Conformal Vortex Crystals

Defects in conformal crystals: Discrete versus continuous disclination models

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 nonuniform densities in the clusters. Ground states in such systems 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 nonuniform density to the formation of disclination defects, one of which assumes a continuum distributions of defects, 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 study 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 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.

Formation and stability of conformal spirals in confined 2D crystals

We investigate the ground-state and dynamical properties of nonuniform two-dimensional (2D) clusters of long-range interacting particles. We demonstrate that, when the confining external potential is designed to produce an approximate 1/r²density profile, the particles crystallize into highly ordered structures featuring spiral crystalline lines. Despite the strong inhomogeneity of the observed configurations, most of them are characterized by small density of topological defects, typical of conformal crystals, and the net topological charge induced by the simply-connected geometry of the system is concentrated near the cluster center. These crystals are shown to be robust with respect to thermal fluctuations up to a certain threshold temperature, above which the net charge is progressively redistributed from the center to the rest of the system and the topological order is lost. The crystals are also resilient to the shear stress produced by a small nonuniform azimuthal force field, rotating as a rigid body (RB). For larger forces, topological defects proliferate and the RB rotation gives place to plastic flow.

Self-assembled vortex crystals induced by inhomogeneous magnetic textures

We investigate the self-assembly of vortices in a type-II superconducting disk subjected to highly nonuniform confining potentials produced by inhomogeneous magnetic textures. Using a series of numerical experiments performed within the Ginzburg-Landau theory, we show that vortices can arrange spontaneously in highly nonuniform, defect-free crystals, reminiscent of conformal lattices, even though the strict conditions for the conformal crystal are not fulfilled. These results contradict continuum-limit theory, which predicts that the order of a nonuniform crystal is unavoidably frustrated by the presence of topological defects. By testing different cooling routes of the superconductor, we observed several different self-assembled configurations, each of which corresponding to one in a set of allowed conformal transformations, which depends on the magnetic and thermal histories of the system.

Disordering, clustering, and laning transitions in particle systems with dispersion in the Magnus term

We numerically examine a two-dimensional system of repulsively interacting particles with dynamics that are governed by both a damping term and a Magnus term. The magnitude of the Magnus term has one value for half of the particles and a different value for the other half of the particles. In the absence of a driving force, the particles form a triangular lattice, while when a driving force is applied, we find that there is a critical drive above which a Magnus-induced disordering transition can occur even if the difference in the Magnus term between the two particle species is as small as one percent. The transition arises due to the different Hall angles of the two species, which causes their motion to decouple at the critical drive. At higher drives, the disordered state can undergo both species and density phase separation into a density-modulated stripe that is oriented perpendicular to the driving direction. We observe several additional phases that occur as a function of drive and Magnus force disparity, including a variety of density-modulated diagonal-laned phases. In general, we find a much richer variety of states compared to systems of oppositely driven overdamped Yukawa particles. We discuss the implications of our work for skyrmion systems, where we predict that even for small skyrmion dispersities, a drive-induced disordering transition can occur along with clustering phases and pattern-forming states.