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We measure the trajectories of macroscopic magnetic particles pulled against gravity between twisted alternating magnetic square patterns in a superposed homogeneous magnetic field normal to both patterns. The two patterns are built from a set of magentic cubes having a distribution of magnetization. The magnetic potential between the patterns is a sum of three contributions: two being periodic on two lattices with different magnitude and orientation, and the third random contribution arising from the distribution of magnetization of the cubes. As one varies the twist angle between the two patterns each time the twist angle coincides with a magic twist angle one of the two periodic lattices becomes a sublattice of the other lattice. Simulations of particles moving through patterns with a precise cube magnetization produce pronounced mobility peaks near magic twist angles that are associated with flat channels. Weak random fluctuations of the cube magnetization in the experiment and the simulations cause enhanced random disorder of the potential and reduce the mobility by scattering particles into the interior of the twisted Wigner Seitz cells. The mobility undergoes an Anderson transition from magic to generic behavior as the magnetization disorder increases beyond half of a percent of the cube magnetization.

We simulate the dynamics of paramagnetic colloidal particles that are placed above a magnetic hexagonal pattern and exposed to an external field periodically changing its direction along a control loop. The conformation of three colloidal particles above one unit cell adiabatically responds with half the frequency of the external field creating a time crystal at arbitrary low frequency. The adiabatic time crystal occurs because of the non-trivial topology of the stationary manifold. When coupling colloidal particles in different unit cells, many body effects cause the formation of topologically isolated time crystals and dynamical phase transitions between different adiabatic reversible and non-adiabatic irreversible space-time-crystallographic arrangements., Comment: submission to SciPostPhys. 5 videos in the ancillary directory

Detailed control over the motion of colloidal particles is relevant in many applications in colloidal science such as lab-on-a-chip devices. Here, we use an external magnetic field to assemble paramagnetic colloidal spheres into colloidal rods of several lengths. The rods reside above a square magnetic pattern and are transported via modulation of the direction of the external magnetic field. The rods behave like bipeds walking above the pattern. Depending on their length, the bipeds perform topologically distinct classes of protected walks above the pattern. We demonstrate that it is possible to design parallel polydirectional modulation loops of the external field that command up to six classes of bipeds to walk on distinct predesigned paths. We use such parallel polydirectional loops to induce the collision of reactant bipeds, their polymerization addition reaction to larger bipeds, the separation of product bipeds from the educts, the sorting of different product bipeds, and also the parallel writing of a word consisting of several different letters.

Single and double paramagnetic colloidal particles are placed above a magnetic square pattern and are driven with an external magnetic field precessing around a high symmetry direction of the pattern. The external magnetic field and that of the pattern confine the colloids into lanes parallel to a lattice vector of the pattern. The precession of the external field causes traveling minima of the magnetic potential along the direction of the lanes. At sufficiently high frequencies of modulation only the doublets respond to the external field and move in direction of the traveling minima along the lanes, while the single colloids cannot follow and remain static. We show how the doublets can induce a coordinated motion of the single colloids building colloidal trains made of a chain of several single colloids transported by doublets.

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., Comment: 9 page, 5 figures

Edge currents of paramagnetic colloidal particles propagate at the edge between two topologically equivalent magnetic lattices of different lattice constant when the system is driven with periodic modulation loops of an external magnetic field. The number of topologically protected particle edge transport modes is not determined by a bulk-boundary correspondence. Instead, we find a rich variety of edge transport modes that depend on the symmetry of both the edge and the modulation loop. The edge transport can be ratchet-like or adiabatic, and time or non-time reversal symmetric., Comment: 16 pages, 7 figures

Topological protection allows robust transport of localized phenomena such as quantum information, solitons, and dislocations. The transport can be either dissipative or non-dissipative. Here, we experimentally demonstrate and theoretically explain the topologically protected dissipative motion of colloidal particles above a periodic hexagonal magnetic pattern. By driving the system with periodic modulation loops of an external and spatially homogeneous magnetic field, we achieve total control over the motion of diamagnetic and paramagnetic colloids. We can transport simultaneously and independently each type of colloid along any of the six crystallographic directions of the pattern via adiabatic or deterministic ratchet motion. Both types of motion are topologically protected. As an application, we implement an automatic topologically protected quality control of a chemical reaction between functionalized colloids. Our results are relevant to other systems with the same symmetry.