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.
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.
We study the defect dynamics in a colloidal spin ice system realized by filling a square lattice of topographic double well islands with repulsively interacting magnetic colloids. We focus on the contraction of defects in the ground state, and contraction or expansion in a metastable biased state. Combining realtime experiments with simulations, we prove that these defects behave like emergent topological monopoles obeying a Coulomb law with an additional line tension. We further show how to realize a completely resettable "NOR" gate, which provides guidelines for fabrication of nanoscale logic devices based on the motion of topological magnetic monopoles. DOI: 10.1103/PhysRevLett.117.168001 Geometric frustration is a complex phenomenon which encompasses a broad range of systems, from magnetic materials [1], to ferroelectrics [2], trapped ions [3], confined microgel particles [4], and folding proteins [5]. It emerges when the spatial arrangement of the system elements cannot simultaneously minimize all interaction energies, and leads to exotic phases of matter with a low-temperature degenerate ground state, such as spin ice [6][7][8]. Artificial spin ice systems (ASI) are lattices of interacting nanoscale ferromagnetic islands, recently introduced as a versatile model to investigate geometrically frustrated states [9,10], including the role of disorder [11,12], thermalization [13][14][15], and the excitation dynamics [16][17][18][19][20]. In opposition to bulk spin ice such as pyrochlore compounds, ASI allow us to directly visualize the spin textures and to tailor the spatial arrangement of the system elements.An intriguing aspect in ASI, which is attracting much theoretical interest, is the dynamics of defects [21][22][23][24][25][26][27][28]. The interactions between pairs of defects is one of the distinctive features between three dimensional (3D) and two dimensional (2D) spin ice. In a 3D pyrochlore compound, the spins are located on a lattice of corner-sharing tetrahedra, and can point either towards the tetrahedra center (spin in), or away from it (spin out). Thus the ground state (GS) follows the "ice rules," with two spins coming in and two going out of each vertex in order to decrease the vertex energy. At finite temperature, defects that behave like "magnetic monopoles" [29,30] can emerge when a spin flips, producing a local increase of the magnetic energy. A way to overcome the system complexity is to use the "dumbbell" model [31], which only considers the magnetic charge distribution at the vertices of the lattice. Within this formalism, it was shown that in 3D spin ice, a pair of defects connected by strings of flipped spins only interact through a magnetic Coulomb law at low temperature. In contrast, numerical simulations show that for a 2D square ASI, i.e., a projection of the 3D ice system on a plane, such a string requires an additional energetic term in the form of a line tension [21]. The reason is that, while in a 3D system all spin configurations that satisfy the ice rules have equal ener...
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.
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