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In the context of critical phenomena, there are several critical exponents commonly used to characterize the singular behaviour at the point of phase transition. The Fisher exponent $\eta$ is defined through \begin{equation} C(T,x) = \frac{ \tilde{C} (x/\xi)}{|x|^{d-2+\eta}} \end{equation} where $C$ is order parameter correlation function with typical correlation length given by $\xi$, $T$ is the control parameter (eg temperature in Ising model), $x$ is spatial coordinate in $d$ dimensions, and $\tilde{C}$ a scaling function.

I have seen it is stated in multiple places that a non-zero value for $\eta$ implies the correlated regions in the system have fractal structures, but I have no clue what the basis for that statement could be. I'd appreciate it if someone can explain this, and also possibly a physical intuition into how the Fisher exponent appears in the correlation function.

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  • $\begingroup$ I wonder if by "fractal" they mean scale-invariant. $\endgroup$
    – Diego
    Commented May 5, 2021 at 21:18
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    $\begingroup$ Not sure if this counts as "physical" intuition, but the appearance of $\eta$ is due to the existence of a microscopic length scale that has been neglected. For a field with dimensions $[\phi^2] = L^{d-2}$ dimensional analysis says $C=\langle \phi(x)\phi(0) \rangle$ must scale like $\phi^2$ times a function of dimensionless ratios, e.g., $C = x^{-d+2} f(x/\xi,a/x)$, where $\xi$ is corr length and $a$ is lattice spacing. For $\xi \gg x \gg a$ we might hope we can set $x/\xi = a/x = 0$, so that $C \sim 1/x^{d-2}$, but in many cases $f(x/\xi,a/x) \sim (a/x)^\eta$ and so $C \sim 1/x^{d-2+\eta}$ $\endgroup$
    – bbrink
    Commented May 8, 2021 at 14:42
  • $\begingroup$ @bbrink thank you, your comment is very helpful. Does this only happen at a critical point, or are there more general circumstances where $\eta$ appears? $\endgroup$
    – SaMaSo
    Commented May 9, 2021 at 19:51
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    $\begingroup$ @SaMaSo This is known as 'incomplete similarity,' which occurs in other contexts not necessarily/typically thought of as critical points in an RG flow. e.g., intermediate asymptotics of differential equations. The scaling of the perimeter of a fractal follows incomplete similarity. See, e.g., the text 'Scaling' by Barenblatt. I don't know an explicit argument that incomplete similarity of correlation functions implies incomplete similarity and fractal boundaries of 'correlated regions,' however. $\endgroup$
    – bbrink
    Commented May 11, 2021 at 0:14

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