Figure 1

A composite vortex reflects the hierarchy of the order-parameter space. The size of a region increases as the energy scale of its order parameter decreases. In the white, gray, and black regions the order parameter moves from ${\mathcal{M}}_q$ to $\mathcal{M}$ to $\mathcal{H}$. The charges of the subvortices are tetrahedral charges, represented by $\Gamma$, and the charge of the composite vortex is a field-aligned charge.Reuse & Permissions

Figure 2

Geometrical representation of the cyclic phase. Two orientations of a tetrahedron, corresponding to the spin nodes of the ground-state spinor, are illustrated and the symmetry axes are labeled. The magnetic field is along the $z$ axis (see the coordinate axes at left). (a) An orientation with the magnetic field along an order-3 axis, corresponding to the ground state $\sqrt{n_0}{\chi }_3$ for $c>4$. (b) An orientation with the magnetic field along an order-2 axis, corresponding to the ground state for $c<4$, $\sqrt{n_0}{\chi }_2$ [see Eq. (34)]. Note that the three order-2 axes $A,B,$ and $C$ are orthogonal so they can be used as a set of body axes.Reuse & Permissions

Figure 3

The order parameter at infinity of a vortex winds around a loop $\Gamma$ in the order-parameter space $\mathcal{N}$. (The space illustrated here is the surface of a solid with an arch and two openings which the order parameter passes through.) Realistic order-parameter spaces are usually more symmetrical.Reuse & Permissions

Figure 4

Two tetrahedral vortices and their topological charges including the rotational and phase portions. The axes of the rotations, $\hat{\mathit{r}}$, $\hat{\mathit{c}}$, point along the symmetry axes of the tetrahedron which are labeled with $R$ and $C$ in Fig. 2. The first vortex is an aligned vortex if the magnetic field is perpendicular to the plane and $c>4$.Reuse & Permissions

Figure 5

A group of vortices is combined into the core of a composite vortex $X$. The energy outside the blacked-out core is calculated from the resultant winding number of the vortices inside and the energy inside the core is calculated by looking inside the core to see which vortices are actually there.Reuse & Permissions

Figure 6

An illustration of the composite vortex $\left(A,0\right)\ast \left(A,0\right)$. The magnetic field is perpendicular to the figure, so the tetrahedra prefer to be oriented with a face or a vertex facing up (since $c>4$). The two small circles enclose the $180°$ vortices, with symmetry axes indicated by black dots. Large circle indicates the transition region where the axis of the $360°$ rotation changes relative to the tetrahedra, from the $A$ to the $R$ axis. The figure uses a functional form with a rapid jump for $\mu \left(r\right)$ [unlike in Eq. (64)] for simplicity, so that the transition region does not overlap the cores. For the probably more realistic form given by Eq. (64), the tetrahedra are already tipped at the centers of the vortices so that the local rotation axes ${\hat{\mathit{n}}}_{\mathit{i}}^{\prime }$ do not have the standard orientations illustrated in Fig. 2a.Reuse & Permissions

Figure 7

Breakup of a molecule (a) which satisfies the binding criteria, but whose charge is not absolutely stable, into other composite vortices (b). If $Q=Q_3{Q}_2{Q}_1$ and $I_E\left(Q\right)>I_E\left(Q_1\right)+I_E\left(Q_2\right)+I_E\left(Q_3\right)$ then this process conserves charge and releases energy, but the breakup may still not occur spontaneously. If the component vortices in (b) are different from the component vortices in (a), then the vortices making up $Q_1$, $Q_2$, and $Q_3$ would have to be produced in a “chemical” reaction from the components of $Q$.Reuse & Permissions

Figure 8

Densities in the five spinor components around an order-3 vortex, with a randomly oriented local axis ${\hat{\mathit{n}}}^{\prime }$. On the left is the pattern one might observe experimentally; the lighter regions correspond to regions with fewer atoms. On the right are plotted the percentage of atoms for each value of $F_z$ at some fixed distance from the vortex core. The phases and amplitudes of these oscillations should help to determine the direction of the local axis.Reuse & Permissions

Figure 9

The evolution of vortex molecules as the magnetic field is increased. The three curves illustrate how the sizes of the molecules from Table , for $c>4$, might change as the strength of the magnetic field is increased. The sizes decrease as $\frac{1}{B^2}$. At a certain magnetic field, an absolutely stable vortex might become rotationally symmetric (molecule 3). Metastable vortices will become unstable when a certain magnetic field is reached, indicated by the $x$’s terminating the curves corresponding to molecules 1 and 4.Reuse & Permissions

Figure 10

Changes in the charge assigned to a moving vortex. (a) The convention for assigning vortex charges in the initial configuration. Tethers are drawn directly from the origin to an anchor just below each vortex. As long as the tethers are moved continuously, the correspondence between the tetrahedral state at the anchor and the standard tetrahedral state does not change, so the charges do not change. [(b) and (c)] How the charge of vortex 2 changes when vortex 1 moves below it. Dashed lines in (a) and the dashed-dotted-dashed lines in (c) are the tethers before and after vortex 1 moves. (b) focuses on the tether of vortex 2, showing how the original tether gets pushed to the side by vortex 1 and is replaced by a new tether. Continuing the labeling of the tetrahedron vertices around the original path changes the labeling of the tetrahedron just below vortex 2. Hence the charge of vortex 2 is identified differently in (c).Reuse & Permissions

Figure 11

Catalysis by conjugation; the vortex on the right moves between the other two vortices changing their repulsion to attraction.Reuse & Permissions