I just realized that anomalous dimensions in quantum/statistical field theory is not that different from fractal dimensions of objects. They both describe how quantitaive objects transform under a scale transformation (renormalization group transformation in the QFT case and dilation of the mesh/ruler when computing the perimeter of a fractal). Is there a more profound link between the two? I have't read much about the subject but it seems that for any statistical model at criticality the fractal dimension of the clusters becomes a function of the full dimension of the field. Is it a general rule?
This connection was much discussed in the early days after Ken Wilson's explanation that the continuum limit of a quantum field theory is mathematically the same thing as a critical point of a statistical mechanics system. In particular the typical field configuration in a continuum QFT path integral, or at the critical point in the thermal system, is self-similar under scale transformations and therefore has a fractal character and power-law correlation functions.
Likely not. Note that fractional dimension is already possible without any anomalous dimensions.
$$S=\int d\omega dk\, \psi^\dagger(\omega, k) [i\omega - \epsilon(k)] \psi(\omega, k) $$
It is easy to verify the scaling dimension of $\psi$ is $-3/2$, which is fractional. A field scaling with $-3/2$ power of the linear size is much less exotic than a shape doing so. The former does not have a direct correspondence with any fractal.