# Conformal Vortex Crystals

## TL;DR

This paper demonstrates that under specific conditions, vortices in superconductors can self-organize into conformal crystal structures, which are topologically ordered and mathematically related to traditional vortex lattices, with potential for experimental observation.

## Contribution

It introduces the concept of conformal vortex crystals in superconductors, showing their formation, stability conditions, and methods for engineering such structures, extending understanding of vortex matter.

## Key findings

- Vortex systems form conformal crystals under suitable force fields.
- Conformal vortex crystals are topologically ordered and related to Abrikosov lattices.
- These structures become unstable above a certain temperature, leading to polycrystalline phases.

## Abstract

We investigate theoretically globally nonuniform configurations of quantized-flux vortices in clean superconductors trapped by an external force field that induces a nonuniform vortex density profile. Using an extensive series of numerical simulations, we demonstrate that, for suitable choices of the force field, and bellow a certain transition temperature, the vortex system self-organizes into highly inhomogeneous conformal crystals in a way as to minimize the total energy. These nonuniform structures are topologically ordered and can be mathematically mapped into a triangular Abrikosov lattice via a conformal transformation. Above the crystallization temperature, the conformal vortex crystal becomes unstable and gives place to a nonuniform polycrystalline structure. We propose a simple method to engineer the potential energy profile necessary for the observation of conformal crystals of vortices, which can also be applied to other 2D particle systems, and suggest possible experiments in which conformal or quasi-conformal vortex crystals could be observed in bulk superconductors and in thin films.