PERCOLATION THEORY IN THE POTTS MODEL

The Potts model may be used to describe several systems, as e.g. the magnetic behaviour of some materials. In this model, the magnetic moment of each atom, known as spin, can assume a range of possible values and orientations which depend on the material under consideration. The Hamiltonian of this model is given by:

\begin{center}
$H=-J\sum\limits_{i,j} cos\theta_{ij}$
\end{center}

where $\theta_{ij}$ is the angle between two nearest neighbours spins and J is the energy exchange. This model may exhibit a magnetic phase transition from ferromagnetism to paramagnetism at a critical temperature Tc. This temperature can be estimated via computational methods like, for example, the Monte Carlo method.

Fractal lattices are scale-free structures, very commonly present in nature, that have interesting topological features, such as lacunarity and fractal dimension. These geometrical properties can give a better perspective of the topological behavior of magnetism in general lattices.

In this work, we have shown that the ferromagnetic phase transition is related to the geometrical phase transition by the percolation theory. This theory studies the connection among clusters within the lattice and, through techniques such as image analysis, provides a mean for obtaining the Tc faster than some other methods. Additionally, we compared the phase transition order for different number of orientations and its relation to the geometry of general structures. These results presents a deep connection between geometry and phase transition theory, opening new possibilities for many models and possible magnetic phases.