# What are the symmetry criteria for continuous phase transitions in Landau theory?

My understanding is that within Landau theory, a continuous phase transition is only possible if certain symmetry rules are satisfied. (These rules represent necessary but not sufficient conditions for a continuous phase transition.) One criterion is that the symmetry group of one phase must be a subgroup of the other phase's symmetry group (cf. p. 782 of Rep. Prog. Phys. 50, 783). What are the other criteria, and where can I find a derivation of them?

There are many symmetry-breaking transitions which are not continuous. For example, it is well-known that $n$-state Potts model has a thermal symmetry-breaking phase transition, and when $n>4$ it is first order. For a more realistic one, I think melting transition is first order. Basically there is no way to tell whether a transition is continuous or not just from the symmetry. It really depends on the details a lot.
• Then it basically depends on what kind of terms you include in the free energy functional. Usually one just have $\Delta^2$ and $\Delta^4$, where $\Delta$ is the order parameter. If one further includes $\Delta^3$ or $\Delta^6$, the theory can be used to describe first-order transition. – Meng Cheng Jun 18 '15 at 0:00