# Ground States of Crystalline Caps: Generalized Jellium on Curved Space

## TL;DR

This paper investigates the structure and elastic energy of crystalline caps on spherical surfaces, revealing continuous symmetry-breaking transitions and the emergence of icosahedral symmetries influenced by curvature and geometry.

## Contribution

It introduces a comparative analysis of continuum and discrete models for defective caps and uncovers novel ground state behaviors driven by curved geometry.

## Key findings

- Transition from defect-free to disclination states is continuous.
- Icosahedral symmetries appear across various curvatures.
- Curved geometry leads to unique ground state behaviors not seen in flat models.

## Abstract

We study the structure and elastic energy of the ground states of crystalline caps conforming to a spherical surface. These ground states consist of positive disclination defects in structures spanning from flat and weakly curved crystals to closed shells. We compare two different continuum theories and one discrete-lattice approach for the elastic energy of defective caps. We first investigate the transition between defect-free caps to single-disclination ground states and show that it is characterized by continuous symmetry-breaking transition. Further, we show that ground states with icosahedral subgroup symmetries in caps arise across a range of curvatures, even far from the closure point of complete shells. Thus, while superficially similar to other models of 2D `jellium' (e.g. superconducting disks and 2D Wigner crystals), the interplay between the free edge of crystalline caps and the non-Euclidean geometry of its embedding leads to non trivial ground state behaviors, without counterpart in planar jellium models.