Hard topological versus soft geometrical magnetic particle transport

TL;DR

This paper compares topologically protected magnetic particle transport with geometrical transport, showing how increasing particle number shifts the transport mechanism from topological to geometrical, supported by experiments and simulations.

Contribution

It introduces a detailed analysis of how particle number affects the transition from topological to geometrical transport in magnetic systems.

Findings

Topological transport is characterized by discrete displacements and winding numbers.

Transport becomes geometrical and loses topological protection with many particles.

Simulations confirm the transition from topological to geometrical transport as particle number increases.

Abstract

The transport on top of a periodic two-dimensional hexagonal magnetic pattern of (i) a single macroscopic steel sphere, (ii) a doublet of wax/magnetite composite spheres, and (iii) an immiscible mixture of ferrofluid droplets with a perfluorinated liquid is studied experimentally and analyzed theoretically. The transport of all these magnetic objects is achieved by moving an external permanent magnet on a closed modulation loop around the two-dimensional magnetic pattern. The transport of one and also that of two objects per unit cell is topologically protected and characterized by discrete displacements of the particles as we continuously scan through a family of modulation loops. The direction and the type of transport is characterized by the winding numbers of the modulation loops around special objects in control space, which is the space of possible directions of the external magnetic field. The number of winding numbers necessary for characterizing the topological transport increases with the number of particles per unit cell. The topological character of the transport is destroyed when transporting a large collection of particles per unit cell like it is the case for a macroscopic assembly of magnetic nano particles in a ferrofluid droplet for which the transport is geometrical and no longer topological. To characterize the change in the transport from topological to geometrical, we perform computer simulations of the transport of an increasing number of particles per unit cell.

Equations

$$\mathbf{H}={\mathbf{H}}_p+{\mathbf{H}}_{\text{ext}}.$$

$$U({\mathbf{x}}_{\mathcal{A}})\propto -{\mathbf{H}}^2,$$

$$\text{(R,G,B)}=\frac{8}{A_{\text{UC}}}(A_{+},(A_{+}+A_{-})/2,A_{-}),$$

$$\mathbf{\Delta }=\frac{{\mathbf{\Delta }}_{+}{A}_{+}+{\mathbf{\Delta }}_{-}{A}_{-}}{A_{+}+A_{-}},$$

$$U({\mathbf{x}}_{\mathcal{A}},{\mathbf{H}}_{\text{ext}}(t))\propto -{\mathbf{H}}_{\text{ext}}(t)\cdot {\mathbf{H}}_p({\mathbf{x}}_{\mathcal{A}}),$$

$$\varphi (r)=\{\begin{array}{ll}4ϵ\left({\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6+\frac{1}{4}\right)& r\le 2^{1/6}\sigma \\ 0& r>2^{1/6}\sigma \end{array},$$

Full text

I Introduction

When a system is driven adiabatically, its state changes slower than any relaxation time. A state of a classical system then follows the same path independently of the speed of the driving. If driven adiabatically at different speeds, a state of a quantum system also follows the same path up to a global dynamic phase TI of its wave function that cannot be measured. Measurable quantities are geometrical in the adiabatic limit since they can be deduced from the path without knowledge of the particular time table with which one drives the system along this path. For a periodically driven system the transport of particles over a period then is proportional to a geometric quantity of the loop of the driving field GPP1 ; Berry ; Hannay . For example, autonomous low Reynolds number swimmers propel by a distance proportional to the area of the driving control loop in shape space Wilczek1 ; Wilczek2 . Adiabatic quantum two-band electrons propel by an amount proportional to the area enclosed by the SU(2) respectively SO(3)$\cong$SU(2)$/\mathbb{Z}_{2}$ matrices of the periodic control loop induced by the external field Hasan ; TI . The area of a loop is a geometric quantity that continuously changes when the driving control loop is altered. When symmetries or other conditions constraint the driving loop, geometrical properties might become discrete global properties called topological invariants. Then, the transport no longer changes continuously with the loop since families of loops share the same topological invariant. The transport changes discretely between two families of loops with different values of the topological invariant Hasan . The transport is robust because it is topologically protected. For example, in a nucleus one nucleon must rotate by multiples of $2\pi$ when it propels by one lattice constant above the lattice of the crystallized nucleons that form the rest of the nucleus Harland2018 .

Understanding how a system changes from geometrical towards topological is important because it might help to change a geometrical and thus fragile system towards a topological and thus robust system. For example, in the quantum Hall effect steps between the plateaus in the conductivity can be created either by lowering the temperature or by using clean systems with fewer impurities Klitzing . Both methods decrease the probability of exciting unoccupied bulk Landau levels and thus make the system topological. For many quantum and classical systems Mao ; Paulose ; Rechtsman ; Nash ; Kane ; Huber the transition from geometric towards topological transport can be understood via the amount of dissipation occurring due to the scattering between states. It has, however, been shown that there exist non-Hermitian quantum and dissipative classical topological transport systems Murugan ; Xiong ; Yao ; Kunst ; Loehr where it is precisely the dissipation that causes the topological character of the transport. For these systems the transition from topological towards geometrical must be different.

Here, we show experimentally and with computer simulations that a macroscopic topological magnetic particle pump Rossi , that transports paramagnetic or soft magnetic particles across a magnetic lattice, is topological when transporting a small number of particles per unit cell. The transport is robust for those modulation loops of a driving homogeneous external field that share the same topological invariant. Subclasses of modulation loops appear for a loading with two or more particles per unit cell increasing the number of discrete steps. However, for loadings with a macroscopic ensemble of magnetic nanoparticles, such as a ferrofluid droplet, the topological nature of the transport is destroyed and becomes geometrical.

II Topogeometrical pump

The system consists of a two-dimensional hexagonal magnetic pattern made of up- and down-magnetized magnets, see Fig. 1a. The pattern creates a two-dimensional magnetic potential that acts on paramagnetic objects located above the pattern at fixed elevation. The potential is a function of the position $\mathbf{x}_{\cal A}\in\cal A$ of the paramagnetic object in action space $\cal A$ , which is the plane parallel to the pattern in which the objects are located. A uniform external magnetic field is also applied to the system. Hence, the total potential depends parametrically on the direction of the superimposed external magnetic field. Paramagnetic objects, such as soft magnetic spheres and ferrofluid droplets move in action space when we adiabatically modulate the total potential by changing the direction of the uniform external field.

Our two dimensional magnetic hexagonal lattice is built from an arrangement of NbB-magnets Rossi . The lattice (Fig. 1a) consists of large (l) and small (s) cylindrical magnets of height $h=2$ mm, diameters $d_{l}=3$ mm and $d_{s}=2$ mm, and remanences $\mu_{0}M_{l}=1.19$ T and $\mu_{0}M_{s}=1.35$ T, with $\mu_{0}$ the permeability of free space. The large magnets are magnetized upwards (silver magnets in Fig 1a and red regions in Fig. 1b) and are surrounded by six small magnets that are magnetized downwards (black magnets in Fig 1a and green regions in Fig. 1b) and that touch the large magnets. The primitive unit cell of the lattice is a six fold symmetric $C_{6}$ hexagon with corners centered within the smaller magnets, see Fig. 1b. Each unit cell thus contains one large magnet and two small magnets. The resulting lattice with two primitive lattice vectors of length $a=4.33$ mm is mechanically metastable in zero external magnetic field. Therefore, we need to fix the metastable arrangement with an epoxy resin placed in the voids and in the two dimensional surroundings of the pattern. This stabilizes the pattern also in the presence of an external field. The pattern is put on a support and covered with a transparent PMMA and white-illuminated spacer of thickness $z=1.5$ mm (Fig. 1c).

The total magnetic field is the sum of the pattern ${\mathbf{H}}_{p}$ and the external ${\mathbf{H}}_{\text{e}xt}$ contributions

[TABLE]

The potential energy of a paramagnetic object in the total magnetic field ${\rm H}$ is proportional to the square of the magnetic field

[TABLE]

and it can be decomposed into a discrete Fourier series of contributions from reciprocal lattice vectors Loehr ; delasHeras . The Fourier series of the potential evaluated in a plane above the pattern and parallel to it is the square of the Fourier series of the magnetization of the pattern augmented by the external field. As a function of the elevation, the higher Fourier coefficients are attenuated more than those with lower reciprocal vectors. At the experimental elevation (comparable to the length of the unit cell of the pattern), only the ”universal” contributions to the potential from the lowest non-zero reciprocal lattice vectors remain relevant Loehr ; delasHeras . The purpose of the spacer (Fig. 1c) is thus to render the potential universal such that only the symmetry and not the fine details of the pattern are important.

We either place one steel sphere of diameter $2r=1$ mm or two spheres of diameter $2r=0.5$ mm consisting of $10:1$ weight percent mixture of wax and magnetite on top of the spacer. Alternatively, we spray the spacer with PTFE, place two fluids, a nonmagnetic fluid (Galden), and an aqueous ferrofluid immiscible with the Galden at a volume ratio Galden/ferrofluid of $152:1$ on top of the PTFE and close the compartment with a transparent lid. The magnetic pattern with the transported paramagnetic object on top is then placed in the center of a goniometer which is set up at an angle of $45$ degrees to ensure that the relevant motion is not affected by the restrictions of motion of the goniometer caused by the support. Both a sketch and an actual picture of the setup are shown in Figs. 1d and 1e, respectively. The goniometer holds two NbB-magnets (diameter $d_{\text{ext}}=60$ mm, thickness $t_{\text{ext}}=10$ mm, and remanence $\mu_{0}M_{\text{ext}}=1.28$ T) that generate the external field. The magnets are aligned parallel to each other at a distance $2R=120$ mm and create an external magnetic field of magnitude $\mu_{0}H_{\text{ext}}=45$ mT that penetrates the two-dimensional pattern, the steel sphere, the wax/magnetite spheres, and the ferrofluid droplets. Dipolar interactions between two wax/magnetite spheres or between ferrofluid nano particles are weak as compared to the interaction with the pattern and the external field.

The gradient of the magnitude of the external field at the position of the transported objects ${\boldmath\nabla H_{\text{ext}}}\approx M_{\text{ext}}t_{\text{ext}}d_{\text{ext}}^{2}/R^{4}$ is at least two orders of magnitude smaller than that of the magnetic field of the pattern ${\boldmath\nabla H_{p}}\approx(M_{l}+M_{s})/a$. Hence, the field created by the external magnets is effectively uniform. The two external magnets can be oriented to produce an arbitrary direction of the external magnetic field with respect to the pattern. A laser pointing along ${\mathbf{H}}_{\text{ext}}$ is mounted on the goniometer, see Fig. 1e, to create a stereographic projection of the instantaneous external magnetic field direction on a recording plane.

III Topologically nontrivial transport loops

The parametric dependence of the potential acting on a paramagnetic object (equ. (2)), was studied in detail in Ref. Loehr . The potential has hexagonal symmetry and the number of minima per unit cell of the potential can be one or two, depending on the orientation of the uniform external field. The set of possible orientations of the external field forms a sphere that we call the control space $\cal C$ (see Fig. 2b). Two minima exist in the excess region of $\cal C$ (see Fig. 2b) in which the orientation of the external field is roughly antiparallel to the magnetization $\mathbf{M}_{l}$ of the silver magnets of Fig 2a. Only one minimum of the potential exists for orientations of the external field roughly parallel to the magnetization of silver magnets (see green region of control space in Fig. 2b). The boundary in $\cal C$ between the excess region and the region of one single minimum is a closed curve in $\cal C$ that we call the fence $\cal F$. The fence consists of twelve segments (red and blue in Fig. 2b) meeting in twelve bifurcation points. These bifurcation points (${\cal B}_{\pm a_{i}}$, and ${\cal B}_{\pm Q_{i}}$) are located in the southern hemisphere of $\cal C$ on longitudes running through the directions $\pm\mathbf{a}_{i}$ and $\pm\mathbf{Q}_{i}$ ($i=1,2,3$) of the primitive unit vectors of the direct and reciprocal lattice, respectively (see Figs. 2a and 2b). The fence segments are of two types $+\mathbf{Q}$-segments (red segments in Fig. 2b) and $-\mathbf{Q}$-segments (blue segments in Fig. 2b).

We reorient the external magnets by moving along a closed reorientation loop that starts and ends at the same orientation. (See the black point in Fig. 2b marked as starting point between the $\mathbf{Q}_{2}$ and the $\mathbf{a}_{3}$ longitude in the southern unique minimum region.) As a result of the reorientation loop of the external field, the steel sphere, the wax/magnetite spheres, and the ferrofluid droplets move above the magnetic lattice. A topologically trivial motion of the steel sphere is that where the sphere responds to a closed reorientation loop with a closed loop on the lattice. Not every closed reorientation loop causes such a trivial response of the steel sphere. There are topologically nontrivial trajectories, where the steel sphere ends at a position differing from the initial position by one unit vector of the magnetic lattice. Nontrivial closed reorientation loops in control space are those loops that have loop segments in both, the excess region and the region of the unique minimum Loehr ; Rossi . Here, the reorientation loop enters the excess region with a longitude $\phi_{\text{entry}}$ between the $\mathbf{Q}_{2}$ and the $-\mathbf{a}_{2}$ directions and exits the excess region between the $\mathbf{Q}_{2}$ and the $\mathbf{a}_{3}$ longitudes at $\phi_{\text{exit}}=4.4\pi/6$. Schematic reorientation loops in control space are depicted in Fig. 2b.

III.1 Single steel sphere transport

We have reported in Refs. Loehr ; Rossi ; Loehr2 how the transport of a single paramagnetic particle changes as we move the entry longitude $4.4\pi/6<\phi_{\text{entry}}<8.5\pi/6$ of the reorientation closed loop. Here we briefly repeat the findings which are important for this work. The steel sphere adiabatically returns to its initial position (performs a closed loop above the lattice) if the reorientation loop enters and exits the excess region via the same fence segment. As an example of such a loop we have drawn the loop a in Fig. 2b for which $\phi_{\text{entry}}=\phi_{\text{exit}}=4.4\pi/6$. The sphere also returns to the same position when the reorientation loop encloses only the bifurcation point ${\cal B}_{\mathbf{a}_{3}}$ or the two bifurcation points ${\cal B}_{\mathbf{a}_{3}}$ and ${\cal B}_{-\mathbf{Q}_{1}}$ in $\cal C$ such in the case of loops b-e ($5\pi/6<\phi_{\text{entry}}<7\pi/6$ and $4\pi/6<\phi_{\text{exit}}<5\pi/6$) in Fig. 2b. However, as the modulation loop encloses ${\cal B}_{\mathbf{a}_{3}}$ the motion is no longer adiabatic. Instead, an irreversible ratchet jump occurs as the modulation loop exits the excess region through the fence. We have measured the position of the fence in control space via these ratchet jumps. The blue and red spheres in Fig. 2b are the experimental data of those measurements. The motion is always adiabatic if the control loop enters and exits the excess region via the same type of fence segments, either $+\mathbf{Q}$-segments or $-\mathbf{Q}$-segments Loehr ; Rossi . A ratchet occurs if the entry and the exit fence segments are of different type. The motion of the steel sphere becomes adiabatic when enclosing the third bifurcation point ${\cal B}_{-\mathbf{a}_{1}}$ (loops f-i $7\pi/6<\phi_{\text{entry}}<9\pi/6$ in Fig. 2b) with a total displacement of the sphere by one unit vector $-\mathbf{a}_{2}$. Since this reorientation loop is adiabatic, the time reversed loop (e.g. the inverse loop -h with $4\pi/6<\phi_{\text{entry}}<5\pi/6$ and $7\pi/6<\phi_{\text{exit}}<8\pi/6$) transports into the opposite, i.e. the $\mathbf{a}_{2}$ direction than the direct loop.

III.2 Ferrofluid droplet transport

We consider next the motion of ferrofluid droplets since they mimic the transport in a system with hundreds of tiny particles per unit cell. Hence, the transported object can no longer be considered a point particle. To understand the motion we need to consider the equipotential lines around the minima of the total magnetic potential that drives the motion. In Fig. 3 we collect images that show the motion of a ferrofluid droplet along some of the control modulation loops displayed in Fig. 2. The loops enclose from zero up to four bifurcation points. At the starting point the steel sphere and the ferrofluid droplets reside above the central magnet of the unit cell. The sphere/droplet moves away from this location as the external magnetic field enters into the excess region of ${\cal C}$. Nothing special occurs when the external field crosses the fence, and nothing particular happens to the single steel sphere as the field moves deeper into the excess region. The ferrofluid droplet, however, deforms into a dogbone-like shape and eventually splits in two smaller droplets when the modulation loop crosses a droplet splitting line in ${\cal C}$. Some of the shapes of such droplets are shown in Fig. 2c. Their shape and size agree very well with the shape and size of the lemniscates, which are simply the equipotential lines of the colloidal potential passing through the saddle point between both minima, see Fig. 2c. The two separated ferrofluid droplets reside in a region above the two different minima that the potential has in the excess region of ${\cal C}$. The droplets are in general transported into different directions. When the modulation loop is closed (returns to the starting point) the two split ferrofluid droplets must return to either the original position above the same central silver magnet or to an equivalent position in a different unit cell. The transport over one period is therefore the coexistence of two different types of transport directions. The total transport is the sum of the two coexisting displacements weighted with the two areas of the droplets when they split.

The splitting of a ferrofluid droplet occurs either in an adiabatic way ($\mathbf{a}$-splitting) or irreversibly ($\mathbf{Q}$-splitting). Both types of splitting are schematically represented in Fig. 4. The ferrofluid droplet covers a certain area $A$ of action space when the external field enters the excess region of ${\cal C}$. A ”minor” excess minimum and an excess saddle point are created in the magnetic potential upon the entry of the external field into the excess region Loehr . The equi-potential line passing through the excess saddle point is a lemniscate that first winds around the preexisting ”major” minimum, then passes through the saddle point, and next winds around the minor excess minimum. Hence, the lemniscate defines a closed curve of area $L=L_{+}+L_{-}$ where each of the two sub areas, $L_{+}$ and $L_{-}$, surrounds a minima of the potential. At the fence in ${\cal C}$, the sub lemniscate area of the minor minimum, $L_{+}=0$ ($L_{-}=0$) for a $-\mathbf{Q}$- fence segment ($+\mathbf{Q}$-fence segment) vanishes. At the fence $L_{+}=0$ or $L_{-}=0$ and the area occupied by the ferrofluid droplet can be either larger $A>L_{+}+L_{-}$ or smaller $A<L_{-}$ ($A<L_{+}$) than that of the preexisting major minimum.

In the case $A>L_{+}+L_{-}$ ($\mathbf{a}$-splitting), which occurs if the loop enters the excess region of ${\cal C}$ close to a ${\cal B}_{a}$ bifurcation point, the ferrofluid droplet assumes the shape of an equipotential line containing both minima (see Fig. 2c bottom and Fig. 4a). When the loop enters deeper into the excess region of $\cal C$, the area of the lemniscate grows, Figs. 4a1 and 4a2. At the point where the area of the lemniscate is the same as the area of the droplet, $L=L^{a}=A$, both lemniscate subareas $L^{a}_{+}$ and $L^{a}_{-}$ are fully filled with ferrofluid, Fig. 4a3. When the area of the lemniscate grows beyond that of the droplet, $L>A$, then the droplet splits into two droplets of areas $A_{+}=L^{a}_{+}$ and $A_{-}=L^{a}_{-}$, Fig. 4a4. The areas of both droplets do not change until the droplets coalesce again, i.e. no further splitting occurs. The splitting is reversible if the control loop is reversed and crosses the splitting line $L^{a}=A$ at exactly the same point. In Fig. 2b we have color coded the splitting ferrofluid droplets with the a normalized RGB-color given by the triplet

[TABLE]

where $A_{\text{UC}}$ is the area of the unit cell, and the factor eight accounts for the fact that the maximum subarea of a lemniscate is one eighth of the area of the unit cell (see Fig. 4b4 ). We have also color coded the excess region of ${\cal C}$ in Fig. 2b with the same criterion but replacing the subareas $A_{\pm}$ with the subareas of the lemniscates $L_{\pm}$. Hence, the color of the experimental data points at the splitting lines in Fig. 2b match the color of control space only if $A_{\pm}=L_{\pm}$, i.e., if the subareas of the theoretical lemniscates and those of the droplets are equal. In Fig. 2b the color of the experimental $\mathbf{a}$-splitting points are darker than the background indicating that the experimental droplets split later than predicted by the theory. This is presumably because adhesive forces of the droplet prevent an early splitting. The color discrepancy of the data below the ${\cal B}_{-a_{1}}$ bifurcation points is likely due to scatter in the magnetization of the $NiB$ magnets forming the pattern. The finite size of the pattern and the elevation of the particles above the pattern have also an effect on the experimental measurements.

In the case $A<L_{-}$ ($-\mathbf{Q}$-splitting), which occurs if the control loop enters the excess region close to a ${\cal B}_{-Q}$ bifurcation point, the ferrofluid droplet assumes the shape of an equi-potential line surrounding only the preexisting major minimum but not the excess minor minimum. We show in Fig. 2c top the picture of a droplet just after a $-\mathbf{Q}$-splitting (see also Fig. 4b). The largest areas of $L_{-}$ occur if the external field points at the south pole and at the ${\cal B}_{-Q}$ bifurcation points of ${\cal C}$. Hence, a locally minimal area $L_{-}^{\text{min}}$ occurs for external fields pointing along a longitude that connects the south pole and one of the ${\cal B}_{-Q}$ bifurcation points. The subarea $L_{-}$ of the preexisting minimum of the lemniscate shrinks as the control loop moves from the fence towards the south pole, Fig. 4b1. At some point, the subarea of the lemniscate $L_{-}$ equals that of the droplet $L_{-}^{Q}=A$ (provided that $L_{-}^{\text{min}}<A<L_{-}$), Fig 4b2. There, the fluid completely fills the subarea $L_{-}^{Q}$ while the other subarea $L_{+}$ is completely empty. When the major area of the lemniscate shrinks below the droplet area, i.e. $L_{-}<A$, then the droplets split into two droplets of areas $A_{-}=L_{-}$ and $A_{+}=A-L_{-}$, Fig. 4b3. The fluid in $L_{-}$ is expelled from the droplet through the saddle point and flows down the path of steepest descend into the basin of the minor excess minimum. The areas of the droplet change until the decrease of $L_{-}$ stops, Fig. 4b4. The splitting process is irreversible and both droplets cannot be rejoined in a reversible way since the fluid in the excess minimum cannot flow up the path of steepest decent back into the preexisting minimum. We have placed $\pm\mathbf{Q}$-splitting experimental points at the location where the splitting starts. The points are colored according to Eq. 3 with $A_{\pm}$ being here the subareas when interchange of fluid between $A_{+}$ and $A_{-}$ stops. The agreement with the theoretical prediction given by the areas of the lemniscate (colored background) is excellent.

The splitting lines $L^{a}=A$, $L^{Q}_{-}=A$ and $L^{Q}_{+}=A$ are segments of a closed curve that are joined at the fence of $\cal C$. For $A>L_{-}^{\text{min}}$ any closed curve of a modulation loop in $\cal C$ that penetrates the excess region deep enough must also pass the splitting curve. A nontrivial modulation therefore causes nontrivial transport that is the coexistence of two topological displacements weighted with the two split areas of the droplet. The splitting areas $A_{-}$ and $A_{+}$ continuously change along the splitting lines. A video clip (adfig3.mp4) showing more details of the ferrofluid transport is provided in the supplementary information.

III.3 Doublet transport

We have also studied experimentally the transport of two particles per unit cell. The area $A$ enclosed by the two wax/magnetite spheres is smaller than the local minimum area, i.e. $A<L_{-}^{\text{min}}$. Hence, in contrast to the ferrofluid droplet the $\mathbf{Q}$-splitting cannot occur for these doublets. Both, the ferrofluid droplet and the doublet exhibit $\mathbf{a}$-splitting. A measurement of the $\mathbf{a}$-splitting line $L^{a}=A$ is shown for the wax/magnetite doublets in Fig. 2b. The doublets are transported together within the major minimum if the control loop enters the excess region in the vicinity of a ${\cal B}_{\pm Q}$ bifurcation point. If the loop enters the excess region in the vicinity of a ${\cal B}_{a}$ bifurcation point, then an $\mathbf{a}$-splitting occurs and separates both spheres. One sphere is transported within the major minimum and the other one within the minor minimum. Two transport directions coexist.

Hence, depending on $\phi_{\text{entry}}$ there are two different transport modes for the doublets: (i) no splitting and (ii) $\mathbf{a}$-splitting. The transition from one transport mode towards the other happens at doublet bifurcation points ${\cal B}_{\text{doublet}}$ that are the intersections of the fence with the $\mathbf{a}$-splitting line $L^{a}=A$ for doublets. Since the spheres have the same size, the areas of the $\mathbf{a}$-splitting transport are equal $A_{-}=A_{+}=A/2$. Hence, the experimental data points for doublet $\mathbf{a}$-splitting have all the same color, see Eq. (3) and Fig. 2b. The color is darker than the theoretical background color of the lemniscates. Hence, the splitting happens later in the experiment than predicted by the theory, presumably due to dipolar attraction between the two spheres as well as due to friction with the bottom surface. Like the transport of a single sphere, the transport of doublets is discrete and therefore topological.

Net displacement We analyze next the net displacement after the completion of one entire control loop for all three types of objects: single spheres, doublets, and ferrofluid droplets. For all objects, we define the vector of the net displacement $\mathbf{\Delta}$ as the area averaged sum of the two possible displacements. That is

[TABLE]

where $\mathbf{\Delta}_{\pm}$ are the net displacement vectors of the two minima in one control loop. Hence, $\mathbf{\Delta}_{\pm}$ are always lattice vectors and $\mathbf{\Delta}$ moves along a straight line between the two lattice vectors.

In Fig. 5a we plot the net displacement of all magnetic objects as a function of $\phi_{\text{entry}}$ for a family of loops with $\phi_{\text{exit}}=4.4\pi/6$ (loops similar to those in Fig. 2b). A video of the motion is presented in the supplemental material (adfig5.mp4). The net displacement is zero when the loop encloses no bifurcation point. The displacement moves along the straight lines connecting the sequence of lattice vectors $\mathbf{0}$,$-\mathbf{a}_{2}$, $-\mathbf{a}_{3}$, $\mathbf{a}_{1}$, and $\mathbf{0}$ (see the gray arrows in the center of Fig. 5a) The areas $A_{+}$ and $A_{-}$ are continuous functions of $\phi_{entry}$ for the ferrofluid droplet. In contrast, the transported areas $A_{\pm}$ for one steel sphere (a wax/magnetite doublet) can be only integer multiples $n=0,1$ ($n=0,1,2$) of the area of one sphere. Contrary to the ferrofluid transport, the sphere and doublet net displacement change discretely with $\phi_{\text{entry}}$. The number of discrete steps for the doublets are twice that of a single steel sphere. Hence the transport of one or two spheres is topological while the transport of the ferrofluid is geometrical.

III.4 Multi particle transport

What is the minimum number of particles required for having geometrical transport? To address this fundamental question, we have simulated the transport of multiple spheres using overdamped Langevin dynamics (note that inertial effects are negligible). Each unit cell is filled with exactly $n$ particles. Each particle is subject to the magnetic potential

[TABLE]

where $\mathbf{H}_{\text{ext}}(t)$ is the external field at time $t$ and $\mathbf{H}_{p}(\mathbf{x}_{\mathcal{A}})$ is the magnetic field created by the pattern at position of the particle $\mathbf{x}_{\mathcal{A}}$ in action space $\mathcal{A}$, see Refs. delasHeras ; Loehr2 .

The particles interact via the purely repulsive Weeks-Chandler-Andersen potential

[TABLE]

where $r$ is the distance between the particles, $\epsilon$ fix our unit and energy, and $\sigma$ is the effective particle length that we fix to $\sigma/a=0.2$ which is the same as in the experiments (transport of single spheres). We integrate the equations of motion with a time step $dt/T=10^{-5}$, with $T$ the period of a modulation loop. Fig. 5b shows the net displacement of $n=2,3,4,5$ spheres. The number of plateaus in the displacement of $n$ particles per unit cell is $n$ times the number of plateaus of a single sphere. This is true provided that each unit cell is filled with precisely $n$ particles since the results depend on the initial distribution of particles among the different unit cells. Our simulation results suggest therefore that the change from topological towards geometrical transport is determined by the precision of the experimental setup and measurements.

Note that the $\phi_{\text{entry}}$ of the doublet bifurcation points (at which the net displacement jumps) differs in experiments, Fig. 5a, and simulations, Fig. 5b. In the experiment, the spheres do not only interact via excluded volume interactions, but are also subject to long range dipolar interactions. We tried to minimize the effect of dipolar interactions by using the wax/magnetite composite spheres. The transport properties depend also on the size of the particles, and hence a depth understanding of how the transport changes from topological to geometrical requires further studies.

IV Conclusions

We have studied experimentally and with computer simulations the transport of paramagnetic particles on top of a magnetic lattice and driven by a uniform and time-dependent external magnetic field. The external field performs periodic closed loops. We have shown that increasing the number of particles within the unit cell of the lattice changes the transport from topological towards geometrical.

The transport as a function of a parameter that continuously characterizes a family of control loops is discrete for low particle densities and continuous for a macroscopic number of particles per unit cell (ferrofluid droplet). The possibility to split or disjoin soft matter assemblies increases the number of transport modes and bifurcations, effectively changing the transport from topological towards geometrical.

V Acknowledgments

Scientific discussion with Michel Lönne is highly appreciated.