Sun Honghui, Yao Zhenwei
School of Physics and Astronomy, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China.
Phys Rev E. 2023 Aug;108(2-2):025001. doi: 10.1103/PhysRevE.108.025001.
Understanding the organization of matter under the long-range electrostatic force is a fundamental problem in multiple fields. In this work, based on the electrically charged tethered membrane model, we reveal regular structures underlying the lowest-energy states of inhomogeneously stretched planar lattices by a combination of numerical simulation and analytical geometric analysis. Specifically we show the conformal order characterized by the preserved bond angle in the lattice deformation and reveal the Poincaré-Klein mapping underlying the electrostatics-driven inhomogeneity. The discovery of the Poincaré-Klein mapping, which connects the Poincaré disk and the Klein disk for the hyperbolic plane, implies the connection of long-range electrostatic force and hyperbolic geometry. We also discuss lattices with patterned charges of opposite signs for modulating in-plane inhomogeneity and even creating 3D shapes, which may have a connection to metamaterials design. This work suggests the geometric analysis as a promising approach for elucidating the organization of matter under the long-range force.
理解长程静电力作用下物质的组织形式是多个领域的一个基本问题。在这项工作中，基于带电束缚膜模型，我们通过数值模拟和解析几何分析相结合的方法，揭示了非均匀拉伸平面晶格最低能量状态下的规则结构。具体而言，我们展示了晶格变形中以保持键角为特征的共形序，并揭示了静电驱动的非均匀性背后的庞加莱 - 克莱因映射。庞加莱 - 克莱因映射的发现，它将双曲平面的庞加莱圆盘和克莱因圆盘联系起来，意味着长程静电力与双曲几何之间的联系。我们还讨论了带有相反符号图案电荷的晶格，用于调节面内非均匀性甚至创建三维形状，这可能与超材料设计有关。这项工作表明几何分析是阐明长程力作用下物质组织形式的一种有前景的方法。
