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.

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Cited by in corpus (5)

• Universal work statistics in long-range interacting quantum systems
• Continuously varying critical exponents in an exactly solvable long-range cluster XY mode
• Slow decay rate of correlations induced by long-range extended Dzyaloshinskii-Moriya interactions
• One-dimensional long-range Ising model: two (almost) equivalent approximations
• Entropy production and statistical relaxation of dipolar bosons and fermions in interaction quench dynamics