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Concept# Critical phenomena

Summary

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, and ergodicity breaking. Critical phenomena take place in second order phase transitions, 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.
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 appear. 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 .

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