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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?

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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.

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  • $\begingroup$ 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 '15 at 8:09
  • $\begingroup$ It is not necessary. Search for "topological phase transitions." $\endgroup$ – Meng Cheng Jun 17 '15 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 '15 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 '15 at 0:00

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