regime, the response resembles the single particle limit and θ sk approaches the intrinsic skyrmion Hall value of θ int sk = 5.7 • . In Fig. 14 we show the dynamic phase diagram as a function of F D versus ρ for the Magnus dominated system with α m /α d = 9.95. There is still a pinned regime, but the region in which there is finite motion with θ sk = 0.0 • is absent or too small to detect. In the blue region, the velocity is significantly boosted beyond the single particle limit and the skyrmion Hall angle increases with increasing drive. In the orange region, both θ sk and the velocity approach the single particle limit. Between the damping dominated and Magnus dominated limits, the dynamic phase diagram includes a combination of the features of the phase diagrams in Fig. 13 and Fig. 14.

V. VARIED MAGNUS TO DAMPING RATIO AND VELOCITY-FORCE CURVES
We next fix the density ρ and vary the ratio of the Magnus and damping terms while maintaining the normalization relation α 2 d + α 2 m = 1.0. In Fig. 15(a) we plot V || versus F D for systems with ρ = 0.5 at α m /α d = 9.95 and 0.1. Figures 15(b) and (c) show the corresponding values of V ⊥ and θ sk versus F D , respectively. For the Magnus dominated case of α m /α d = 9.95, V || and V ⊥ become finite at almost the same value of F D , which also corresponds to the appearance of a nonzero θ sk . For the damping dominated sample with α m /α d = 0.1, V ⊥ and θ sk remain zero up to F D ≈ 1.0.
The ratio of the Magnus to the damping term determines the shape of the velocity-force curves. For α m /α d = 0.1, both V || and V ⊥ increase monotonically with F D according to the linear behavior V ∝ (F D − F c ), where F c is the threshold for motion in either the parallel or the perpendicular direction. When α m /α d = 9.95, V ⊥ still increases linearly with increasing F D above the depinning threshold; however, V || exhibits a nonmonotonic behavior in which it initially rises rapidly with increasing F D but then decreases again. In this regime,   Fig. 15 with ρ = 0.5 and α 2 m + α 2 d = 1.0 at αm/α d = 9.95 (green) and 0.1 (red). The dashed lines are the expected velocity-force curves in the single particle limit for these two cases. (b) The corresponding d V || /dFD versus FD, where there is a region of negative differential conductivity with d V || /dFD < 0.0 for the αm/α d = 9.95 sample. the velocity decreases even though the drive is increasing, giving d V || /dF D < 0, which is known as negative differential conductivity 21 . In Fig. 16(a) we plot V || versus F D for the samples from Fig. 15. The green dashed line shows the expected value for V || in the single particle limit at α m /α d = 9.95, which increases linearly according to V || = 0.995F D . For the α m /α d = 0.1 sample, the red dashed line is the single particle limit V || = 0.1F D . In the damping dominated system with α m /α d = 0.1, V || is below the single particle limit, while in the Magnus dominated system with α m /α d = 9.95, V || is higher than the single particle limit due to the boosting effect. In Fig. 16(b) we show the corresponding values of d V || /dF D , with the single particle limits marked by dashed lines. For the α m /α d = 9.95 system, the initial peak in d V || /dF D corresponding to the depinning transition is followed by a region of negative differential conductivity where d V || /dF D < 0.0. At higher drives, d V || /dF D saturates to the value expected in the single particle limit. For the damping dominated system with α m /α d = 0.1, d V || /dF D is initially below the single particle value due to the increased damping from the surrounding skyrmions, while at higher drives it approaches the single particle limit. The negative differential conductivity in the Magnus-dominated system is affected by the density, and for low ρ the shape of the velocity-force curve approaches that found in the single particle limit.
In Fig. 17(a) we plot V ⊥ versus F D for the system in Fig. 15  system and V ⊥ = −0.995F D for the α m /α d = 9.95 system. In the Magnus dominated regime, | V ⊥ | is slightly larger than it would be in the single particle limit, while in the damping dominated regime it is slightly lower. Most of the velocity boost in the Magnus dominated regime is parallel to the driving direction since the induced density gradient is perpendicular to the drive. In Fig. 17(b) we show the corresponding d V ⊥ /dF D versus F D curves where, unlike the parallel velocity plotted in Fig. 16, there is no regime of negative differential mobility. There is still a peak in |d V ⊥ /dF D | near the depinning threshold, while at high drives the curves approach the single particle limit.
In Fig. 18(a) we plot V || versus F D for the system in Fig. 15 with fixed ρ = 0.5 and α 2 m + α 2 d = 1.0 at α m = 0, 0.1, 0.3, 0.5, 0.7, 0.8, 0.9, 0.97, 0.995, and 0.998. Figures 18(b) and (c) show the corresponding V ⊥ and θ sk , respectively, versus F D . We find negative differential conductivity for α m > 0.9 or α m /α d > 5.0, indicating that this effect appears only when the Magnus force is sufficiently large. There is no negative differential conductivity in the V ⊥ versus F D curves, while the plots of θ sk versus F D indicate that as α m decreases, the threshold drive above which the skyrmion Hall angle becomes finite increases.
From the features in Fig. 18 we can construct a dynamic phase diagram as a function of F D versus α m , as shown in Fig. 19. Here we outline the pinned phase, the flowing regime with θ sk = 0.0 • , the region in which θ sk increases with increasing F D , and the saturation regime where there is little change in θ sk . For drives above the dashed line, the velocities are boosted compared to the single particle limit. The boost is strongly reduced in the saturation regime. Figure 19 indicates that the depinning threshold is approximately constant as a function of increasing α m , while the regime in which the velocity is locked in the driving direction grows in extent with decreasing α m . At α m = 0, the onset of saturation coincides with the point where the velocity-force curves start to grow linearly with F D . The transition demarcating the onset of a velocity boost shifts to lower α m as ρ increases.
We next relax the constraint of α 2 d + α 2 m = 1.0 and instead hold either the Magnus or damping term constant while varying the other quantity. For a fixed drive, this means that in the single particle limit, |V | obeys Fig. 20(a) we plot V || and V ⊥ versus α d for a system with ρ = 0.05, F D = 0.5, and α m = 1.0. Here V || is zero for α d = 0.0, increases to a maximum value near α d = 1.0, and then decreases with increasing α d , while V ⊥ gradually approaches zero as α d increases. In the single particle limit, , plotted as a dashed line. In this case, V || = 0.0 when α d = 0.0, and the parallel velocity also approaches zero at large α d . Similarly, in the single particle limit, V ⊥ = −F D /(1 + α 2 d ), so that when α d = 0.0, V ⊥ = −F D = −0.5. For ρ = 0.05, the density is low enough that the behavior is close to the single particle limit. In Fig. 20(b) we plot |V | versus α d for the system in Fig. 20(a), where the dashed line is the single particle limit of |V | = F D /(1 + α 2 d ) 1/2 . There is a small boost in |V | for α d < 1.0, while |V |  The same but at ρ = 0.5, where the skyrmion Hall angle is smaller and the velocity of the driven skyrmion is higher. drops below the free particle limit for α d > 1.5. Figure 20(c) shows the corresponding θ sk versus α d for the ρ = 0.05 system, where the magnitude of θ sk gradually decreases with increasing α d . The dashed line is the single particle limit, θ sk = −F D arctan(1/α d ). The measured skyrmion Hall angle is smaller in magnitude than the single particle value due to the collisions with the background skyrmions. In Fig. 21(a)  In Fig. 20(d) we show V || and V ⊥ versus α d for the same system in Fig. 20(a) but at a higher density of ρ = 0.5. The dashed lines are the expected behavior in the single particle limit. Here there is a boost in V || for α d < 0.75, while V || drops below the single particle limit at higher α d . There is also a small boost in V ⊥ for α d < 0.3. We note that unlike the low density case, V || is finite at α d = 0.0 due to collisions with the background skyrmions. Figure 20(e) shows the corresponding |V | versus α d and the expected single particle limit, indicating that there is a net velocity boost for α d < 0.5 followed by increased damping for α d > 0.5. In Fig. 20(f) we plot the corresponding θ sk versus α d and the single particle limit. The skyrmion Hall angle rapidly decreases in magnitude with increasing α d , reaching a saturation for α d > 0.5. Figure 21(b) shows the positions and trajectories of the driven and bath skyrmions for the system in Fig. 20(d) at ρ = 0.5 and α d = 0.3, where the skyrmion Hall angle is much smaller than that found in the ρ = 0.05 system illustrated in Fig. 21(a); however, the velocity of the driven skyrmion is much larger for the ρ = 0.5 system.
In Fig. 22(a) we plot V || and V ⊥ versus α m for a system with fixed α d = 1.0, ρ = 0.05, and F D = 0.5. and V ⊥ = −F D α m /(1 + α 2 m ). We find that V || monotonically decreases and changes from being slightly higher than the free particle limit for α m < 5.0 to being slightly lower than the free particle limit for α m > 5.0. V ⊥ starts from zero at α m = 0.0, reaches a maximum value near α m = 1.0, and gradually drops back to zero with increasing α m , closely following the single particle limit. In Fig. 22(b) we show the corresponding |V | versus α m along with a dashed line indicating the single particle limit of |V | = F D /(1 + α 2 m ) 1/2 , which decreases monotonically with increasing α m . There is a small boost in the velocity due to the collisions with the background skyrmions. Figure 21(c) shows θ sk versus α m for the same system as well as the single particle limit of θ sk = − arctan(α m ). Here the measured skyrmion Hall angle is slightly smaller in magnitude than the single particle limit due to the collisions with bath skyrmions.
In Fig. 22(d) we plot V || , V ⊥ , and the single particle limits for the same system from Fig. 22(a) but at a higher density of ρ = 0.5. Here V || is considerably below the single particle limit at α m = 0.0 due to the increased frequency of collisions; however, for α m > 2.5, it is considerately higher than the single particle limit. V ⊥ is close to zero for α m < 7.0 but begins to increase at larger α m . In Fig. 22(e) we show the corresponding |V | versus α m . |V | falls below the dashed line representing the single particle limit up to α m = 7.0 and exhibits a boost for higher α m . Figure 22(f) illustrates the corresponding θ sk versus α m and the single particle limit. θ sk is close to zero for α m < 7.0, while for higher α m , the magnitude of θ sk increases and a velocity boost appears. For higher values of F D , all of the quantities gradually approach the single particle limit.
In Fig. 23(a) we plot V || and V ⊥ versus ρ for a sys-tem with fixed α m = 1.0, α d = 0.3, and F D = 0.5. The single particle limit for these parameters gives V || = 0.14 and V ⊥ = −0.48. V || initially increases with increasing ρ to a peak value of 0.65, a strong boost that is four times larger than the single particle limit. As ρ increases further, V || gradually decreases; however, even at the high density of ρ = 3.0, V || is still almost twice as large as the single particle limit, indicating the continuing effectiveness of the boost effect. V ⊥ decreases monotonically in magnitude with increasing density, approaching a value close to zero for ρ > 0.7. In Fig. 23(b) we show the corresponding |V | versus ρ, where the dashed line is the single particle limit of |V | = F D /(α 2 d + α 2 m ) 1/2 which for these coefficients is |V | = 0.4789. Here there is a boost in the velocity up to ρ = 0.7, while at higher densities the velocities are much more strongly damped. This indicates that it is possible for the parallel velocity to be boosted while the overall velocity is not boosted. Figure 23(c) illustrates the corresponding θ sk versus ρ. For low density, the skyrmion Hall angle is near the intrinsic value of θ int sk = −73.3 • , and as ρ increases, θ sk approaches zero once ρ > 0.7.
In Fig. 23(d) we plot V || and V ⊥ versus ρ for a system with a larger α d = 3.0 at α m = 1.0 and F D = 0.5. For these parameters, the single particle limit gives V || = 0.15 and V ⊥ = −0.05. We find that both V || and V ⊥ monotonically decrease in magnitude with increasing ρ. In Fig. 23(e) we show the corresponding |V | versus ρ, where the dashed line is the single particle limit of |V | = 0.158. Here there is no boost and |V | drops off rapidly with increasing ρ, showing a change in slope near ρ = 0.3 to a slower decline. Figure 23(f) illustrates the corresponding θ sk versus ρ, which starts off near θ sk = −18.4 • and approaches zero for ρ > 0.3. The change in the skyrmion Hall angle and velocity across ρ = 0.3 occurs because the density is low enough for ρ < 0.3 that the bath skyrmions act like a fluid which is not strongly coupled to the driven skyrmion, while when ρ > 0.3, the bath skyrmions act more like a solid, increasing the drag on the driven skyrmion.

VI. DISCUSSION
In our work, the driven skyrmion is free to move in any direction. In certain chiral soft matter systems, a similar probe particle could be implemented using a magnetic or charged particle coupled to a uniform magnetic or electric field which does not couple to the remaining particles, allowing the probe particle to move at any angle. In skyrmion systems, it is possible to have samples containing multiple species of skyrmions, some of which could couple more strongly than others to an externally imposed field. The closest experimental realization of our system for skyrmions would be to drag a single skyrmion using some form of localized trap. Such an arrangement would constrain the driven skyrmion to move only in the direction the trap is being translated, and would not al-low the driven skyrmion to move at a speed greater than that of the trap. In this case, changes in the effective viscosity could be deduced by measuring the force or the fluctuations exerted by the skyrmion on the trap. The case of a trap moving at a constant velocity will be studied in another work; however, the results of the present study can be used as a guide to understand which different velocity regimes could arise.
In our work we have only considered a particle-based model, which neglects internal degrees of freedom and shape changes of the skyrmion. Such modes could increase or decrease the damping experienced by the driven skyrmion or change the nature of the skyrmion motion. We have also assumed a simple pairwise repulsion between skyrmions, but it is possible for skyrmions to have more complex interactions, such as competing interactions at different length scales. This could produce additional coupling/decoupling or depinning transitions.
In constant velocity experiments, the force the skyrmion experiences could have a periodic signature if the motion occurs through a skyrmion solid, or a broad band noise signature if the skyrmion is moving through a glass or liquid state. Numerical work 90,92 and experiments 102,103 on collectively moving skyrmions have shown the presence of both broad and narrow band noise, so it would be interesting to study the fluctuations exerted on a single skyrmion as it moves through a bath of other skyrmions. We can also compare our results to active rheology in overdamped chiral granular systems, where a disk with short range repulsive interactions is pushed through an assembly of spinning grains 58 . The granular system generally does not exhibit any velocity boost due to its overdamped nature. The probe particle in the granular system has no intrinsic Hall angle, but as function of driving force shows a finite Hall angle at intermediate drives, with no Hall angle at low or high drives. The finite Hall angle arises as a result of collisions between the probe particle and the spinning disks, which create a deflection of the probe particle perpendicular to the driving direction. This deflection decreases in magnitude as the velocity of the probe particle increases.

VII. SUMMARY
We have numerically examined the active rheology of a single skyrmion driven through a bath of other skyrmions in the absence of quenched disorder. Active rheology has been used to study the changes in drag on driven probe particles in various soft matter and superconducting vortex systems where the dynamics is overdamped. In those systems, the velocity of the probe particle under a constant driving force rapidly decreases with increasing bath particle density due to an increased dragging effect. For skyrmions, which have a strong Magnus force, we find that the behavior differs strongly from what is observed in the damping dominated limit. The driven skyrmion velocity in the driving direction is highly non-monotonic as a function of density, and can increase rather than decreasing when the density is increased. This effect appears as a boost in the net velocity. At higher densities, the velocity decreases with increasing density. The skyrmion Hall angle also decreases as the bath density increases. The magnitude of the velocity boost depends on the system density, the strength of the Magnus term, and the applied drive. For a fixed density, as we increase the driving force we find a critical threshold force below which the driven skyrmion does not move, a regime in which the magnitude of the skyrmion Hall angle increases with drive, and a regime at higher drive where the skyrmion Hall angle saturates. The drive dependence of the skyrmion Hall angle is similar to that observed for skyrmions driven over quenched disorder. If the Magnus force is dominant, a velocity boost appears which is maximum for an intermediate drive and diminishes at higher drives. When the damping force is strong, the velocities are reduced but approach the single particle limit at higher drives. The velocity-force curves in the damped regime have linear or monotonic behavior, while in the Magnus dominated regime, the velocity in the driving direction can decrease with increasing drive, leading to negative differential conductivity. The maximum velocity boost shifts to higher drives with increased density. The velocity boost originates when the driven skyrmion moves at a finite skyrmion Hall angle and creates a localized density inhomogeneity in the background skyrmions, which generate an unbalanced pairwise repulsive force on the driven skyrmion perpendicular to the driving direction. The Magnus force then converts this force into an additional velocity component in the direction of drive. At low bath densities, the localized density fluctuation relaxes quickly and the velocity boost is small, while at high drives the driven skyrmion moves too quickly past the bath skyrmions for a localized density fluctuation to form, so the velocity boost is again reduced. We find regimes in which the skyrmion Hall angle decreases with a simultaneous increase in the skyrmion velocity, suggesting that skyrmion-skyrmion interactions can be useful for producing effects that are of value for use in devices. We discuss possible experimental realizations of this system where a single skyrmion could be driven with some form of tip or optical trap while deflection forces on the skyrmion are measured. Beyond skyrmions, our results should be relevant to any kind of active rheology in systems with gyroscopic forces, such as active chiral matter, fluid vortices, electrons in a magnetic field, fractons, and other gyroscopic systems.

ACKNOWLEDGMENTS
We gratefully acknowledge the support of the U.S. Department of Energy through the LANL/LDRD program for this work. This work was supported by the US Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is oper-