Landau Theory of Phase Transitions

Abstract

A phase transition is the phenomenon that a many-body system may suddenly change its properties in a rather drastic way due to the change of an externally controllable variable. Familiar examples in everyday life are the transitions from gases to liquids or from liquids to solids, due to for example a change in the temperature or the pressure. Another example is the transition from a disordered to a magnetized state in a ferromagnetic material as a function of temperature or magnetic field. One property that all these transitions share is that the order of the system, described for example by the density or the magnetization, differs at each side of the transition. We consider as an example an Ising-like spin system at a low, but nonzero temperature, such that the ferromagnetic state with many spins pointing in the same direction corresponds to an absolute minimum of the free energy. Upon applying a magnetic field in the opposite direction, the equilibrium state may change to a state where most spins point in the opposite direction. As a result, the system is initially in a local minimum of the free energy, and it has to overcome a large energy barrier in order to reach the new equilibrium state. Still, eventually the system reaches the new equilibrium state due to the thermal activation of random spin flips in the system, such that the corresponding transition can be said to be driven by thermal fluctuations. Note that the magnetization makes a large jump by going from one equilibrium state to the other. Such a transition, when the parameter describing the order in the system is discontinuous, we call a first-order phase transition.

Phase transitions can also be continuous, which is the case when the order parameter changes from zero to a nonzero value in a continuous way. Continuous, or second-order, phase transitions can be very spectacular, because we will see that they give rise to a diverging correlation length and hence to behavior known as critical phenomena. The infinite correlation length implies that fluctuations extend over the whole many-body system, such that they are present at each length scale. As a result, the system looks similar at every length scale, i.e. it is scale invariant, which can be used to recursively describe the critical system at increasing wavelengths. This leads to the very powerful renormalization group method, which is able to go far beyond mean-field theory and which is the topic of Chap. 14. Finally we remark that phase transitions can also occur at zero temperature, and are then called quantum phase transitions because they are solely driven by quantum fluctuations. We will see a detailed example of a quantum phase transition in Chap. 16.