Defects in Conformal Crystals: Discrete vs. Continuous Disclination Models

Abstract

We study the relationship between topological defect formation and ground-state 2D 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 systmes have previously been modeled as 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 theoretical descriptions to numerical simulations of discrete particles clusters, we show the influence of interaction range and confining potential on the topological charge, number and distribution of 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. Detailed analysis of the discretized defect theory suggests that that failure of the continuous defect theory in this limit can be attributed to the asymmetry in the preferred 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.

Phonon eigenfunctions of inhomogeneous lattices: Can you hear the shape of a cone?

“…In other words, we cannot "gauge away" the effect of gravitational pressure in the 2d plane. In this situation, the extra strain energy for this wedge-shaped sheet might possibly be reduced by introducing defects, such as a vertical grain boundary at x = 0 [35,36]. However, we show in the main text that, remarkably, this geometric frustration can be removed entirely by wrapping the horizontal direction x around to form a cylinder and thus allowing the sheet to curve into the third dimension.…”