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?


1 Answer 1


This "criteria" is neither sufficient nor necessary.

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.

There are also "exotic" continuous phase transitions between states with the same symmetry, or states with very different symmetry breaking. A famous example of the latter type is the so-called deconfined quantum criticality, which happens in lattice spin models where one phase spontaneously breaks translation symmetry and the other phase spontaneously breaks spin rotation symmetry. Neither is a subgroup of the other. The critical theory is a gauge theory, going beyond the scheme of Landau theory.

  • $\begingroup$ Thanks for your answer. I think that the symmetry criteria I'm thinking of represent necessary but not sufficient conditions, and that they apply only to Landau theory. I've rephrased the question accordingly. As a side note, I think transitions between states of the same symmetry (such as a liquid-gas transition) satisfy the group-subgroup criterion. $\endgroup$
    – Max Radin
    Jun 17, 2015 at 8:09
  • $\begingroup$ It is not necessary. Search for "topological phase transitions." $\endgroup$
    – Meng Cheng
    Jun 17, 2015 at 8:19
  • $\begingroup$ My understanding is that topological phase transitions cannot be described by Landau theory. I'm asking specifically about symmetry criteria in Landau theory, and so I do not expect them to apply to topological transitions. $\endgroup$
    – Max Radin
    Jun 17, 2015 at 23:37
  • $\begingroup$ 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. $\endgroup$
    – Meng Cheng
    Jun 18, 2015 at 0:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.