Defects in Conformal Crystals: Discrete vs. Continuous Disclination Models

We study the relationship between topological defect formation and ground-state packings in a model of repulsions in external confining potentials. Specifically we consider screened 2D Coulombic repulsions, which conveniently parameterizes the effects of interaction range, but also serves as simple physical model of confined, parallel arrays of polyelectrolyte filaments or vortices in type-II superconductors. The countervailing tendencies of repulsions and confinement to, respectively, spread and concentrate particle density leads to an energetic preference for non-uniform densities in the clusters. Ground states in such systems have previously been modeled as {\it conformal crystals}, which are composed of locally equitriangular packings whose local areal densities exhibit long range gradients. Here, we assess two theoretical models that connect the preference for non-uniform density to the formation of disclination defects, one of which assumes a continuum distributions of defect, while the second considers the quantized and localized nature of disclinations in hexagonal conformal crystals. Comparing both models to numerical simulations of discrete particles clusters, we study the influence of interaction range and confining potential on defects in ground states. We show that treating disclinations as continuously distributable well-captures the number of topological defects in the ground state in the regime of long-range interactions, while as interactions become shorter range, it dramatically overpredicts the to growth in total defect charge. Analysis of the discretized defect theory suggests that limitations of the continuous defect theory can be attributed to the asymmetry in the placement of positive vs. negative disclinations in the conformal crystal ground states, as well as a strongly asymmetric dependence of self-energy of disclinations on sign of topological charge.