Multifractality can be a universal signature of phase transitions

Macroscopic systems often display phase transitions where certain physical quantities are singular or self-similar at different (spatial) scales. Such properties of systems are currently characterized by some order parameters and a few critical exponents. Nevertheless, recent studies show that the multifractality, where a large number of exponents are needed to quantify systems, appears in many complex systems displaying self-similarity. Here we propose a general approach and show that the appearance of the multifractality of an order parameter related quantity is the signature of a physical system transiting from one phase to another. The distribution of this quantity obtained within suitable (time) scales satisfies a $q$-Gaussian distribution plus a possible Cauchy distributed background. At the critical point the $q$-Gaussian shifts between Gaussian type with narrow tails and L$\acute{\text{e}}$vy type with fat tails. Our results suggest that the Tsallis $q$-statistics, besides the conventional Boltzmann-Gibbs statistics, may play an important role during phase transitions.