Critical Phenomena - The Critical Point of The 2D Ising Model

The Critical Point of The 2D Ising Model

Let us consider a square array of classical spins which may only take two positions: +1 and −1, at a certain temperature, interacting through the Ising classical Hamiltonian:

where the sum is extended over the pairs of nearest neighbours and is a coupling constant, which we will consider to be fixed. There is a certain temperature, called the Curie temperature or critical temperature, below which the system presents ferromagnetic long range order. Above it, it is paramagnetic and is apparently disordered.

At temperature zero, the system may only take one global sign, either +1 or -1. At higher temperatures, but below, the state is still globally magnetized, but clusters of the opposite sign appears. As the temperature increases, these clusters start to contain smaller clusters themselves, in a typical Russian dolls picture. Their typical size, called the correlation length, grows with temperature until it diverges at . This means that the whole system is such a cluster, and there is no global magnetization. Above that temperature, the system is globally disordered, but with ordered clusters within it, whose size is again called correlation length, but it is now decreasing with temperature. At infinite temperature, it is again zero, with the system fully disordered.