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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?

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I answered a very similar question here, but in the context of quantum field theory rather than statistical field theory. The point is that it is impossible to have a nontrivial fixed point classically (i.e. without accounting for quantum/thermal fluctuations) for exactly the reason you stated: the dimensionful coefficients will define scales.

We already know that quantum/thermal fluctuations can break scale invariance, e.g. through the phenomenon of dimensional transmutation, where a quantum theory acquires a mass scale which wasn't present classically. And what's going on here is just the same process in reverse: at a nontrivial critical point the classical scale-dependence of the dimensionful coefficients is exactly canceled by quantum/thermal effects. Of course this cancellation is very special, which is why critical points are rare.

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  • $\begingroup$ I'm not sure I follow your answer. Are you saying that the statistical field theories are not scale-invariant? @knzhou $\endgroup$ – SRS Mar 28 '18 at 9:44
  • $\begingroup$ @SRS They are scale-invariant at critical points. At such points the apparent scale-dependence of a polynomial is precisely canceled by thermal effects (i.e. loop diagrams) when you perform RG. $\endgroup$ – knzhou Mar 28 '18 at 9:45
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My background is experimental, which may not be relevant, but I wonder if the following helps... Below a boiling point some liquid at the surface will vapourize, but at the boiling point vapour, or perhaps gas, can be formed throughout the liquid. -- This is how I interpret the scale invariant nature of changes at phase transition points.

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.

I realize this may all be just blindingly obvious and/or irrelevant, but hope it may be helpful and perhaps part of the answer you are looking for.

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