Landau theory of charge order: symmet...

Landau theory of charge order: symmetry breaking

Landau theory is a good way to describe the symmetry breaking of of charge order using group theory.

Suppose the charge pattern can be described as

\rho(\vec{r})=\rho_0(\vec{r})+\delta\rho(\vec{r})

where ρ is the total charge density, ρ₀ is the charge density of the high temperature phase corresponding to symmetry group G₀ and δρ correspond to one or more irreducible representations (except the identity) of G₀.

At high temperature δρ=0, giving the high symmetry phase. While at low temperature, δρ≠0, giving the low symmetry phase.

A good example of cubic to tetragonal structural phase transition can be very revealling.

In this case:

\rho(\vec{r})=\sum_i\rho_0(\vec{r}-\vec{R}_i)+\rho_1(\vec{r}-\vec{R}_i+z_0)

where R_i represents the cubic lattice and z₀ is the distortion along z direction.

One can look at the character table (below) and recognize that the z₀ distortion corresponds T_1u irreducible representation (IR).



A closer look at the character table shows that only C₄, σ_h and σ_d leave z₀ invariant. Therefore, in the low temperature (low symmetry phase), the group has E, C₄, σ_d and σ_h, which gives a new group C_4v, corresponding to the symmetry of the low T phase.