Dynamics and Topology of Symmetry Breaking with Skyrmions

We observe that pretransitional order parameter fluctuations of a skyrmion-forming chiral nematic liquid crystal are slowed down for 4 orders of magnitude, if confined to $\lesssim 100\mathrm{nm}$ thin layers. Fluctuating fragments of half-skyrmions are observed in a narrow temperature interval and are explained by thermally activated hopping between the various energy states. Skyrmion fluctuations are accompanied by imbalanced topological charge: positive charges appear at higher temperatures and dominate in the fluctuating region until skyrmions fully condense and negative charges appear at lower temperatures.

Figure 1

(a) Each panel shows the same region of BPI LC in a wedge cell at different temperatures compared to the I-BPI transition in bulk. The thickness ranges from 60 nm (left) to 90 nm (right). The dashed vertical line denotes $d_f$, where fluctuations are first observed at the transition. The top inset shows the structure of a HS. Scale bar $5\mathrm{\mu m}$, images are taken with right-handed circularly polarized light and rescaled for contrast. Sequence is shown in SV 1. (b) Subfigures 1-3 depict enlargements of regions in (a) at the thickness of (1) $\sim 80–82\mathrm{nm}$, (2) $\sim 72–74\mathrm{nm}$, and (3) $\sim 62–64\mathrm{nm}$. Scale bar $1\mathrm{\mu m}$.

Figure 2

(a) Different types of events occurring in a single second within the fluctuating region of a highly chiral LC. Scale bar 500 nm. (b) Autocorrelation functions (left) for three different temperatures at $\overline{q}=6.7{\mathrm{\mu m}}^{-1}$, showing critical slowing down of skyrmion fluctuations due to symmetry breaking when cooling from the I phase. The images on the right were rescaled for contrast. Scale bar 500 nm. (c) Temperature dependence of effective relaxation rates ($1/{\tau }_{\mathrm{eff}}$) at two different image wave vectors: $\overline{q}=6.7{\mathrm{\mu m}}^{-1}$ (long wavelength limit) and $\overline{q}=22.7{\mathrm{\mu m}}^{-1}$ (comparable to HS size). The inset shows relaxation rate dispersion at three temperatures in (b), also marked with vertical dashed lines in (c).

Figure 3

(a) $(T,d)$ phase diagram. The blue gradient represents a gradual change of phase transition nature with thickness and the fluctuations are observed roughly in the square-patterned region. The inset shows theoretically predicted phase diagram with critical point (red dot). (b) Schematic of three phase transition types observed in confined LCs (above) and schematic free energy $g(s)$ OP dependencies of different thicknesses. (c) Temperature dependence of OP relaxation rate in the I phase at different thicknesses. (d) Temperature dependencies of the hopping rate between the two equilibrium states at $d=85\mathrm{nm}$ ($d_c=79\mathrm{nm}$ in this model).

Figure 4

Formation of (a) a single and (b) two HSs from the I phase. Note the circular shape of the HS in image 5 of (a), indicating the absence of two $-1/2$ disclination lines, representing excess of positive charge of $+1$ per HS. Note the appearance of the two $-1/2$ disclination lines when two connected HSs are formed in image 5 of (b), keeping the total topological charge of a merged cluster to $+1$. (c) Merging of HS upon cooling, which partially compensates the total topological charge of all the HSs. Because of $-1/2$ disclinations, the total winding of an island remains $+1$ (encircled red), regardless of merging with other HSs. Additional HSs within the islands (dashed red ellipse) can form by splitting a HS into two, with simultaneous appearance of $-1/2$ disclinations. All images were taken in the longer pitch LC. Scale bars in (a)–(c) represent 200 nm. All sequences are from SV 3.