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Emergent structure of multi-dislocation ground states in curved crystals

We study the structural features and underlying principles of multi-dislocation ground states of a crystalline spherical cap. In the continuum limit where the ratio of crystal size to lattice spacing $W/a$ diverges, dislocations proliferate and ground states approach a characteristic sequence of structures composed of radial grain boundaries ("neutral scars"), extending radially from the boundary and terminating in the bulk. Employing a combination of numerical simulations and asymptotic analysis of continuum elasticity theory, we prove that an energetic hierarchy gives rise to a structural hierarchy, whereby dislocation number and scar number diverge as $a/W \to 0$ while scar length and dislocation number per scar become {\it independent} of lattice spacing. We characterize a secondary transition occurring as scar length grows, where the $n$-fold scar symmetry is broken and ground states are characterized by polydisperse, forked-scar morphologies.