Continuous transformation of a $-1/2$ wedge disclination line to a $+1/2$ one

It is known that, in the order-parameter space $S^2/Z_2$ (a typical example being a uniaxial nematic liquid crystal in three dimensions), a $-1/2$ wedge disclination line and a $+1/2$ one are topologically equivalent and can thus be transformed continuously into each other. Here we report the realization of this transformation in a simulation of a cholesteric blue phase under an electric field.

Figure 1

(Color online) Schematic illustration of the topological equivalence of $-1/2$ and $+1/2$ wedge disclinations. Here the director $\mathit{n}$ is indicated by a cylinder and the disclination by a sphere. By a rotation of $\mathit{n}$ about an axis indicated in the figure, a $-1/2$ disclination transforms continuously into a $+1/2$ disclination via a twist disclination.

Figure 2

(Color online) The director profile at a cross section at $t=0$, just after the application of the electric field $\mathit{E}$ perpendicular to the plane of the figure. The director is shown by a cylinder, and the curved surfaces (green) are the isosurfaces with $\text{Tr}{Q}^2=0.7$, indicating the position of disclination lines. The isosurfaces are shown only in a region sandwiched by two planes whose distance from the cross section is $\sqrt{3}/6$. The location of the cross section with respect to the unit cell of BP I is shown by a gray plane in the inset. To clarify that all the disclination lines are of topological charge $-1/2$, we also show the director profiles around two disclination lines at a different cross section perpendicular to the disclination line. The circle (red) indicates the contour along which the director variation is shown in Fig. 5.

Figure 3

(Color online) The director profile at $t=18.9$. The local director profile around one disclination line is shown separately to emphasize that it is a twist disclination.

Figure 4

(Color online) The director profile at $t=38.5$. In this region three disclination lines of charge $+1/2$ and five ones of charge $-1/2$, which are almost perpendicular to the cross section, can be seen. The number of $+1/2$ disclination is the same as that of $-1/2$ disclination in the infinite system not shown here.

Figure 5

(Color online) The director variations along the counterclockwise contours in Figs. 2, 3, 4, shown as a contour on a unit sphere. Two figures from different viewpoints are presented for clarity. The starting point of the contour of the director is shown by a small sphere. The plane of Figs. 2, 3, 4, or the $xy$ plane, is presented by a black circle, and the axis perpendicular to the $xy$ plane, or the $z$ axis, is indicated by a straight line. The direction of the horizontal axis of Figs. 2, 3, 4, or the $x$ axis, is shown by a short line. The figure below is drawn from the $+z$ point as in Figs. 2, 3, 4, and the $z$ axis cannot be seen. This illustration follows the idea of Fig. 12.15 in [3].