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

TL;DR

This paper rigorously constructs a two-dimensional weakly interacting Bose gas model at zero temperature, analyzing both quantum critical and condensate phases using renormalization group methods, and challenges previous fixed point hypotheses.

Contribution

It provides a complete, rigorous renormalization group construction of the 2D Bose gas at zero temperature, extending understanding of quantum criticality and condensate phases.

Findings

Constructed the quantum critical point in both UV and IR regimes.

Solved the ultraviolet problem for the condensate phase.

Identified violations of Ward Identities due to regularization, questioning fixed point assumptions.

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 $(\lambda^3 \rho_0)^{-1/2}$ (here $\lambda$ is the interaction strength and $\rho_0$ 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.