  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.
