Saturday, 23 June 2018

statistical mechanics - 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?




Answer



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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