Title: Orientational melting in a mesoscopic system of charged particles

Abstract: A mesoscopic system of a few particles exhibits behaviors that strongly differ from those of a macroscopic system. While in a macroscopic system phase transitions are universal, a change in the state of a mesoscopic system depends on its specific properties, like the number of particles, to the point that changes of state can be disfavored for specific magic numbers. A transition that has no counterpart in the macroscopic world is orientational melting, in which localized particles with long-range repulsive interactions forming a two-dimensional crystal become delocalized in common circular or elliptical trajectories. Orientational melting has been studied extensively with computer simulations and witnessed in a few pioneering experiments. However, a detailed experimental investigation fully revealing its non-universal nature has been missing so far. Here we report the observation of orientational melting in a two-dimensional ensemble of up to 15 ions with repulsive Coulomb interaction. We quantitatively characterize orientational melting, and compare the results with a Monte Carlo simulation to extract the particles kinetic energy. We demonstrate the existence of magic numbers, and control locally the occurrence of melting by adding a pinning impurity. Our system realizes a fully-controllable experimental testbed for studying the thermodynamics of small systems, and our results pave the way for the study of quantum phenomena in systems of delocalized ions, from the emergence of quantum fluctuations and quantum statistics, to the control of multi-shell quantum rotors.