Instantons and fractals in critical systems

The geometry of the critical fluctuations for a general system undergoing a thermal second-order phase transition is investigated. Adopting a generalized free-energy for the local description of the fluctuations of the order parameter at the critical temperature it is shown that instanton-like configurations corresponding to the minima of the free-energy functional, build up clusters with fractal geometry characterizing locally the critical fluctuations. For sufficiently small coarse-graining scales these configurations dominate the partition function and provide an exact description of the critical system. The connection between the corresponding (local) fractal dimension and the critical exponents is derived. Implications of these ideas to chiral QCD phase transition in relativistic heavy ion collisions are also discussed. (author)