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In AlQuemist's answer to this PSE question:223892 and Thomas' recent answer to one of my questions. There is a mention of the application of the Ginzburg criterion and in general the Ginzburg-Landau theory restricted to second order-phase transitions. This appears to be a consistent theme throughout the literature.

I cannot see why this restriction is in place - i.e. why we don't consider the Ginzburg-Landau theory and Ginzburg criterion to hold in the case of first-order phase transitions.

As far as I can tell everything we do with these involves either taking a saddle point approximation or Gaussian approximation around the saddle point - either of which seem to not actually rely on the nature of the transition at all.

Thus my question is; Why is this restriction to second order phase transitions in place in the context of Ginzburg-Landau theory and the Ginzburg criterion?

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The usual Landau-Ginzburg potential can be slightly generalized to

$$W(\phi) = t \phi^2 + a \phi^4 + \phi^6.$$

The phase transition is at $t = 0$, and it is continuous if $a > 0$ and discontinuous if $a<0$. $a = 0, t = 0$ is a multicritical point.

This basic observation begins to explain why it's so hard in practice to distinguish between a continuous and discontinuous phase transition, since it just comes down to a sign of one phenomenological parameter.

On the other hand, if the phase transition is not continuous, and there is no diverging correlation length, it's not so easy to justify something like mean field theory (or even effective field theory), since lattice scale fluctuations could be very important.

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