论文标题
锥体上的$ P $液晶的缺陷吸收和排放
Defect absorption and emission for $p$-atic liquid crystals on cones
论文作者
论文摘要
我们研究了固定弯曲表面上$ p $ - 液晶的基态配置。我们专注于固有的几何形状，并表明等温坐标明确编码对弹性电位的几何贡献时，它们特别方便。在具有半角度$β$的锥体的特殊情况下，Apex会产生$ -OB $的有效拓扑费用，其中$2πχ=2π（1- \sinβ）$是锥的缺陷角度，并且充电$σ$的拓扑缺陷表现得像有效的拓扑$ q_ \ qudrm $ quts $ quts $ qudy $ quts $ quant $ fecter $ fecter $ fect $ fect n 2/2/顶点的有效电荷会导致锥顶点的缺陷吸收和发射，因为锥的缺陷角度变化。 对于总拓扑缺陷充电1，例如通过边缘的切向边界条件强加，我们发现对于磁盘，基态配置由$ p $缺陷，每个电荷$+1/p $均等地均等放在半径$ d =（\ frac {p-1} {p-1} {3p-1} {3p-1} {3p-1} {3p-1}）^{\ frac {\ frac {\ frac {1} {1} {2p} {在底座处有切向边界条件的锥体的情况下，我们发现三种类型的基态构型作为圆锥角的函数：（1）对于锋利的锥体，所有$+1/p $缺陷均由顶点吸收； （2）在中间锥角，$+1/p $缺陷中的某些缺陷被顶点吸收，其余的则沿着侧面的同心环平均间隔； （3）对于几乎平坦的锥体，所有$+1/p $缺陷都沿着侧面的同心环平均间隔。在这里，缺陷位置和吸收过渡都大致取决于$ p $以及我们在分析中计算的缺陷角度。我们通过数值模拟检查这些结果，以了解一组相称的锥角，并找到出色的一致性。
We investigate the ground state configurations of $p$-atic liquid crystals on fixed curved surfaces. We focus on the intrinsic geometry and show that isothermal coordinates are particularly convenient as they explicitly encode a geometric contribution to the elastic potential. In the special case of a cone with half-angle $β$, the apex develops an effective topological charge of $-χ$, where $2πχ= 2π(1-\sinβ)$ is the deficit angle of the cone, and a topological defect of charge $σ$ behaves as if it had an effective topological charge $Q_\mathrm{eff} = (σ- σ^2/2)$ when interacting with the apex. The effective charge of the apex leads to defect absorption and emission at the cone apex as the deficit angle of the cone is varied. For total topological defect charge 1, e.g. imposed by tangential boundary conditions at the edge, we find that for a disk the ground state configuration consists of $p$ defects each of charge $+1/p$ lying equally spaced on a concentric ring of radius $d = (\frac{p-1}{3p-1})^{\frac{1}{2p}} R$, where $R$ is the radius of the disk. In the case of a cone with tangential boundary conditions at the base, we find three types of ground state configurations as a function of cone angle: (1) for sharp cones, all of the $+1/p$ defects are absorbed by the apex; (2) at intermediate cone angles, some of the $+1/p$ defects are absorbed by the apex and the rest lie equally spaced along a concentric ring on the flank; and (3) for nearly flat cones, all of the $+1/p$ defects lie equally spaced along a concentric ring on the flank. Here the defect positions and the absorption transitions depend intricately on $p$ and the deficit angle which we analytically compute. We check these results with numerical simulations for a set of commensurate cone angles and find excellent agreement.
