**Authors:**
Serena Cenatiempo, Alessandro Giuliani

**Title:** *
Renormalization theory of a two dimensional Bose gas: quantum critical point and quasi-condensed state
*

**Abstract: **
We present a renormalization group construction of a weakly interacting Bose gas at zero temperature in the two-dimensional continuum, both in the quantum critical regime and in the presence of a condensate fraction. The construction is performed within a rigorous renormalization group scheme, borrowed from the methods of constructive field theory, which allows us to derive explicit bounds on all the orders of renormalized perturbation theory. Our scheme allows us to construct the theory of the quantum critical point completely, both in the ultraviolet and in the infrared regimes, thus extending previous heuristic approaches to this phase. For the condensate phase, we solve completely the ultraviolet problem and we investigate in detail the infrared region, up to length scales of the order
(λ^{3}ρ_{0})
(here λ
is
the
interaction strength
and
ρ_{0}
the condensate density), which is the largest length scale at which the problem is perturbative in nature.
We exhibit violations to the formal Ward Identities, due to the momentum cutoff used to regularize the theory, which suggest that
previous proposals about the existence of a non-perturbative non-trivial fixed point for the infrared flow should be reconsidered.

**Keywords:**
Competing interactions, long range interactions, periodic patterns, striped state

S. Cenatiempo

Institut of Mathematics, University of Zurich

E-mail address: serena DOT cenatiempo AT math DOT uzh DOT ch

A. Giuliani

Dipartimento di Matematica e Fisica, Università di Roma Tre

E-mail address: giuliani AT mat DOT uniroma3 DOT it