[Physics] Is this Landau’s other critical phenomena mistake

statistical mechanicssymmetrythermodynamics

There was an old argument by Landau that while the liquid gas transition can have a critical point, the solid-liquid transition cannot. This argument says that the solid breaks translational symmetry, and it is impossible to do this in a second order transition.

But this argument is subtly false. Second order transitions break symmetries, which can be discrete, like in the Ising model, or continuous, like in the x-y model. The reason Landau said it is because it is hard to imagine breaking all the translational and rotational symmetries all at once to make a second order liquid-solid point.

But nowadays we know about nematics, and we can imagine the following chain of second-order transitions:

fluid (I)-> fluid with broken x-y-z rotational symmetry with a z-directional order (II)
-> fluid broken translational symmetry in the same direction
-> broken x-y direction rotational symmetry in the y-direction
-> broken y- direction translational symmetry
-> broken x-direction translational symmetry

Each of these transitions can be second order, and together, they can make a solid from a fluid. the question is, how badly does Landau's argument fail.

  • Are there any two phases which cannot be linked by a second order phase transition?
  • Are there always parameters (perhaps impossible to vary in a physical system) which will allow the second order points to be reached?
  • Is it possible to make the second order transitions collide by varying other parameters, to bring them to one critical point (in the example, a critical point between fluid and solid).
  • Do these critical points exist in any system?

Best Answer

Good point, I think, but when we look at your progression, from (ii) -> (iii), you break two unrelated symmetries (lattice rotation, then lattice translation). This is fine, and you would have two distinct second order transitions. But when you try to bring the two critical points together, you correctly notice that this is going to require fine tuning of the parameters. Now, that's not to say such points don't exist. I think they tend to go under the name "multicritical points," and I seem to recall work, possibly by Dagotto, arguing that CMR transitions were governed by some sort of bicriticality (not my expertise, unfortunately.)

The point though, and I think the point Landau is making, is that that sort of fine tuning is unlikely to be relevant for real phase transitions, with only a handful of knobs (That is, you write down the action density and it's something like $a \phi^2+b \psi^2+$interactions etc. but for reasons beyond our control, $a(g)$ and $b(g)$ must both go to zero at the same value of $g$... On the other hand, some folks believe there are "deconfined" quantum critical points, where no fine-tuning is necessary to drive two unrelated order parameters to zero at the same "tuning" parameter point.)

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