Skyrmion Spin Ice in Liquid Crystals

We propose the first skyrmion spin ice, realized via confined, interacting liquid crystal skyrmions. Skyrmions in a chiral nematic liquid crystal behave as quasiparticles that can be dynamically confined, bound, and created or annihilated individually with ease and precision. We show that these quasiparticles can be employed to realize binary variables that interact to form ice-rule states. Because of their unique versatility, liquid crystal skyrmions can open entirely novel avenues in the field of frustrated systems. More broadly, our findings also demonstrate the viability of liquid crystal skyrmions as elementary degrees of freedom in the design of collective complex behaviors.

Figure 1

Skyrmion structure. (a) Midlayer of a vectorized skyrmion in LC cell with homeotropic anchoring and mapping of directors on to the surface of a sphere. (b) External electric field (coupling coefficient $\alpha$) and surface anchoring (coupling coefficient $K$) can be used in various proportions to sustain long lived skyrmions. Different skyrmion shapes appear depending on $K$ and $\alpha$. The ratio of the cell thickness to the cholesteric pitch is $N_z/p\approx 0.36$. (c) Top view of 2D skyrmions ($K=0$) stabilized by background field (stability range $5.5>\alpha >3$ as shown in [25]). Reducing field strength results in increased skyrmion size and increased deformability near obstacles produced by strong vertical alignment (blue circles).

Figure 2

Constructing binary variables out of trapped skyrmions. (a) Dumbbell traps with closed ends would suppress skyrmion-skyrmion interaction, hence, are modified to final design with open ends. (b) Schematics of vertex configuration types in order of decreasing energy for square skyrmionic ice. Red circles represent the skyrmions, and vertex types are listed in order of decreasing energy. Ice-rule configurations are the two type IV vertices corresponding to two skyrmions in the vertex, and two out of the vertex. (c) Vertex configurations in order of decreasing energy for hexagonal skyrmionic ice. Ice-rule configurations are the three type II vertices (two skyrmions in, one out) and three type III (one in, two out) vertices.

Figure 3

Relaxed skyrmion spin ices obey the ice rule. Top: Skyrmions initially swollen and then deswollen, relax to lower energy states. Bottom: 2D square and hexagonal ice with surface anchoring or field relax to ice-rule-obeying states with defects. The square geometry shows an ordered state where domains of different “antiferromagnetic” orientations of ice rule, type IV vertices are separated by domain walls (green lines) of ice-rule-obeying type III (blue) and ice-rule-violating type II (yellow) and type V (white). The hexagonal geometry converges to a disordered manifold of ice-rule-obeying type II (two in and one out) and type III (one in and two out) vertices. Excitations are ice-rule-violating type I (three in, yellow) or type IV (three out, white) vertices. White lines superimposed on the lattices highlight the midpoint of the traps. Aspect ratios of the traps are 0.52 for square and 0.41 for hexagonal lattices. Other simulation parameters are listed in the Supplemental Material [24].

Figure 4

Limits of the ice-rule validity. (a),(b) Defect ratio vs size of the central circle in the trap for square and hexagonal lattices. Purple region: $\omega$ values too large for trap asymmetry. Pink region: $\omega$ values too small to allow skyrmion motion. Green region: ice behavior observed. (c) Suitable obstacle size depends on the type of wall mechanism and its strength. Results for obstacles produced by light and by various strengths of surface anchoring $K$ inside the obstacles are shown. (d) Skyrmion size also influences how traps work. Ice rule is not observed for very large or very small skyrmions. (e) Defect counts vs random noise added to the width of traps. The noise is uniformly distributed between $\pm \mathrm{\Delta }\omega$ with $D=60$ and $\omega /D=0.6$. Other simulation parameters are listed in the Supplemental Material [24].