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Dimensional correspondences have a long history in critical phenomena. Here, we review the effective dimension approach, which relates the scaling exponents of a critical system in d spatial dimensions with power-law decaying interactions r^d+ to a local system, i. e. , with finite range interactions, in an effective fractal dimension dₑff. This method simplifies the study of long-range models by leveraging known results from their local counterparts. While the validity of this approximation beyond the mean-field level has been long debated, we demonstrate that the effective dimension approach, while approximate for non-Gaussian fixed points, accurately estimates the critical exponents of long-range models with an accuracy typically larger than 97\%. To do so, we review perturbative RG results, extend the approximation's validity using functional RG techniques, and compare our findings with precise numerical data from conformal bootstrap for the two-dimensional Ising model with long-range interactions.
Solfanelli et al. (Thu,) studied this question.