Supersolidity and disorder

A solid is called 'supersolid' if it exhibits superfluid properties. Supersolidity is a paradoxical phenomenon whose understanding has become a major challenge since 2004, when Kim and Chan first observed what could be mass superflow through solid helium 4. In this review, we describe how successive experiments indicated that what was observed in helium 4 was not intrinsic properties of the crystalline state as originally proposed 35 years before. Disorder coming from how the solid is grown (dislocations, grain boundaries and other interfaces, liquid or glassy regions, impurities...) was shown to play an essential role. However, one does not know yet which type of disorder is involved or by which mechanism it leads to the observed properties. Furthermore, all the experimental features probably cannot be explained by a common mechanism. Recent measurements of the shear modulus of helium 4 crystals could even be explained without the need of any superfluidity. In fact, many theoretical predictions need to be checked experimentally, so the whole issue is far from understood. Even some crucial experiments would need to be repeated more systematically. The present review of the experimental observations and theoretical scenarios raises a series of questions which call for answers. (topical review)