Journal of Statistical Physics

, Volume 157, Issue 4–5, pp 755–829 | Cite as

Renormalization Theory of a Two Dimensional Bose Gas: Quantum Critical Point and Quasi-Condensed State

  • S. Cenatiempo
  • A. Giuliani


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.


Two-dimensional Bose gas Quantum criticality Bose–Einstein condensation Bogoliubov theory Renormalization group Ward identities 



The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme ERC Starting Grant CoMBoS (Grant Agreement No. 239694). We thank Giuseppe Benfatto and Vieri Mastropietro for several discussions and illuminating comments. S.C. acknowledges the Hausdorff Center for Mathematics in Bonn for financial support.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Universität ZürichZürichSwitzerland
  2. 2.Università degli Studi Roma TreRomaItaly

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