In a classical solid, thermal fluctuations produce a small density of vacancies. They diffuse slowly. In solid He4, a quantum solid, large quantum fluctuations generate a finite density of vacancies, even at T = 0. The lightness of He4 atoms make diffusion of `zero point' vacancies coherent and quantum mechanical; at low temperatures vacancies can even bose condense. This brings in the possibility of a superfluid like behavior hidden in a solid ! This is a supersolid .
Recent experiments of Kim and Chan  claim to see supersolid behavior, nearly 35 years after the original prediction. In addition, there are other fascinating properties such as crystallization waves and anomalous mass diffusion exhibited by solid He4. I will review the physics of this quantum matter and briefly discuss my recent theory . In this theory, superfluid phase stiffness and an atomic scale spontaneous superflow circulation of a periodic pattern gives a solid like rigidity to an otherwise superfluid. Many properties get explained in a natural way and some interesting consequences follow.
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 G. Baskaran, And quiet flows the supersolid, Cond-mat/0505160
(to appear in J Phys: Cond-Mat)