具有反扭转边界条件的圆锥上的液晶基态
Liquid Crystal Ground States on Cones with Anti-Twist Boundary Conditions
AI总结：
该研究结合共形映射方法与模拟，探索圆锥上具p重旋转对称液晶基态，揭示相关边界条件下有限尺寸效应，给出反扭转边界（总拓扑电荷-1）下基态新结果，发现顶点尖锐化时的缺陷行为及电荷屏蔽现象。
摘要：
在凝聚态物理中，几何与拓扑在决定二维表面上的图案形成方面起着基础性作用。例如，二维表面的局部正高斯曲率会吸引受限在曲面上的液晶相中的正拓扑缺陷，同时排斥负拓扑缺陷。尽管圆锥几何在侧面是平坦的，但圆锥顶点处集中的高斯曲率会在任意远处几何上挫败液晶取向场。顶点充当未量化的伪缺陷，与侧面的拓扑缺陷相互作用。通过利用F. Vafa等人的共形映射方法，我们探索了一个简单的理论框架来理解圆锥上具有p重旋转对称性的液晶基态，并揭示了将正负缺陷都限制在圆锥侧面的边界条件下基态的重要有限尺寸效应。通过结合理论和模拟，我们给出了圆锥底部具有反扭转边界条件（强制总拓扑电荷为-1）的圆锥上液晶基态的新结果。我们发现，随着圆锥顶点通过缺陷解束缚过程变得更尖锐，侧面会产生额外的量化负缺陷，使得等效数量的量化正缺陷被困在顶点，从而部分屏蔽顶点电荷（其大小是圆锥角的连续函数）。
英文摘要：
Geometry and topology play a fundamental role in determining pattern formation on 2D surfaces in condensed matter physics. For example, local positive Gaussian curvature of a 2D surface attracts positive topological defects in a liquid crystal phase confined to the curved surface while repelling negative topological defects. Although the cone geometry is flat on the flanks, the concentrated Gaussian curvature at the cone apex geometrically frustrates liquid crystal orientational fields arbitrarily far away. The apex acts as an unquantized pseudo-defect interacting with the topological defects on the flank. By exploiting the conformal mapping methods of F. Vafa et al., we explore a simple theoretical framework to understand the ground states of liquid crystals with $p$-fold rotational symmetry on cones, and uncover important finite size effects for the ground states with boundary conditions that confine both plus and minus defects to the cone flanks. By combining the theory and simulations, we present new results for liquid crystal ground states on cones with anti-twist boundary conditions at the cone base, which enforce a total topological charge of $-1$. We find that additional quantized negative defects are created on the flank as the cone apex becomes sharper via a defect unbinding process, such that an equivalent number of quantized positive defects become trapped at the apex, thus partially screening the apex charge, whose magnitude is a continuous function of cone angle.
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