The topologically protected transport of colloidal particles on top of periodic magnetic patterns is studied experimentally, theoretically, and with computer simulations. To uncover the interplay between topology and symmetry we use patterns of all possible two dimensional magnetic point group symmetries with equal lengths lattice vectors. Transport of colloids is achieved by modulating the potential with external, homogeneous but time dependent magnetic fields. The modulation loops can be classified into topologically distinct classes. All loops falling into the same class cause motion in the same direction, making the transport robust against internal and external perturbations. We show that the lattice symmetry has a profound influence on the transport modes, the accessibility of transport networks, and the individual transport directions of paramagnetic and 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 two-fold symmetric case. The three-fold symmetric case however consists of a whole family of patterns that continuously vary with a phase variable. 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. We discuss and experimentally demonstrate the topological transition between both classes. All three-fold symmetric patterns support independent transport directions of paramagnetic and diamagnetic particles. 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.

The transport of a steel sphere on top of two dimensional periodic magnetic patterns is studied experimentally. Transport of the sphere is achieved by moving an external permanent magnet on a closed loop around the two dimensional crystal. The transport is topological i.e. the steel sphere is transported by a primitive unit vector of the lattice when the external magnet loop winds around specific directions. We experimentally determine the set of directions the loops must enclose for nontrivial transport of the steel sphere into various directions.

Geometrical displacement of transported ferrofluid droplets (red) versus topological displacement of transported doublets and single spheres.

A mixture of an ester based ferrofluid with silicone oil and 2,6-lutidine is exposed to an external magnetic field. We find a region of composition of the ternary mixture, where weak magnetic fields of the order of a few kA m induce a modulated phase with a pattern characterized by equilibrium size droplets of the minority phase immersed into the extended majority phase. While the pattern resembles in many ways the pattern of immiscible magnetic fluids, the dependence of the characteristic parameters of the pattern on the magnetic field are completely different than in immiscible fluids. We theoretically explain the pattern formation as a magnetic field induced polymerization of magnetic particles into magnetic chains that goes along with a reduction of the entropy of mixing. This entropy reduction causes the Ostwald ripening of chains into mesoscopic droplets the size of which is limited by repulsive dipolar interactions between the chains.