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.

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.Topologically protected quantum edge states arise from the non trivial topology (non-vanishing Chern number) of the bulk band structure [1]. If the Fermi energy is located in the gap of the bulk band structure, like in an ordinary insulator, edge currents might propagate along the edges of the bulk material. The edge currents are protected as long as perturbations to the system do not cause a band gap closure. The topological mechanism at work is not limited to quantum systems but has been shown to work equally well for classical photonic [2,3], phononic [4,5], solitonic [6], gyroscopic [7], coupled pendulums [8], and stochastic [9] waves. It is also known that the topological properties survive the particle limit when the particle size is small compared to the width of the edge. In the semi-classical picture of the quantum Hall effect, the magnetic field enforces the electrons to perform closed cyclotron orbits in the bulk of the material.

Magnetic Janus particles (MJPs), fabricated by covering a non-magnetic spherical particle with a hemispherical magnetic in-plane exchange-bias layer system cap, display an onion magnetization state for comparably large diameters of a few microns. In this work, the motion characteristics of these MJPs will be investigated when they are steered by a magnetic field landscape over prototypical parallel-stripe domains, dynamically varied by superposed external magnetic field pulse sequences, in an aqueous medium. We demonstrate, that due to the engineered magnetization state in the hemispherical cap, a comparably fast, directed particle transport and particle rotation can be induced. Additionally, by modifying the frequency of the applied pulse sequence and the strengths of the individual field components, we observe a possible separation between a combined or an individual occurrence of these two types of motion. Our findings bear importance for lab-on-a-chip systems, where particle immobilization on a surface via analyte bridges shall be used for low concentration analyte detection and a particle rotation over a defined position of a substrate may dramatically increase the immobilization (and therefore analyte detection) probability.

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Hemispherical caps of in-plane exchange biased IrMn/CoFe layer systems have been fabricated on top of regularly arranged spherical silica particles by magnetron sputtering, creating magnetic Janus particles. In this thin film layer system cap, the magnetic shape anisotropy of the topographically non-flat hemispheres competes with the unidirectional anisotropy induced by the exchange bias. The magnetic properties of this non-trivial system have been investigated by longitudinal magneto-optical Kerr effect magnetometry, where a characterization method has been developed considering both the curved layer system and the signal contributions of flat parts of the sputtered thin film system. Both remagnetization curves, from Kerr magnetometry and the magnetic force microscopy images, reveal an onion state in the magnetic caps of the ensemble. Additional micromagnetic simulations show a stabilization of the onion state due to the introduced unidirectional anisotropy also in individual hollow hemispheres as compared to the vortex state exhibited by purely ferromagnetic caps.