Phase transition vs. critical phenomena

Question

Just trying to get some clarity in terminology: is phase transitions synonymous with critical phenomena? At the first glance they mean the same thing, but I am not sure whether phase transitions really include such phenomena as Anderson localization and percolation, which are not thermally driven. However, what then about quantum phase transitions, which are not thermally driven either?

Clarification
To clarify the above, here are the options:

• phase transitions and critical phenomena are the same thing
• phase transitions are a subset of critical phenomena - e.g., we may consider Anderson localization, percolation, topological transitions, Mott transition to be critical phenomena, but not phase transitions - in this case one needs to define the difference between the two.
• Finally, there may be simply no clearly established terminology, as is suggested by expressions such as quantum phase transition, which likely should not be called phase transition under the classification proposed in the previous bullet.

Answer 1

is phase transitions synonymous with critical phenomena

I don't know and I don't think there's a consensus.

A book on condensed matter, for example, says:

The term critical phenomena refers to the peculiar behaviour of a substance when it is at or near the point of a continuous-phase transition, or the critical point

It would be easier to disprove your statement rather than proving it, just by finding a counterexample. I cannot think of one, though.

How can we define whether two phases are different from each other?

One may distinguish phases based on:

• physical properties, such as compressibility and rigidity (liquids vs gas);
• distinct symmetries associated with each phase, and a symmetry breaking to go from one to the other;
• an unequivocally defined mathematical quantity (an invariant) that has a different values in either phase, such as a topological invariant like a Chern number or a Pfaffian;
• the presence of a phase transition resulting from a gradual change in some parameter (order parameter)

None of these points is satisfactory though. Because one can still change phases via crossovers and not transitions, in which there are no abrupt changes. So once again I think there is no consensus (yet), and it much easier to say what is not a phase transition or a distinct phase, rather than the opposite.

The order parameter is usually temperature (Ising model, melting points, Bose-Einstein condensation), but does not have to be (e.g. quantum phase transitions). Highly recommend this answer for quantum phase transitions.

The special case of Anderson localisation and other disorder-induced phenomena

When comparing two phases, one usually assumes that both of them are at equilibrium. That is, they had time to settle, and re-arrange their microscoping structure. I could just melt everything and then quench it, freezing its messy internal configuration, and calling it a new "phase" because it looks different than before (this is actually what a glass is).

And usually by equilibrium one talks about thermal equilibrium: that is, let's go over the transition slowly enough that the whole system can thermalise, and there are not small pockets or the other phase that survive (see Kibble-Zurek mechanism).

In this context, disorder-induced "phases" such as an Anderson insulator are tricky. Because, by definition, a disorder-induced phase does not thermalise, thus it cannot reach thermal equilibrium. So you cannot use temperature as the order parameter, and cannot look for a discontinuity in thermodynamic quantities and/or in the free energy.

Hopefully, though, we can agree that the phases "look different" in the Anderson localised and extended regime. Hence why I, personally, still refer to them as distinct phases. Turns out that non-random disorder (Anderson is random) does give you a critical disorder strength for the transition, reminiscent of "usual" transitions with an order parameter (e.g. quasiperiodic disorder).

Answer 2

When it comes to equilibrium systems there is no universal definition of the concept of a phase. However, phase transitions are very well defined and can occur even within the same phase under some definitions (for example liquid and gas are often considered to be the same phase).

Phase transitions are characterized by the non analyticity of a thermodynamic potential which occurs at a critical point in a thermodynamic limit. Mathematically, as you approach those two limits some quantities in your system will diverge leading to critical phenomena. Of course, strong imprints of this behaviour survive in finite physical systems. How quantities behave when you approach this limit is characterized by critical exponents, finite size exponents and scaling functions. All these quantities characterize criticality and define universality classes. This is interesting because you can find the same kind of phase transitions in very different fields that have nothing to do with each other. It gives a structure to how macroscopic behaviour emerges from the underlying microscopic physics.

This is to put in contrast to crossovers where you have a transition between two phases without criticality or divergences.

So I would say that phase transitions imply critical phenomena. What is the nature of criticality depends on you transition. I don't know about the other way around though. Are there other kind of divergences not linked to phase transitions that could also be dubbed critical phenomena? I guess why not if there are mathematical similarities.

To complete this answer, quantum phase transitions are the same exact thing when your thermodynamic potential becomes the ground state energy of the system. A system in equilibrium in its ground state is by definition at zero temperature. Therefore conceptually, the phase change can be thought of as driven by quantum fluctuations instead of thermal fluctuations but one needs to be careful with this statement.

Answer 3

In thermodynamics, phase transition means the transition from one phase (solid, liquid, gas, or other phase) to another phase. Also in thermodynamics, critical point is the transition from where two separate phases exist to where only one phase exists. Beyond the critical point, only one phase exists.

Approaching the critical point, the two phases become ever more similar, at pressures or temperatures beyond the critical point, there is no longer a dividing line, no longer two phases but one. https://en.m.wikipedia.org/wiki/Critical_point_(thermodynamics)

Quantum phase transitions are completely different and unrelated. For example, solid and liquid are not quantum phases.