Phase transitions and renormalization group: From theory to numbers

During the last century, in two apparently distinct domains of physics, the theory of fundamental interactions and the theory of phase transitions in condensed matter physics, one of the most basic ideas in physics, the decoupling of physics on different length scales, has been challenged. To deal with such a new situation, a new strategy was invented, known under the name of renormalization group. It has allowed not only explaining the survival of universal long distance properties in a situation of coupling between microscopic and macroscopic scales, but also calculating precisely universal quantities. We here briefly recall the origin of renormalization group ideas; we describe the general renormalization group framework and its implementation in quantum field theory. It has been then possible to employ quantum field theory methods to determine many universal properties concerning the singular behaviour of thermodynamical quantities near a continuous phase transition. Results take the form of divergent perturbative series, to which summation methods have to be applied. The large order behaviour analysis and the Borel transformation have been especially useful in this respect. As an illustration, we review here the calculation of the simplest quantities; critical exponents. More details can be found in the work J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press 1989, (Oxford 4th ed. 2002).