Non Perturbative Renormalization Group and BoseEinstein 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 swave 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 eprints
 Pub Date:
 December 2007
 arXiv:
 arXiv:0801.0009
 Bibcode:
 2008arXiv0801.0009B
 Keywords:

 Condensed Matter  Statistical Mechanics
 EPrint:
 Lectures given at the 2006 ECT* School "Renormalization Group and Effective Field Theory Approaches to ManyBody Systems", Trento, Italy. Late submission to arXiv