向列相中向错线之间相互作用的 Morse 理论与 Meron 介导机制
Morse Theory and Meron Mediated Interactions Between Disclination Lines in Nematics
AI总结：
本文提出基于 Morse 理论的框架，利用 meron 在二维投影下表现出的奇异性，通过奇点理论、分割曲线和手术理论来表征向列相织构，并分类向错线的链接、重连与交叉等三维行为，为缺陷电荷提供新视角。
摘要：
向列相液晶的拓扑理解传统上集中于奇点（即缺陷）及其通过同伦理论的分类。然而，这一方法最终被证明不足以恰当捕捉三维中已报道的一系列复杂行为。为了解决这一问题，我们认为需要一种更精细的拓扑理解，其中非奇异但非平凡的拓扑孤子——即所谓的 meron——在介导向错线之间的相互作用中发挥核心作用。我们提出了一个全面的框架来捕捉此类行为，该框架大量借鉴了 Morse 理论的各个方面；其关键思想是，meron 在投影到二维曲面上时表现为奇异的。这使得可以利用奇点理论和分割曲线通过层析成像来表征向列相织构，并可以通过手术理论理解拓扑转变。我们利用这些思想来理解和分类复杂的三维行为，例如向错线的链接、重连和交叉，并为缺陷电荷提供一个新的视角。
英文摘要：
The topological understanding of nematic liquid crystals is traditionally centered on singularities, or defects, and their classification via homotopy theory. However, this approach has ultimately proved insufficient to properly capture a range of complex behaviours that have been reported in three dimensions. To address this, we argue that a finer understanding of topology is required, in which non-singular but non-trivial topological solitons - so-called merons - play a central role in mediating interactions between disclination lines. We present a comprehensive framework for capturing such behaviour that draws heavily on aspects of Morse theory; the key notion being that merons appear singular under projection onto a two-dimensional surface. This permits the use of singularity theory and dividing curves to characterise nematic textures via tomography, as well as an understanding of topological transitions via surgery theory. We use our ideas to understand and classify complex three-dimensional behaviours, such as the linking, rewiring and crossing of disclination lines, as well as to provide a new perspective on defect charge.
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