Universality in long-range interacting systems: the effective dimension approach

TL;DR

This paper reviews the effective dimension approach for relating critical exponents of long-range interacting systems to local models in a fractal dimension, demonstrating high accuracy in estimating critical behavior.

Contribution

It provides a comprehensive review and validation of the effective dimension method, extending its applicability beyond mean-field theory using functional RG and numerical comparisons.

Findings

Effective dimension approach estimates critical exponents with over 97% accuracy.

The method simplifies studying long-range models by relating them to local models.

Validation against numerical data confirms the approach's reliability.

Abstract

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 $r^{d+\sigma}$ to a local system, i.e., with finite range interactions, in an effective fractal dimension $d_\mathrm{eff}$. 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.