Power law behaviour at phase transition point & presence of fluctuations of all length scales

Question

In physics, exponentials such as $\exp(-r/\xi) $ typically come with a natural length scale $\xi$ while power laws such as $\sim 1/r^n$ don't have a definite length scale (at least not readily identifiable).

At the phase transition point, (i) the correlation functions exhibit power-law behavior, and (ii) fluctuations of all length scales are said to be present.

Question But if power laws do not have a length scale, how are the facts (i) and (ii) be reconciled? Is having no definite length scale in a power-law is the same as having all length scales present?

Answer 1

You pretty much answered your own question. What you are perhaps looking for is for a reason why people say that all length scales are present. In a second order phase transition, the correlation length goes to infinity sending $e^{-r/\xi} \to 1$, and only the polynomial decaying piece $r^{-n}$ remains. This means that at this phase transition, if you choose two arbitrary points in your system, they will be correlated. Does not matter if these points are very close or very far apart. In particular, if you zoom in on your system you will not see any difference, just by looking at it, you cannot tell which zoom augmentation you selected. You thus conclude that you cannot tell apart small distances from large distances, i.e. all length scales are involved.

Answer 2

Is having no definite length scale in a power-law is the same as having all length scales present?

Yes. In this case “definite” means unique length scale. There are correlations at all length scales, so there’s no single length that you can pick out as the “most important” one. This is not a statement about the presence or absence of correlations at a given length scale considered by itself, but a statement about the correlation function taken as a whole.