This thesis attempts to investigate the phenomenon of supersolidity, where a system exhibits both spontaneously broken translational and U(1) symmetries, in two different two dimensional systems, one fermionic and one bosonic. The fermionic system consists of two parallel GaAs quantum wells that are independently gated. This allows the electron and hole populations of the two layers to be independently varied. Using mean field theory, it will be shown that the zero temperature phase diagram of this system contains a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, analogous to that predicted to occur in superconductors. This phase has spontaneously broken U(1) and translational symmetries, and can therefore be thought of as a supersolid. This mean field analysis will be complemented by a GinzburgLandau approach, which will be used to confirm the results and to calculate the lattice structure of the FFLO order parameter. The bosonic system consists of a thin helium-4 film deposited on graphite. Recent experiments on this system have produced results that suggest the presence of a supersolid phase over a range of helium filling fractions, as well as the lack of a Kosterlitz-Thouless phase transition at finite temperature. An attempt to explain these results is made by applying mean field and Bogoliubov theories to a toy model at zero temperature.

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