Conformal order and Poincar-Klein mapping underlying electrostatics-driven inhomogeneity in tethered membranes
arXiv:2308.06857 · doi:10.1103/PhysRevE.108.024115
Abstract
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 Poincar-Klein mapping underlying the electrostatics-driven inhomogeneity. The discovery of the Poincar-Klein mapping, which connects the Poincar 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.
arXiv comment14 pages, 9 figures
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