Critical phenomena

In physics, critical phenomena is the collective name associated with thephysics 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. In physics, critical phenomena is the collective name associated with thephysics 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 2 D {displaystyle 2D} square array of classical spins which may only take two positions: +1 and −1, at a certain temperature T {displaystyle T} , interacting through the Ising classical Hamiltonian: where the sum is extended over the pairs of nearest neighbours and J {displaystyle J} is a coupling constant, which we will consider to be fixed. There is a certain temperature, called the Curie temperature or critical temperature, T c {displaystyle T_{c}} 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 T c {displaystyle T_{c}} , 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, ξ {displaystyle xi } grows with temperature until it diverges at T c {displaystyle T_{c}} . 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. The correlation length diverges at the critical point: as T → T c {displaystyle T o T_{c}} , ξ → ∞ {displaystyle xi o infty } . This divergence poses no physical problem. Other physical observables diverge at this point, leading to some confusion at the beginning. The most important is susceptibility. Let us apply a very small magnetic field to thesystem in the critical point. A very small magnetic field is not able to magnetize a large coherent cluster, but with these fractal clusters the picture changes. It affects easily the smallest size clusters, since they have a nearly paramagnetic behaviour. But this change, in its turn, affects the next-scale clusters, and the perturbation climbs the ladder until the whole system changes radically. Thus, critical systems are very sensitive to small changes in the environment. Other observables, such as the specific heat, may also diverge at this point. All these divergences stem from that of the correlation length.