Ginsburg-Landau theory of supersolid

We develop a simple Ginsburg-Landau theory to study all the possible phases and phase transitions in $^{4}He $, analyze the condition for the existence of the supersolid (SS) and map out its global phase diagram from a unified framework. If the condition favors the existence of the SS, we use the GL theory to address several experimental facts and also make some predictions that are amenable to experimental tests. A key prediction is that the X-ray scattering intensity from the SS ought to have an additional modulation over that of the NS. The modulation amplitude is proportional to the Non-Classical Rotational-Inertial (NCRI) observed in the torsional oscillator experiments.