The second lecture focused on renormalization in real space. In the case of the Ising model, Kadanoff's original idea is to group spins taking the values ± 1 into blocks (e.g. blocks of 5 spins) and define a renormalized spin for each block equal to the sign of the sum of the spins in the block. The main difficulty with this approach is the profiling of the number of couplings (i.e. the effective interactions between renormalized spins), which makes it impossible to iterate the renormalization transformation exactly. From a theoretical point of view, physicists have been interested in models of spins on hierarchical networks, whose main advantage is to avoid the proliferation of these couplings through renormalization : in a way, hierarchical networks are tailor-made for renormalization. For these networks, the transformation of couplings by renormalization can be written explicitly. In this way, the exact values of the exponents can be deduced, and the critical behavior of the free energy in the vicinity of the transition can be accurately described. Unlike renormalization in real space, for Euclidean networks renormalization in Fourier space allows us to limit the number of couplings during renormalization when we are close to the higher dimension. Taking the example of the Edwards model, which models a polymeric chain in solution, it has been shown how the critical higher dimension dc = 4 appears in this type of problem.
09:30 - 11:00
