# Is the Landau free energy scale-invariant at the critical point?

My question is different but based on the same quote from Wikipedia as here. According to Wikipedia,

In statistical mechanics, scale invariance is a feature of phase transitions. The key observation is that near a phase transition or critical point, fluctuations occur at all length scales, and thus one should look for an explicitly scale-invariant theory to describe the phenomena. Such theories are scale-invariant statistical field theories, and are formally very similar to scale-invariant quantum field theories.

Question I understand that at the critical point the correlation length $\xi$ diverges and as a consequence, the correlation functions $\langle\phi(\textbf{x})\phi(\textbf{y})\rangle$ behave as a power law. Power laws are scale-invariant. But for a theory itself to be scale-invariant (as Wikipedia claims) the Landau Free energy functional should have a scale-invariant behaviour at the critical point. But the free energy functional is a polynomial in the order parameter and polynomials are not scale-invariant.

Then how is the claim that the relevant statistical field theory is scale-invariant justified?

Thus, I expect that the scale-invariance is to do with the phase change being able to occur throughout the gas/liquid or solid so that if you calculate 100 molecules or 10$^6$ molecules you should get the same phase transition at the same temperature.