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

Year: 2024

Authors: Solfanelli A., Defenu N.

Autors Affiliation: SISSA, Via Bonomea 265, Trieste 34136, Italy; INFN, Sez Trieste, Via Valerio 2, I-34127 Trieste, Italy; CNR, INO, Area Sci Pk, I-34149 Trieste, Italy; Swiss Fed Inst Technol, Inst Theoret Phys, Wolfgang Pauli Str 27, Zurich, Switzerland.

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 rd+sigma 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 renormalization-group (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.

Journal/Review: PHYSICAL REVIEW E

Volume: 110 (4)      Pages from: 44121-1  to: 44121-6

More Information: We thank Connor Behan for sharing the conformal boot-strap numerical data for the two-dimensional Ising model with LR interactions. We acknowledge funding by the Swiss National Science Foundation (SNSF) under project funding ID: 200021 207537 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC2181/1-390900948 (the Heidelberg STRUCTURES Excellence Cluster) and by the European Union under GA No. 101077500-QLR-Net. This research was supported in part by grant NSF PHY-230935 to the Kavli Institute for Theoretical Physics (KITP) .
KeyWords: Renormalization-group; Critical Exponents; Models
DOI: 10.1103/PhysRevE.110.044121