Braiding with magnetic octupoles

We simulate the trajectories of magnetic octupole colloids driven by periodic loops of an external magnetic field and placed above a two-dimensional threefold symmetric magnetic pattern. The octupoles avoid the threefold symmetric points above the lattice, either in a topologically trivial way, or by nontrivially winding around these high symmetry points both with respect to their position and with respect to their orientation. We calculate the full dynamical phase space of this winding behavior by changing both lattice symmetries and modulation loops. We further use the nontrivial topology to braid with octupoles and we supply a protocol for both braiding and weaving with such microscopic particles. Our classical external field command should work equally well for quantum mechanical octupoles on magnetic nanopatterns, providing an explicit protocol for the exchange of anyonic quantum particles.

Figure 1

(a) A mathematical braid consists of permutation of strings. There are five strings. The permutation ${\sigma }_1$ moves the string at position 1 to the position 2 above the string that is moved in the opposite direction. The permutation ${\sigma }_4^{-1}$ moves the string at position 4 to the position 5 below the string that is moved in the opposite direction. (b) A trajectory of winding number $w=0$ and $w=1$ around a point $I_1$ in translational space. (c) A closed trajectory of orientational winding number $w=0$ and $w=1/2$ in the space spanned by an orientational angle $\omega$ and a spatial coordinate $x$. For an object with two perpendicular mirror planes, such as a liquid crystal or an octupole, the orientation angle $\omega$ is $\pi$ periodic.

Figure 2

Scheme of a magnetic pattern with magnetization ${\mathbf{M}}_{\mathrm{p}}$ exhibiting a sixfold point symmetry. Paramagnetic (blue) and diamagnetic (red) colloids are placed above the pattern (restricted to move on a plane parallel to the pattern). Individual colloids responding as induced dipoles, bound pairs responding as induced quadrupoles, and a bound quadruple responding as an induced octupole to the external magnetic field are represented.

Figure 3

Side view: Scheme of an octupole built from paramagnetic (blue) and diamagnetic (red) colloidal particles with induced dipole moments (blue and red arrows) above a magnetic pattern with magnetization ${\mathbf{M}}_{\mathrm{p}}$ in an external field ${\mathbf{H}}_{\mathrm{ext}}$ (characterized by ${\varphi }_{\mathrm{ext}},{\vartheta }_{\mathrm{ext}}$). Top view: Same octupole with paramagnetic (diamagnetic) axes ${\mathbf{e}}_{\mathrm{p}}$ (${\mathbf{e}}_{\mathrm{d}}$) and orientation $\omega$ above the threefold symmetric magnetic pattern changing with pattern phase ${\phi }_{\mathrm{p}}$. The color-coded octupole potential (blue = minima, yellow = maxima) is plotted in arbitrary units for an external field ${\mathbf{H}}_{\mathrm{ext}}$ pointing along the ${\mathbf{e}}_z$ direction. The threefold symmetric points $I_1$, $I_2$, and $I_3$ of the pattern (green triangles) are maxima of the potential and hence are avoided by the octupoles. At certain pattern symmetries one of these points acquires a proper (green hexagon) or improper (red hexagon) sixfold symmetry. ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are the primitive lattice vectors. Illustrative unit cells are framed red.

Figure 4

Dynamic phase diagram in the plane of precession angle ${\vartheta }_{\mathrm{ext}}$ and pattern phase ${\phi }_{\mathrm{p}}$. Each dynamical phase is characterized by the number of stable coexisting loops $N$ together with the winding number $w=w_1+w_2+w_3$ around the threefold symmetric points of the lattice, and the orientational winding number $w_{\omega }$. Most phases are adiabatic (solid colors), where the octupole is always quasistable. The ratchet phases (textured) contain trajectories where a potential minimum converts to a saddle point and the octupole jumps into a new stable position with a speed independent of the speed of the external field. The panels (a)–(d) and (f)–(g) show coexisting trajectories in a unit cell. In panels (b) and (f) we show trajectories corresponding to two different values of ${\vartheta }_{\mathrm{ext}}$ as indicated by the two different points in the same phase (the smaller trajectories correspond to the point at smaller ${\vartheta }_{\mathrm{ext}}$). Panels (a), (c), and (d) have insets with magnified trajectories. The location of the threefold symmetric points $I_1,I_2$, and $I_3$ can be found in panel (b). Panel (e) shows a driving external field loop together with the definition of the angles ${\vartheta }_{\mathrm{ext}}$ and ${\varphi }_{\mathrm{ext}}$ characterizing the loop. The trajectories in panels (a)–(d) and (f)-(h) are colored in the same way as the modulation path in (e). The octupole is depicted as a black circle with a paramagnetic director line of orientation $\omega$. We show the orientation of the octupole at six or three positions on each trajectory. A movie showing the dynamics of (a), (c), (d), (f) and (g) is provided with the Supplemental Material [32].

Figure 5

Complex movements of octupoles in the action space $\mathcal{A}$ of a $C_6$ symmetric pattern (${\phi }_{\mathrm{p}}=0$) driven by a complex modulation loop in control space $\mathcal{C}$. (a) Effective movement of the octupoles initially located close to one of the minima $a$, $b$, and $c$ to either $I_1$ or $I_2$ upon a transition of the driving field from the trivial phase in the north, $N,w,w_{\omega }=\text{(3,0,0)}$, to the nontrivial phase in the tropics, $N,w,w_{\omega }=(2,-2,-1/2)$ at different azimuths ${\varphi }_{\mathrm{ext}}$ as shown in (b). The colors of the octupoles in (a) and the transition angles in (b) match. Only two positions survive each transition as minima of the octupole potential. Minima moving towards $I_3$ (indicated by the red crosses) are annihilated. (c) Trajectories of two octupoles a (blue) and b (green) in $\mathcal{A}$ subject to the driving loop in $\mathcal{C}$ (d). The modulation switches between the trivial phase and the nontrivial phase and causes the exchange of the octupoles, i.e., the octupole a moves from the position marked as $t_1^a$ via $t_i^a$, $i\in \{2,...,7\}$ toward $t_1^b$ while the second octupole b moves from the position marked as $t_1^b$ via $t_i^b$, $i\in \{2,...,7\}$ toward $t_1^a$. The trajectories in (c) and the driving loop in (d) are colored in the same way. The unit vector ${\mathbf{e}}_1$ (${\mathbf{e}}_2$) points along the direction of the lattice vector ${\mathbf{a}}_1$ (${\mathbf{a}}_2$). A movie of the motion of both octupoles, a space-time plot together with the driving loop is provided with the Supplemental Material [32].

Figure 6

(a) Trajectories of three octupoles forming a braid using two octupoles a and ${\mathbf{a}}^{\mathbf{\prime }}$ (blue) initially placed in equivalent potential minima of two different unit cells and a third octupole placed in a nonequivalent minimum b (green). Virtual strings colored according to the corresponding octupoles indicate the movement of the octupoles in time. The braid can be seen at the top of the image, where we have deformed the trajectories into the standard braid representation without altering the topology. (b) Space-time plot of the trajectories of a series of equivalent octupoles (blue) woven together by a fourth octupole (green) following a nonequivalent minimum position of the octupole potential. The complex modulation loop in control space $\mathcal{C}$ used to drive the motion is shown on the right via the order of the transition times $t_i$, $i\in \{1,\cdots ,36\}$ between the $N,w,w_{\omega }=\text{(3,0,0)}$ phase and the $N,w,w_{\omega }=(2,-2,-1/2)$ phase. The paths in action space $\mathcal{A}$ are sketched by vanishing lines on top of the $C_6$ symmetric pattern. ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are the lattice vectors of the pattern. The sketches on the left show the schematic exchange or winding of the particles. Movies of the motion of the octupoles, a space-time plot, and the driving loop are provided with the Supplemental Material [32].