Fractal dimensions of states and their application to Ising model

Mandelbrot introduced a new criterion to analyse complex geometrical sets [1]. This criterion is the so-called fractal dimension, which is different from the usual euclidean dimension. Usual fractal theory mostly treats only geometrical set. It is desirable to extend the fractal theory so as to be applicable to some other objects. For this purpose, Ohya introduced a concept of fractal dimension for general states [2, 3, 4]. This fractal dimension provides a new measure for the complexity of states. It can be used to distinguish between two states even if their entropies have the same value. In this paper, we briefly review the fractal dimension of states, and show that Ising model can be characterized by this dimension.