Non Perturbative Renormalization Group and Bose-Einstein Condensation
Abstract
These lectures are centered around a specific problem, the effect of weak repulsive interactions on the transition temperature $T_c$ of a Bose gas. This problem provides indeed a beautiful illustration of many of the techniques which have been discussed at this school on effective theories and renormalization group. Effective theories are used first in order to obtain a simple hamiltonian describing the atomic interactions: because the typical atomic interaction potentials are short range, and the systems that we consider are dilute, these potentials can be replaced by a contact interaction whose strength is determined by the s-wave scattering length. Effective theories are used next in order to obtain a simple formula for the shift in $T_c$: one exploits there the fact that near $T_c$ the physics is dominated by low momentum modes whose dynamics is most economically described in terms of classical fields; the ingredients needed to calculate the shift of $T_c$ can be obtained from this classical field theory. Finally the renormalization group is used both to obtain a qualitative understanding, and also as a non perturbative tool to evaluate quantitatively the shift in $T_c$.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2007
- arXiv:
- arXiv:0801.0009
- Bibcode:
- 2008arXiv0801.0009B
- Keywords:
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- Condensed Matter - Statistical Mechanics
- E-Print:
- Lectures given at the 2006 ECT* School "Renormalization Group and Effective Field Theory Approaches to Many-Body Systems", Trento, Italy. Late submission to arXiv