Symmetry Analysis in the Landau Theory

The Landau theory of second order phase transitions (SOPT) considers a continuous phase transition between high-symmetry phase of crystal of symmetry G and low-symmetry phase of symmetry H'. In this theory two functions are used:

1.

The thermodynamical potential $\Phi$, common for all phases. The different phases are associated with non-equivalent minima of $\Phi$.

2.

The density function $\rho=\rho(\vec r)$, which changes continuously in the phase transition.

In order to describe the physical property which causes the phase transition, Landau introduced the variable $\eta$,called order parameter, such that $\eta \equiv 0$ in G phase and $\eta \neq 0$ in H' phase. This parameter can be either a scalar or a tensor $\vec \eta = \{ \eta_i \}$.

Minimizing $\Phi(p, T, \eta_i)$ with $\eta_i$ as variational parameters one can find the values $\eta_i^0$ for the equilibrium states, i.e. one can construct the density function $\rho_i \propto \eta_i^0$ and determine the structure and the symmetry group H_i of the phases.

Fortunately, the main result of the Landau theory of SOPT follows directly from the problem symmetry [1,2]. The results can be obtained using only a set of group-subgroup criteria [3] without a minimization of the potential $\Phi$. From the symmetry analysis we can get [4] Symmetry group G of higher-symmetry phase; The full set of all possible phases (by their symmetry groups); The transformation properties of the order parameter $\eta$ (by the irreducible representations) suggesting its physical origin; Information about type of the phase transition - first or second order, is the lower-symmetry phase is commensurate or not, etc.

From analysis we can get compatible triad (G, H', D_G^j). Each triad defines one allowed phase transition $G \longrightarrow H'$. In the PT analysis in crystals G is among the 230 crystallographic space groups (infinite, with infinite number of subgroups and infinite number of irreducible representations). Using the so called group criteria [4] one can select a small number of triad (G₀, H₁, D_G0^j) which a basis for more detailed investigation. In the frame of infinite space groups this procedure is quite nontrivial and needs deep knowledge of theory of space groups and their irreducible representations [3]. Probability to get wrong conclusion or at least to miss some of the phases is not negligible.