Supersolid phase of matter: exit helium, enter cold atoms

The search for the elusive "supersolid" phase of matter has engaged condensed matter physicists for decades. Solid helium-4 has long been regarded as the condensed matter system most likely to display such a phase, and indeed a lot of enthusiasm was generated in 2004, with the observation by Kim and Chan of a low temperature anomaly in the rotational inertia of solid Helium-four, possibly pointing to a normal to supersolid transition.Understanding that effect has proven a theoretical and experimental challenge, but consensus is now building that in fact such an anomaly has a more mundane explanation than supersolidity, i.e., the stiffening of the helium crystal at low temperature.Still, the investigative effort of the past nine years has yielded a lot of novel and interesting information about the supersolid phase and solid helium. In particular, theoretical progress has been afforded by large-scale computer simulations of realistic models of condensed matter systems (solid helium), which have elucidated and ruled out some of the most accredited scenarios of supersolidity, e.g., that based on point defects, and identified novel mechanisms, based on extended defects, that may still lead to supersolid behaviour in Helium-4.In the meantime, technological progress achieved in the area of cold atoms has paved the way to the possible observation of the supersolid phase in that context.In this talk, the most important theoretical results regarding a putative supersolid phase of helium will be reviewed, with a particular emphasis on the physics of extended defects, such as dislocations. Moreover, novel supersolid droplet crystal phase will be described, predicted by simulation for a system of cold atoms with a modified interaction, whose experimental realization appears feasible.