Universality in long-range interacting systems: the effective dimension approach
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.14651
- arXiv:
- arXiv:2406.14651
- Bibcode:
- 2024arXiv240614651S
- Keywords:
-
- Condensed Matter - Statistical Mechanics