Landau Theory

Landau theory in physics was introduced by Lev Landau in an attempt to formulate a general theory of second-order phase transitions. He was motivated to suggest that the free energy of any system should obey two conditions: that the free energy is analytic, and that it obeys the symmetry of the Hamiltonian.

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. For example, the Ising model free energy may be written as the following:

where the parameter

for physical reasons. The variable is the coarse-grained field of spins, known as the order parameter or the total magnetization.

This theory of Landau first raised the order parameter to prominence. Note that the Ising model exhibits the following discrete symmetry: If every spin in the model is flipped, such that, where is the value of the spin, the Hamiltonian (and consequently the free energy) remains unchanged. This symmetry is reflected in the even powers of in .

Landau theory has been extraordinarily useful. While the exact values of the parameters and were unknown, critical exponents could still be calculated with ease, and only depend on the original assumptions of symmetry and analyticity. For the Ising model case, the equilibrium magnetization assumes the following value below the critical temperature :

At the time, it was known experimentally that the liquid-gas coexistence curve and the ferromagnet magnetization curve both exhibited a scaling relation of the form, where was mysteriously the same for both systems. This is the phenomenon of universality. It was also known that simple liquid-gas models are exactly mappable to simple magnetic models, which implied that the two systems possess the same symmetries. It then followed from Landau theory why these two apparently disparate systems should have the same critical exponents, despite having different microscopic parameters. It is now known that the phenomenon of universality arises for other reasons (see Renormalization group). In fact, Landau theory predicts the incorrect critical exponents for the Ising and liquid-gas systems.

The great virtue of Landau theory is that it makes specific predictions for what kind of non-analytic behavior one should see when the underlying free energy is analytic. Then, all the non-analyticity at the critical point, the critical exponents, are because the equilibrium value of the order parameter changes non-analytically, as a square root, whenever the free energy loses its unique minimum.

The extension of Landau theory to include fluctuations in the order parameter shows that Landau theory is only strictly valid near the critical points of ordinary systems with spatial dimensions of higher than 4. This is the upper critical dimension, and it can be much higher than four in more finely tuned phase transition. In Mukhamel's analysis of the isotropic Lifschitz point, the critical dimension is 8. This is because Landau theory is a mean field theory.

This theory does not explain all the non-analyticity at the critical point, but when applied to superfluid and superconductor phase transition, Landau's theory provided inspiration for another theory, the Ginzburg-Landau theory of superconductivity.