# Ginzburg-Landau theory for first-order phase transitions?

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?

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