Landau theory

Landau theory (also known as Ginzburg–Landau theory, despite the confusing name) in physics is a theory that Lev Landau introduced in an attempt to formulate a general theory of continuous (i.e., second-order) phase transitions. It can also be adapted to systems under externally-applied fields, and used as a quantitative model for discontinuous (i.e., first-order) transitions. Although the theory has now been superseded by the renormalization group and scaling theory formulations, it remains an exceptionally broad and powerful framework for phase transitions, and the associated concept of the order parameter as a descriptor of the essential character of the transition has proven transformative.

Landau was motivated to suggest that the free energy of any system should obey two conditions:

Given these two conditions, one can write down (in the vicinity of the critical temperature, T_c) a phenomenological expression for the free energy as a Taylor expansion in the order parameter.

Consider a system that breaks some symmetry below a phase transition, which is characterized by an order parameter ${\displaystyle \eta }$. This order parameter is a measure of the order before and after a phase transition; the order parameter is often zero above some critical temperature and non-zero below the critical temperature. In a simple ferromagnetic system like the Ising model, the order parameter is characterized by the net magnetization ${\displaystyle m}$, which becomes spontaneously non-zero below a critical temperature ${\displaystyle T_{c}}$. In Landau theory, one considers a free energy functional that is an analytic function of the order parameter. In many systems with certain symmetries, the free energy will only be a function of even powers of the order parameter, for which it can be expressed as the series expansion

In general, there are higher order terms present in the free energy, but it is a reasonable approximation to consider the series to fourth order in the order parameter, as long as the order parameter is small. For the system to be thermodynamically stable (that is, the system does not seek an infinite order parameter to minimize the energy), the coefficient of the highest even power of the order parameter must be positive, so ${\displaystyle b(T)>0}$. For simplicity, one can assume that ${\displaystyle b(T)=b_{0}}$, a constant, near the critical temperature. Furthermore, since ${\displaystyle a(T)}$ changes sign above and below the critical temperature, one can likewise expand ${\displaystyle a(T)\approx a_{0}(T-T_{c})}$, where it is assumed that ${\displaystyle a>0}$ for the high-temperature phase while ${\displaystyle a<0}$ for the low-temperature phase, for a transition to occur. With these assumptions, minimizing the free energy with respect to the order parameter requires

The solution to the order parameter that satisfies this condition is either ${\displaystyle \eta =0}$, or

It is clear that this solution only exists for ${\displaystyle T<T_{c}}$, otherwise ${\displaystyle \eta =0}$ is the only solution. Indeed, ${\displaystyle \eta =0}$ is the minimum solution for ${\displaystyle T>T_{c}}$, but the solution ${\displaystyle \eta _{0}}$ minimizes the free energy for ${\displaystyle T<T_{c}}$, and thus is a stable phase. Furthermore, the order parameter follows the relation

below the critical temperature, indicating a critical exponent ${\displaystyle \beta =1/2}$ for this Landau mean-theory model.