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We study mixtures of spinless bosons and not spin-polarized fermions loadedin two dimensional optical lattices. We approach the problem of the groundstate stability within the framework of the linear response theory; by the meanof an iterative procedure, we are able to obtain a relation for the dependenceof boson-boson effective interaction on the absolute temperature of the sample.Proceeding from such a formula, we write down analyitical expressions forSupersolid (SS) and Phase Separation (PS) transition temperatures, and plot thephase diagrams.

the european physical journal. d, atomic, molecular and optical physics/the european physical journal. d, atomic, molecular, optical and plasma physics

How the effective boson-boson interaction works in Bose-Fermi mixtures in periodic geometries