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We study in random-phase approximation the newly discovered supersolid phaseof ${}^4$He and present in detail its finite temperature properties. ${}^4$Heis described within a hard-core quantum lattice gas model, with nearest andnext-nearest neighbour interactions taken into account. We rigorously calculateall pair correlation functions in a cumulant decoupling scheme. Our resultssupport the importance of the vacancies in the supersolid phase. We show thatin a supersolid the net vacancy density remains constant as function oftemperature, contrary to the thermal activation theory. We also analyzed indetail the thermodynamic properties of a supersolid, calculated the jump in thespecific heat which compares well to the recent experiments.

Finite Temperature Properties of a Supersolid: a RPA Approach