Conformal order and Poincar$\rm{\acute{e}}$-Klein mapping underlying electrostatics-driven inhomogeneity in tethered membranes

TL;DR

This paper uncovers the geometric principles, including conformal order and Poincaré-Klein mapping, that govern the organization of tethered membranes under electrostatic forces, linking hyperbolic geometry with inhomogeneous lattice structures.

Contribution

It introduces the Poincaré-Klein mapping as a novel geometric framework underlying electrostatics-driven inhomogeneity in tethered membranes.

Findings

Identification of conformal order via preserved bond angles.

Discovery of Poincaré-Klein mapping connecting electrostatics and hyperbolic geometry.

Potential for designing inhomogeneous and 3D-shaped lattices with patterned charges.

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$\rm{\acute{e}}$-Klein mapping underlying the electrostatics-driven inhomogeneity. The discovery of the Poincar$\rm{\acute{e}}$-Klein mapping, which connects the Poincar$\rm{\acute{e}}$ 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.