Magnetic field effects and transverse ratchets in charge lattices coupled to asymmetric substrates

We examine a charge lattice coupled to a one-dimensional asymmetric potential in the presence of an applied magnetic field, which induces gyrotropic effects in the charge motion. This system could be realized for Wigner crystals in nanostructured samples, dusty plasmas, or other classical charge-ordered states where gyrotropic motion and damping can arise. For zero magnetic field, an applied external ac drive can produce a ratchet effect in which the particles move along the easy flow direction of the substrate asymmetry. The zero field ratchet effect can only occur when the ac drive is aligned with the substrate asymmetry direction; however, when a magnetic field is added, the gyrotropic forces generate a Hall effect that leads to a variety of new behaviors, including a transverse ratchet motion that occurs when the ac drive is perpendicular to the substrate asymmetry direction. We show that this system exhibits commensuration effects as well as reversals in the ratchet effect and the Hall angle of the motion. The magnetic field also produces a nonmonotonic ratchet efficiency when the particles become localized at high fields.


Introduction
When overdamped particles are coupled to an asymmetric substrate, a ratchet effect can occur when an ac drive is applied, leading to a net dc transport in the direction of the substrate asymmetry.Ratchet effects have been studied for various systems in the single particle limit [1,2,3], including colloidal systems [4,5], biological systems [6], active matter [7], granular matter [8], cold atoms [9,10], superconducting vortices [11,12] and quantum systems [13].For a single overdamped particle, the ratchet effect produces directed motion in the easy direction of the substrate asymmetry.When collective interactions become important, new behaviors emerge, including commensuration effects that appear when the spacing between the particles matches the substrate length scale.As a result, peaks or dips appear in the ratchet efficiency, as observed in interacting disk systems [14], active matter systems [15] and superconducting vortices [16,17,18,19,20,21,22,23,24,25,26,27].Collective effects can also produce ratchet reversals in which the net dc flux is in the hard direction of the substrate asymmetry [16,17,18,19,15].In many of these systems, the interactions are short-range; however, collective ratchet effects can also occur in systems with longer-range interactions where the particles would naturally form a triangular lattice in the absence of a substrate, such as vortices in type-II superconductors [16,19] or magnetic colloids [28].Ratchet effects can also appear in charged lattices, which arise in solidstate systems including Wigner crystals [29,30,31,32], charged colloids with weak screening [33], optically trapped ions [34], and dusty plasmas [35].
Charged systems form crystalline states when the Coulomb energy dominates over the thermal or kinetic energy, and they can exhibit ratchet effects when coupled to an asymmetric substrate [35,36].Little is known, however, about how the ratchet effect is modified when a magnetic field is applied that causes the moving charges to undergo cyclotron-like displacements [37,38].Previous work on so-called electron pinball states showed that cyclotron motion can strongly influence the transport of charged systems coupled to a periodic substrate, resulting in electron localization when the cyclotron orbits become commensurate with the periodicity of the substrate or permitting delocalized and chaotic motion for incommensurate orbits [39,40].When a dc drive is applied in addition to the magnetic field, the particles move with a finite Hall angle, and in the presence of a two-dimensional (2D) periodic substrate, the motion becomes locked to specific symmetry directions of the substrate, producing a quantization of the Hall angle as a function of the magnetic field [41,42].Recently, it was shown that when a Wigner crystal in a magnetic field is driven over random disorder, the sliding dynamics occur at a finite Hall angle that is drive dependent since the charges are more strongly scattered by the disorder at low drives [43].
For particles coupled to one-dimensional (1D) asymmetric substrates, a ratchet effect only occurs when the ac drive is applied parallel to the substrate asymmetry direction; however, once a Hall effect and cyclotron motion are introduced, it should be possible for a ratchet effect to appear even for ac drives applied perpendicular to the substrate asymmetry direction, with the ratcheting motion consisting of a combination of motion parallel and perpendicular to the drive.In magnetic skyrmions, where the Magnus force can have a strong impact [44,45] and can cause the skyrmion motion to obey the same dynamics as charges in magnetic fields [46,44], there have been several studies showing that new kinds of gyrotropic-induced ratchet effects arise [25,47,48,49].A transverse skyrmion ratchet was also observed in which the ac driving can be applied at any angle with respect to the substrate asymmetry direction [50,47].Since the chiral motion itself breaks a symmetry, ratchet effects in skyrmion systems can even occur when the substrate is symmetric [51].Ratchet effects have been found in other systems with chiral motion, even when the substrates are symmetric [52,53,35].This suggests that similar gyrotropic ratchet effects can arise in nonskyrmion systems of charged particles in a magnetic field.
Previous work on skyrmions interacting with an asymmetric 1D substrate focused only on the single skyrmion limit [25], so collective effects on the motion of gyrotropic particles over an asymmetric substrate have not yet been addressed.Additionally, the interactions for skyrmions generally involve either short-range or intermediate repulsion.Open questions include how a strongly interacting system with gyrotropic dynamics would behave on an asymmetric substrate, and what impact collective effects would have on the behavior.
Here, we consider a 2D assembly of charged particles in the presence of an asymmetric 1D substrate under an ac drive applied either parallel or perpendicular to the substrate asymmetry direction.We specifically study the effect of an applied magnetic field, which creates velocity components that are perpendicular to the net force experienced by a charged particle.When the magnetic field is zero, a ratchet effect occurs only when the ac drive is applied parallel to the substrate asymmetry direction, as found in previous work [36].For finite magnetic fields, we find that ratchet effects can occur when the ac driving is either parallel or perpendicular to the substrate asymmetry direction.In general, the direction of motion is at an angle with respect to the direction of the ac drive.The efficiency of the transverse and other magnetic ratchet effects is generally non-monotonic as a function of field since the moving charges become localized at higher fields and execute small cyclotron orbits that remain confined within a single pinning trough.We also observe commensuration effects that arise when the cyclotron orbit radius matches the substrate periodicity, as well as a number of ratchet reversals that produce a reversal in the Hall angle.In addition to providing examples of new types of ratchet effects, our results suggest that a transverse ratchet may be used to detect the presence of Wigner crystals in a magnetic field.

Simulation
We consider a two-dimensional system with periodic boundary conditions in the x and y directions containing N e particles with repulsive Coulomb interactions.The particles also interact with a one-dimensional asymmetric substrate.The sample is of size L × L with L = 36 and the particle density is ρ = N e /L 2 .The overdamped equation of motion for charge i is given by Here α d is the damping constant, the particle-particle interaction potential is U (r ij ) = q/r ij , r i and r j are the positions of particles i and j, respectively, r ij = |r i − r j | is the distance between particles, and q is the particle charge which we set to unity.For computational efficiency, we employ a Lekner method for calculating the long-range Coulomb interactions, as used in previous work on charged particles interacting with a substrate [86,87,81].The second term on the right hand side of Eq.
(1) describes the Magnus force produced by a finite magnetic field B = Bẑ applied perpendicular to our 2D sample.
The substrate force F sub = ∇U (x i ) is determined by the x coordinate of particle i, where the substrate potential U (x) has the form This potential produces an asymmetric force profile with a larger force in the negative x-direction.We measure forces in terms of the parameter A p = U 0 /2π, and we obtain a maximum force in the hard or negative x direction of F hard p = 1.5A p and a maximum force in the easy or positive x direction of F easy p = 0.75A p .We have previously used this potential to study Wigner crystal ratchet effects in the absence of a magnetic field [36].We concentrate on a system size of L = 36 with a substrate lattice spacing of a = 2.117.
The ac driving force has the form F AC = F AC sin(ωt)α, where F AC is the amplitude and α = x or y for driving parallel or perpendicular to the substrate asymmetry direction, respectively.We use a time step of dt = 0.005 and set ω = 0.0000754, so our typical ac drive cycle spans 10 5 simulation time steps.If we decrease ω to a lower frequency we find no changes in the behavior; thus, our results are of relevance to the low frequency limit.In Section 5 we consider the effects of thermal fluctuations by including the term F T , which represents Langevin kicks with the properties ).We measure the average velocity per charge in the x and y directions, ⟨V x ⟩ = N −1 e Ne i ⟨v i • x⟩ and ⟨V y ⟩ = N −1 e Ne i ⟨v i • ŷ⟩, where the time average is taken over 50 ac drive cycles.We also measure |⟨V ⟩| = ⟨V x ⟩ 2 + ⟨V y ⟩ 2 .When B ̸ = 0 and a magnetic field is present, the charges move at a Hall angle with respect to the drive, which we measure according to θ H = arctan(⟨V y ⟩/⟨V x ⟩).
The system also has an intrinsic Hall angle, given by θ int H = arctan(qB/α d ), which increases with increasing B. Interactions with the substrate can cause θ H to be different from θ int H , as shown in previous work [43].Wiersig et al. [41] considered charges moving over a 2D periodic array of obstacles under applied magnetic fields that would produce intrinsic Hall angles as large as θ int H = 60 • .In Fig. 1(a), we show an image of the system with arrows indicating the ac driving direction that is either parallel to the substrate asymmetry direction, giving a longitudinal ratchet effect, or perpendicular to the substrate asymmetry direction, giving a transverse ratchet effect.Figure 1(b) shows a detail of the substrate potential U (x) for a sample where the substrate lattice period is a = 1.

Results
In Fig. 2(a) we plot ⟨V x ⟩ and ⟨V y ⟩ versus qB/α d for a system with ρ = 0.208, A p = 0.75, and F AC = 1.0.Here, the ac drive is applied along x, parallel to the substrate asymmetry direction.For this system, the maximum substrate forces are F hard p = 1.125 and F easy p = 0.5625.When B = 0.0, we find a pronounced longitudinal ratchet effect with ⟨V x ⟩ ̸ = 0, as studied in previous work [36].Figure 2(b) shows the average velocity |⟨V ⟩| versus qB/α d for the same sample.In Fig. 3(a), we illustrate the particle locations and substrate along with the trajectory of a representative particle during 10 ac drive cycles.The same type of trajectory is followed by all of the other particles, each of which moves only along the x-direction over time with a net motion occurring in the +x direction.In Fig. 2(c), we plot |θ H |, the absolute value of the measured Hall angle, versus qB/α d .As B increases, ⟨V y ⟩ becomes nonzero when the particles begin to move along the negative y-direction, and a finite Hall angle emerges.With a further increase of B, |⟨V y ⟩| reaches its maximum value near qB/α d = 0.8 and then diminishes again.The overall velocity |⟨V ⟩| generally decreases with increasing B. For qB/α d = 0.5, the particles move close to an angle of θ H = 30 • , as illustrated in Fig. 3(b), while at qB/α d = 2.0, Fig. 3(c) shows that the particles move along 62 • since the Hall angle increases with increasing B. The dashed line in Fig. 2(c) indicates what the Hall angle would be if A p = 0.0, as determined by the expression θ H = arctan(qB/α d ).Near qB/α d = 1.7, we find a local peak in the velocities, corresponding to the field at which the orbit size partially matches the periodicity of the substrate.For qB/α d ≥ 2.5, the drift velocities drop to zero because the particle motion becomes localized, as illustrated in Fig. 3(d  particles.For larger B, the particle orbits become more compact and remain confined in a single pinning trough. In Fig. 4 we plot ⟨V x ⟩ and ⟨V y ⟩ versus qB/α d for the same system as in Fig. 2 but with the ac drive applied perpendicular to the substrate asymmetry direction.For qB/α d < 0.6, there is no ratchet effect and both ⟨V x ⟩ and ⟨V y ⟩ are zero.Figure 5(a) shows the trajectory of a single particle at qB/α d = 0.5, where the particles only move along the y direction and remain localized.There is a large transverse ratchet effect for 0.6 ≤ qB/α d < 1.9, a reduced ratchet effect appears for 1.9 ≤ qB/α d < 3.9, and for qB/α d ≥ 3.9, the motion is localized again.In Fig. 5(b) at qB/α d = 1.2, the orbit of a single particle consists of a combination of sliding motion along y and jumps along x, giving a net motion that is in the positive x and negative y direction.For 1.9 ≤ qB/α d < 3.9, the motion is much more chaotic and produces a gradual drift, as illustrated in Fig. 5(c) at qB/α d = 2.5. Figure 5(d) shows that at qB/α d = 4.0, the motion has become localized within the pinning troughs.In Fig. 4(c) we plot |θ H | versus qB/α d along with a dashed line that indicates the behavior expected in a substrate-free system.A dip appears when 1.9 < qB/α d < 2.5, where the motion is primarily chaotic with a reduced velocity, as is also visible in the plot of |⟨V ⟩| versus qB/α d in Fig. 4(b).Our results demonstrate that a magnetic field can induce a ratchet effect even when the ac drive is applied perpendicular to the substrate asymmetry direction, and that this ratchet effect is non-monotonic as a function of the field.The maximum net velocity for the transverse ratchet is close to that of the longitudinal ratchet found for ac driving applied parallel to the substrate asymmetry direction.We find that there is a critical field that must be applied in order for the traverse ratchet to occur, and that the value of this field depends on the strength of the substrate.In Fig. 6(a), we plot ⟨V x ⟩ and ⟨V y ⟩ versus A p for the same system from Fig. 2 with F AC = 1.0 and qB/α d = 1.2 under driving perpendicular to the substrate asymmetry direction.Figure 6(b) and (c) show the corresponding |⟨V ⟩| and |θ H | versus A p .There is no ratchet effect at low A p since the system forms a lattice that effectively floats above the substrate.We find several regions where the ratchet effect goes to zero because the particle orbits become localized.For A p > 1.5, the system is pinned.In Fig. 6(c), the Hall angle in the ratcheting regimes is close to the value expected for a pin-free system.Figure 6(d,e,f) shows ⟨V x ⟩, ⟨V y ⟩, |⟨V ⟩|, and |θ H | versus A p for the same system but for ac driving parallel to the substrate asymmetry direction.Here the ratchet effect is lost for both low and high A p values.
In Fig. 7(b), we plot time series of the instantaneous values of V x and V y (red) for the system in Fig. 2 at qB/α d = 0.4 for ac driving applied parallel to the substrate asymmetry direction, where we find velocity spikes in the positive x direction and smaller spikes in the negative y direction.The additional features in the time series correspond to particle configurations that form during specific portions of the ac drive cycle, since the overall structure of the particles can change as the ac drive direction varies throughout the cycle.At qB/α d = 0.0, Fig. 7(a) shows that spikes are still present in V x but are absent for V y .In Fig. 8, we plot the instantaneous V x and V y versus time for the same system with ac driving applied perpendicular to the substrate asymmetry direction.At qB/α d = 0.4 in Fig. 8(a), there are strong oscillations in y but the oscillations are symmetric so ⟨V y ⟩ = 0.There are much smaller oscillations in x but these oscillations are also symmetric and ⟨V x ⟩ = 0.In Fig. 8(b) at qB/α d = 1.2, there are strong spikes in V x in the positive x direction corresponding to net motion of the particles in this direction.Both positive and negative velocity spikes appear in V y , but the spikes are stronger in the negative y direction, leading to ratcheting motion along −y. Figure 8(c) shows that at qB/α d = 4.0, there are velocity oscillations in both directions but ⟨V x ⟩ and ⟨V y ⟩ are both zero.

Ratchet Effects for Varied ac drives
In Fig. 9(a) we plot ⟨V x ⟩ and ⟨V y ⟩ versus F AC for the system in Fig. 6 at A p = 0.75 and qB/α d = 1.2 for driving parallel to the substrate asymmetry direction.Here, the velocities are zero for F AC < 0.6 because the ac drive is not large enough to permit the particles to jump out of the the wells.For 0.6 ≤ F AC < 1.8, there is a strong ratchet effect where the particles move in the positive x and negative y directions, giving θ H ≈ 55 • .We find a series of regions where the ratchet effect drops to zero, corresponding to ac drive amplitudes where the particle orbits become trapped inside the pinning troughs.In other regions, the ratchet effect is reduced but there is still finite ratcheting In both cases, there are regions where the ratchet efficiency drops to zero.
motion corresponding to orbits in which the particles move one lattice constant every other ac cycle or every nth ac cycle.For ac driving applied perpendicular to the substrate asymmetry direction, Fig. 9(b) shows a similar pattern of regions of reduced or zero ratcheting; however, the F AC values at which zero velocities appear are shifted relative to the parallel driving case since the particle orbits are more tilted under perpendicular driving.Behavior similar to that shown in Fig. 9, with oscillations in the ratchet effectiveness as a function of ac amplitude, was observed previously in purely overdamped systems such as superconducting vortices on asymmetric 1D substrates [11].In general, we find that the oscillations in ratchet efficiency are the most pronounced for low particle densities.At higher densities, oscillations appear for fillings at which the system is relatively ordered, while for incommensurate fillings, regions where the ratchet effect drops to zero are replaced with regions of finite but reduced ratchet effect.In Fig. 10(a), we illustrate the particle locations and trajectories for the system in Fig. 9(b) for driving along y, perpendicular to the substrate asymmetry direction, at F AC = 0.5 where the ratchet effect is zero.In this case, the particles oscillate in closed orbits, and there is no hopping from one well to the next.Figure 10(b) shows the trajectory of a single representative particle at F AC = 1.41 where ⟨V x ⟩ = ⟨V y ⟩ = 0.Here each particle moves in a closed orbit spanning three pinning potential minima.At F AC = 1.6, plotted in Fig. 10(c), there is a finite ratchet effect.Here, each particle follows a ratcheting trajectory that covers close to 3.5 pinning troughs per orbit.In Fig. 10(d) at F AC = 1.7, the particles form closed orbits, and Fig. 9(b) indicates that there is zero ratchet effect.In this case, each particle circulates among four pinning minima.The ratchet effect for driving along y, perpendicular to the substrate asymmetry direction, at F AC = 0.5 where there is no ratchet effect.(b,c,d) The same but with the trajectory of only a single representative particle shown.(b) F AC = 1.41,where there is no ratchet effect and the particle orbit is localized.(c) F AC = 1.6,where there is a finite ratchet effect.(d) F AC = 1.7,where the ratchet effect is absent and the particle motion is localized.
is generally lost in orbits where the particles are confined between n pinning minima, while there is a finite ratchet effect for incommensurate orbits.For driving along x, parallel to the substrate asymmetry direction, a similar trapping effect occurs for regions where the ratchet effect is zero; however, the particle orbits are more one-dimensional in character compared to the perpendicular driving case.

Ratchet Reversals for Varied Filling and Substrate Strength
We next consider the effect of varying the charge density ρ.In Fig. 11(a) we plot ⟨V x ⟩ and ⟨V y ⟩ versus ρ for a system with A p = 1.25, qB/α d = 1.2, and F AC = 1.5 for driving perpendicular to the substrate asymmetry direction.For ρ < 1.0, we find a ratchet effect in which the particles move in the positive x and negative y directions.This ratchet motion passes through a local maximum near ρ = 0.5.There is a reversal in the ratchet for ρ > 1.0, where the particles move in the positive y and negative x directions.Figure 11(b) shows the absolute value of the velocity |⟨V ⟩| versus ρ, which peaks near ρ = 0.5 and has a drop near ρ = 1.0 at the point where the ratchet reversal occurs.In the reversed ratchet regime, the absolute value of the velocity is much lower since the motion is produced by a small number of solitons in the lattice that can hop over the substrate barriers in the hard direction.During the +x portion of the ac drive cycle, the particles form a more spread out configuration and the solitons are less well defined.In previous work performed at qB/α d = 0.0, a reversal of the ratchet effect was also observed as a function of increasing filling [36].In Fig. 11(c), we plot ⟨V x ⟩ and ⟨V y ⟩ versus ρ for the same system from Fig. 11(a) but with the ac driving applied perpendicular to the substrate asymmetry direction.For ρ < 1.0, the motion is in the positive x and negative y directions.In the corresponding plot of |⟨V ⟩| versus ρ in Fig. 11(d), there is a dip near ρ = 0.25 due to the formation of a commensurate lattice, which is better pinned by the substrate.For ρ > 1.0, the overall velocity is strongly reduced, and there is a ratchet reversal at ρ ≈ 1.2 where the motion is in the positive y and negative x directions.The magnitude and direction of the ratchet motion depends on both the filling fraction and the substrate strength.For lower fillings of ρ < 0.77, the ratchet motion is generally in the positive x direction; however, for higher fillings, the ratchet motion can be in either the positive or negative x direction.In Fig. 12(a) we plot ⟨V x ⟩ and ⟨V y ⟩ versus F p for a system with ρ = 0.938, F AC = 1.0, and qB/α d = 1.2 under driving along y, perpendicular to the substrate asymmetry, while Fig. 12(b) shows the corresponding |⟨V ⟩| versus F p .For F p < 0.6 there is no ratchet effect.A positive ratchet effect with motion in the positive x and negative y directions appears for 0.6 ≤ F p < 1.3, and is followed by a ratchet reversal for 1.3 ≤ F p < 3.0, where the motion is in the positive y and negative x direction.When the ratchet reverses, the particle motion reverses direction.All of the particles are pinned for F p ≥ 3.0.Both the forward and reverse ratchet effects pass through peak efficiencies, leading to the appearance of a double peak feature in |⟨V ⟩|. Figure 12(c,d) shows ⟨V x ⟩, ⟨V y ⟩, and |⟨V ⟩| versus F p for the same system at a higher density of ρ = 1.27,where only a reversed ratchet effect occurs.In Fig. 13(a,b), we plot heat maps of ⟨V x ⟩ and ⟨V y ⟩ as a function of ρ versus F p for the system from Fig. 12, highlighting both the forward and reverse ratchet effects.For ρ < 0.83, the ratchet motion is primarily in the positive x and negative y directions, with a reversal in the ratchet motion occurring at high densities.It is possible that additional ratchet reversals could occur for higher values of ρ than those considered in this work.

Thermal Effects
We next consider the effect of adding thermal Langevin kicks to the particle motion in order to represent a finite temperature.We use a system with A p = 0.75, ρ = 0.208, and F AC = 1.0, and we report temperature in terms of T /T m , where T m is the temperature at which the substrate-free system thermally disorders.In Fig. 14(a) we plot ⟨V x ⟩ and ⟨V y ⟩ versus T /T m at qB/α d = 0.5 for driving parallel to the substrate asymmetry direction.There is initially a slight increase in the ratchet effect with increasing T /T m followed by a pronounced drop in ratchet motion for T /T m > 1.0.The decrease occurs since the thermal fluctuations reduce the effectiveness of the substrate, and for T /T m > 1.0, thermal hopping is strongly enhanced.Figure 14(b) shows ⟨V x ⟩ and ⟨V y ⟩ versus T /T m for the same system at qB/α d = 3.0, where for T /T m = 0, the magnetic field is large enough to localize the motion and prevent ratchet motion from occurring.In this case, we find that the ratcheting motion reaches a maximum for T /T m > 1.0 and then diminishes; however, the maximum velocity of the ratchet motion is considerably reduced compared to that found at the lower value of qB/α d .In Fig. 14(c) we illustrate ⟨V x ⟩ and ⟨V y ⟩ versus T /T m for the same system at qB/α d = 1.0 but for driving perpendicular to the substrate asymmetry direction.Here the ratchet effect generally decreases with increasing T /T m .At qB/α d = 0.4 in the same system, shown in Fig. 14(d), no ratchet motion occurs when T /T m = 0.0; however, for T /T m > 1.0, thermal hopping of the particles out of the substrate troughs produces a ratchet effect that is gradually destroyed at high temperatures.For qB/α d > 3.0, driving perpendicular to the substrate asymmetry and driving parallel to the substrate both produce similar behavior in which there is finite ratchet motion that appears for T /T m > 1.0 and then diminishes with increasing temperature.These results show that even for systems with strong thermal effects or in a Wigner liquid, the transverse ratchet effects should be robust.

Summary
We have examined ratchet effects for a two-dimensional system of charged particles in a magnetic field interacting with a one dimensional asymmetric substrate under ac driving applied either parallel or perpendicular to the substrate asymmetry direction.When the magnetic field is zero, a ratchet effect only occurs when the ac drive is parallel to the substrate asymmetry direction.Under a finite magnetic field, however, a transverse ratchet effect can occur in which the particles move both parallel and perpendicular to the substrate asymmetry direction due to the finite Hall angle produced by the cyclotron motion of the particles.When the ac drive is applied perpendicular to the substrate asymmetry direction, a transverse ratchet effect can occur.For fixed ac driving amplitude, the ratchet effect drops to zero at higher fields when the charge motion becomes localized inside the pinning troughs.We have investigated this ratchet effect for varied magnetic fields, substrate strengths, ac drive amplitudes, and charge density.We find that there can be reversals of the ratchet effect at higher charge densities, where the particles move against the easy flow direction of the substrate asymmetry.The reversed ratchet motion can occur for driving either parallel or perpendicular to the substrate asymmetry direction.Our results demonstrate that cyclotron motion and the resulting finite Hall angle can produce new kinds of ratchet effects that could be relevant for Wigner crystals, charged colloids, dusty plasmas, and other charged systems coupled to an asymmetric substrate in the presence of a magnetic field.

Figure 1 .Figure 2 .Figure 3 .
Figure1.(a) Image of the simulated 2D assembly of charged particles (red circles) interacting with an asymmetric 1D substrate (green shading).An ac drive is applied along the x direction (blue arrow), parallel to the substrate asymmetry direction, or along the y direction (red arrow), perpendicular to the substrate asymmetry direction.(b) Detail of the substrate potential U (x) for a lattice constant of a = 1.
) where we highlight the trajectories of all of the -

Figure 5 .
Figure 5. (a,b,c) Particle locations (red circles), substrate (green shading), and the trajectory of a single representative particle (line) for the system in Fig. 2 with ρ = 0.208, Ap = 0.75, and F AC = 1.0 for driving along y, perpendicular to the substrate symmetry direction.(a) qB/α d = 0.5 where the motion consists of oscillations along y.(b) qB/α d = 1.2, showing a 2D periodic translating orbit.(c) qB/α d = 2.5, where the motion is more disordered or chaotic.(d) The trajectories for all the particles at qB/α d = 4.0, where the motion is localized.

Figure 9 .
Figure 9. (a) ⟨Vx⟩ (blue) and ⟨Vy⟩ (red) vs F AC for the system from Fig.6with ρ = 0.208 and F AC = 1.0 at Ap = 0.75 and qB/α d = 1.2 for driving along x, parallel to the substrate asymmetry direction.(b) The same for driving along y, perpendicular to the substrate asymmetry direction.In both cases, there are regions where the ratchet efficiency drops to zero.

Figure 10 .
Figure 10.(a) Particle locations (red circles), substrate (green shading), and the trajectories of all the particles (lines) for the system in Fig.9(b) with ρ = 0.208, Ap = 0.75, and qB/α d = 1.2 for driving along y, perpendicular to the substrate asymmetry direction, at F AC = 0.5 where there is no ratchet effect.(b,c,d) The same but with the trajectory of only a single representative particle shown.(b) F AC = 1.41,where there is no ratchet effect and the particle orbit is localized.(c) F AC = 1.6,where there is a finite ratchet effect.(d) F AC = 1.7,where the ratchet effect is absent and the particle motion is localized.

Figure 11 .
Figure 11.(a) ⟨Vx⟩ (blue) and ⟨Vy⟩ (red) vs particle density ρ for a system with qB/α d = 1.2, Ap = 1.25, and F AC = 1.5 for driving along x, parallel to the substrate asymmetry direction.(b) The corresponding |⟨V ⟩| vs ρ.(c) ⟨Vx⟩ (blue) and ⟨Vy⟩ (red) vs ρ for the same system but for driving along y, perpendicular to the substrate asymmetry direction.(d) The corresponding |⟨V ⟩| vs ρ.There is a reversal in the ratchet effect as a function of increasing ρ for both ac driving directions.

Figure 13 .
Figure 13.Heat map of the ratchet velocity as a function of ρ vs Fp for the system in Fig. 12 with F AC = 1.0 and qB/α d = 1.2 for driving along y, perpendicular to the substrate asymmetry direction.(a) ⟨Vx⟩.(b) ⟨Vy⟩.The change in color from red tones to blue tones (or vice versa) indicates a ratchet reversal.