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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.

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 insulators insulate in the bulk but exhibit robust conducting edge states protected by the topology of the bulk material. Here, we design a colloidal topological insulator and demonstrate experimentally the occurrence of edge states in a classical particle system. Magnetic colloidal particles travel along the edge of two distinct magnetic lattices. We drive the colloids with a uniform external magnetic field that performs a topologically non-trivial modulation loop. The loop induces closed orbits in the bulk of the magnetic lattices. At the edge, where both lattices merge, the colloids perform skipping orbits trajectories and hence edge-transport. We also observe paramagnetic and diamagnetic colloids moving in opposite directions along the edge between two inverted patterns; the analogue of a quantum spin Hall effect in topological insulators. We present a new, robust, and versatile way of transporting colloidal particles, enabling new pathways towards lab on a chip applications.

The topologically protected transport of colloidal particles on top of magnetic patterns of all possible single lattice constant two dimensional magnetic point group symmetries is studied experimentally, theoretically, and with numerical simulations. We examine the transport of colloidal particles in response to modulation loops of the external field. We classify the modulation loops into topologically distinct classes causing different transport. We show that the lattice symmetry has a profound influence on the transport modes, the accessibility of transport networks, and the individual addressability of paramagnetic versus diamagnetic colloidal particles. We show how the transport of colloidal particles above a two fold symmetric stripe pattern changes from universal adiabatic transport at large elevations via a topologically protected ratchet motion at intermediate elevations toward a non-transport regime at low elevations. Transport above four fold symmetric patterns is closely related to the transport above two fold symmetric patterns. There exists a family of three fold symmetric patterns that vary as a function of the phase of the pattern. We show how this family can be divided into two topologically distinct classes supporting different transport modes and being protected by proper and improper six fold symmetries. Both classes support individual control over the transport of paramagnetic and diamagnetic particles. We discuss the topological transition when moving the phase from one class of pattern to the other class. The similarities and the differences in the lattice symmetry protected transport of classical over-damped colloidal particles versus the topologically protected transport in quantum mechanical systems are emphasized