Comparing skyrmions and merons in chiral liquid crystals and magnets

When chiral liquid crystals or magnets are subjected to applied fields or other anisotropic environments, the competition between favored twist and anisotropy leads to the formation of complex defect structures. In some cases, the defects are skyrmions, which have ${180}^{\circ }$ double twist going outward from the center, and hence can pack together without singularities in the orientational order. In other cases, the defects are merons, which have ${90}^{\circ }$ double twist going outward from the center; packing such merons requires singularities in the orientational order. In the liquid crystal context, a lattice of merons is equivalent to a blue phase. Here we perform theoretical and computational studies of skyrmions and merons in chiral liquid crystals and magnets. Through these studies, we calculate the phase diagrams for liquid crystals and magnets in terms of dimensionless ratios of energetic parameters. We also predict the range of metastability for liquid crystal skyrmions and show that these skyrmions can move and interact as effective particles. The results show how the properties of skyrmions and merons depend on the vector or tensor nature of the order parameter.

Figure 1

Structure of the modulated liquid-crystal phases studied in this paper: blue phase (meron lattice), cholesteric phase (lattice of walls), and skyrmion lattice. The top row shows schematic views of the director field, and the bottom row shows MC simulation results (with the color scale indicating $|n_z|$).

Figure 2

Phase diagram for chiral liquid crystals in the temperature-chirality plane, with no anisotropy. The insets show structures calculated by the MC simulations. In those structures, the colors represent $|n_z|$, with the same color scale as in Fig. 1.

Figure 3

Two views of the phase diagram for chiral liquid crystals in the temperature-chirality-anisotropy space. (Note that the scales on the axes are different in these two visualizations.) The thick horizontal and vertical arrows show the MC simulation paths discussed in Sec. 3, and the insets show structures calculated by the simulations. In the right column, middle image, the isolated merons resemble the structures studied in Ref. [25]. In all the images of merons, twist walls occur at the dark blue lines where $n_z=0$, and disclinations occur wherever three dark blue lines intersect.

Figure 4

Simulation of a metastable skyrmion. The director field covers all possible orientations on the unit sphere exactly once, giving a skyrmion topological charge of 1. The out-of-plane component $|n_z|$ is indicated by the color scale, while the in-plane component points circumferentially around the center. The yellow line on the bottom shows $n_z$ from $-1$ to 1, as a function of $x$, for fixed $y$ in the center. This structure can be regarded as a $\pi$-wall that is curved into a ring, with vertical nematic in the interior and the exterior.

Figure 5

Static skyrmions as particles: (a) An initially distorted shape quickly evolves into a circular ring. (b) Skyrmions repel each other. (c) A system of many skyrmions forms a lattice.

Figure 6

Skyrmion wall thickness $\delta r$ and average radius $r_{\text{av}}=\frac{1}{2}(r_{\text{in}}+r_{\text{out}})$, as functions of the anisotropy $\mathrm{\Delta }\epsilon E^2$, in units of $\pi /q0$. The points represent simulation results, and the solid lines are the calculation in Sec. 3b. Parameters are $L=0.001,q_0=\pi$, and $S=0.405$.

Figure 7

Skyrmion free energy relative to the vertical nematic state, in arbitrary units. The elastic constant $L$ is varied for fixed $a=-0.1,b=-3$, and $c=3$, with the anisotropy $\mathrm{\Delta }\epsilon E^2$ adjusted to maintain the skyrmion size ($g$ and $\delta r$). The points represent simulation results, and the solid line is the calculation of Sec. 3b for the same $L$ and $\mathrm{\Delta }\epsilon E^2$.

Figure 8

Structure of the different modulated magnetic phases studied here.

Figure 9

Visualization of the phase diagram for chiral magnets in the chirality-field-anisotropy space.

Figure 10

Cross section of the phase diagram for chiral magnets in the field-anisotropy plane for fixed chirality $\kappa =0.5$.