Fractal geometry of critical systems

We investigate the geometry of a critical system undergoing a second-order thermal phase transition. Using a local description for the dynamics characterizing the system at the critical point ${T=T}_c,$ we reveal the formation of clusters with fractal geometry, where the term cluster is used to describe regions with a nonvanishing value of the order parameter. We show that, treating the cluster as an open subsystem of the entire system, new instanton-like configurations dominate the statistical mechanics of the cluster. We study the dependence of the resulting fractal dimension on the embedding dimension and the scaling properties (isothermal critical exponent) of the system. Taking into account the finite-size effects, we are able to calculate the size of the critical cluster in terms of the total size of the system, the critical temperature, and the effective coupling of the long wavelength interaction at the critical point. We also show that the size of the cluster has to be identified with the correlation length at criticality. Finally, within the framework of the mean field approximation, we extend our local considerations to obtain a global description of the system.