Fractal Configurations of the Two- and Three-Dimensional Ising Models at the Critical Point

The fractal structure of the Ising model at the critical point Tc is studied in the present paper. The fractal dimensionality D of the total magnetization at Tc was estimated numerically as D = 1.86 ±O.Ol for the two-dimensional square lattice and D=2.46±O.Ol for the three-dimensional simple cubic lattice by Monte Carlo simulations. These values agree very well with the value D=1.875 obtained from the exact critical exponents and D=2.48 obtained from the known critical exponents, re ~pectively, through the relation D=d-fJ/v=(d+r/v)/2. This fractalness yields the hyperscaling relation dv=2fJ+r. It was also observed how the fractal nature of the relevant system disappears as the system deviates from the critical point. The dimensionality d of the relevant lattice is observed at temperatures lower than Tc and the random percolation value d/2 at higher temperatures.