Non-relativistic interacting bosons at zero temperature exhibit two
interesting critical phases: the celebrated condensate phase and the
critical theory at zero density, known as quantum critical point.
From a theoretical point of view these theories are particularly challenging in
dimension two, which is in both cases critical in the sense of
Renormalization Group (RG). This is the method for understanding from
first principles the emergence of scaling laws in interacting many body
systems at low or zero temperatures, as well as for computing thermodynamic
and correlation functions.
In collaboration with A. Giuliani we proved renormalizability of the
quantum critical point and of the condensed phase in two dimensions, both
in the ultraviolet and in the infrared, and developed a theory valid at all
orders in renormalized perturbation theory, with explicit bounds on the
generic order. In this talk I will present these results and compare them
with the existing literature. While the results obtained for the quantum
critical case match with previous ones and extend them to all orders, we
think that our findings call into question the stability of the two
dimensional condensate at zero temperature.