Critical phenomena

In physics, critical phenomena is the collective name associated with the physics of critical points. Most of them stem from the divergence of the correlation length, but also the dynamics slows down. Critical phenomena include scaling relations among different quantities, power-law divergences of some quantities (such as the magnetic susceptibility in the ferromagnetic phase transition) described by critical exponents, universality, fractal behaviour, ergodicity breaking. Critical phenomena take place in second order phase transition, although not exclusively.

The critical behavior is usually different from the mean-field approximation which is valid away from the phase transition, since the latter neglects correlations, which become increasingly important as the system approaches the critical point where the correlation length diverges. Many properties of the critical behavior of a system can be derived in the framework of the renormalization group.

In order to explain the physical origin of these phenomena, we shall use the Ising model as a pedagogical example.

Let us consider a ${\displaystyle 2D}$ square array of classical spins which may only take two positions: +1 and −1, at a certain temperature ${\displaystyle T}$, interacting through the Ising classical Hamiltonian:

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