Elementary excitations, Spectral weights and Experimental signatures in a Supersolid and a Fulde - Ferrell-Larkin-Ovchinnikov state

We construct a Ginsburg Landau (GL) theory to study the phases of liquid, solid, superfluid, especially a possible supersolid and phase transitions among these phases in a unified framework. In this GL, we put the two competing orders between the solid component and the superfluid component on the same footing. We only introduce two parameters: $ v $ which is the repulsive interaction between the normal component and the local superfluid mode and $ g $ which is a periodically changing chemical potential for the local superfluid mode. The microscopic origins of $ v $ and $ g $ are given. By using this GL action, we study the superfluid to supersolid transition, normal solid to the supersolid transition and analyze the conditions for the existence of the supersolid.