Conformal order and Poincaré-Klein mapping underlying electrostatics-driven inhomogeneity in tethered membranes

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.