Multifractility of Ising models on hierarchical lattices: pure and spin glass cases

The multifractal properties of the order parameter of Ising spin models on hierarchical lattices are investigated by an exact recursion procedure. Both pure and spin glass cases exhibit an order parameter with multifractal structure at the critical point. The connection between the multifractal ƒ(α) function and the critical exponents governing the transition is established. A continuous infinite set of exponents is required to describe the critical behavior of the local order parameter. Scaling relations between these exponents are also obtained.