In most continuous phase transitions, there is a well-defined order parameter $\langle \psi \rangle$ of some observable that is zero above the transition temperature, and continously grows below the transition. In the cases that I am familiar with, this observable is usually the expectation value of some physical quantity (spin, density, momentum, position, number) that is generally a local observable. On the other hand, it has become increasingly clear that there is a lot of non-local physics captured in quantities like entanglement, so perhaps more complex types of non-local order parameters may exist.
My question is, can entanglement entropy, or a similar entanglement witness, ever be an order parameter in a phase transition? If yes, what class of phase transitions do they appear in? If not, can you explain why?
I am interested to hear about both the trivial answer, where an entanglement measure and conventional observable can both be the order parameter, and the non-trivial case where only entanglement can be treated as the order parameter. I suppose at some level this whole discussion must involve quantum phase transitions, not just classical phase transitions.