Theory of hybrid defects, with coupled orientational order parameters, on flat and curved surfaces

Abstract

Many physical systems involve two types of orientational order, which are coupled together. For example, ferroelectric nematic liquid crystals have coupled polar and nematic order, and tilted hexatic phases have coupled polar and hexatic order. In these systems, defect structures can be quite complex. Here, we investigate phases with two types of two-dimensional orientational order, $m$-atic and $n$-atic, where $m$ and $n$ are two distinct integers. We simulate these phases in a flat disk with strong radial anchoring, and on a spherical surface, because both of these geometries require the presence of defects. If the coupling between the two types of order is weak, then the defects are connected by a network of diffuse walls, and the system forms a stable domain structure. As the coupling increases, the domain walls become sharper and shorter. For very strong coupling, the higher-order defects merge into the lower-order defects, forming stretched defect cores.

Figures (7)

• Figure 1: Disk with a single $m$-atic type of orientational order. (a) Polar. (b) Nematic. (c) Tetratic. (d) Hexatic. At each site of the mesh, the $m$ arrows indicate the local orientational order with $m$-fold symmetry. On each triangular cell, the color (yellow to red) indicates the local energy density (average of the three bond energies around the cell).
• Figure 2: Sphere with a single $m$-atic type of orientational order. (a) Polar, shown on a sphere, only one defect visible. (b) Polar, projected onto a cylinder, only one defect visible. (c) Polar, Mercator projection onto flat plane, both defects visible. (d) Nematic, Mercator projection, all four defects visible. (e) Tetratic, Mercator projection, all eight defects visible. (f) Hexatic, Mercator projection, all twelve defects visible. As in Fig. \ref{['fig:disksingleorder']}, the arrows indicate the local orientational order with $m$-fold symmetry, and the color indicates the local energy density.
• Figure 3: Plots of the potential energy $H_{m,n}$ as a function of the relative orientation of neighboring sites, for the four examples considered in this paper. (a) Polar and nematic. (b) Nematic and tetratic. (c) Polar and hexatic. (d) Tetratic and hexatic. In each case, the coefficients are $J_m=J_n=1$.
• Figure 4: Phase with combined polar and nematic order. (a--d) Simulation on disk with $J_2=2$ and $J_1=0.2$, $0.5$, $0.8$, and $1.5$, respectively. (e-h) Mercator projection of sphere with $C=5$, $J_2=1$, and $J_1=0.2$, $0.5$, $0.8$, and $1.5$.
• Figure 5: Phase with combined nematic and tetratic order. (a--d) Simulation on disk with $J_4=2$ and $J_2=0.2$, $0.4$, $0.8$, and $1$, respectively. (e-h) Mercator projection of sphere with $C=5$, $J_4=0.25$, and $J_2=0.05$, $0.1$, $0.2$, and $0.5$.