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.


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...