Long range to short range crossover in one dimension

TL;DR

This paper studies the transition from long-range to short-range interactions in one-dimensional systems, using Monte Carlo simulations to analyze critical exponents and validate theoretical crossover scenarios.

Contribution

It provides numerical evidence supporting Sak's crossover scenario and clarifies the behavior of critical exponents across the LR-SR transition in 1D systems.

Findings

Crossover at  = 1 confirmed

Critical exponents are continuous across the crossover

Deviations from Flory scaling observed

Abstract

This work investigates the critical behavior of one-dimensional systems with long-range (LR) interactions, focusing on the crossover to short-range (SR) universality. Through large-scale Monte Carlo simulations of self-avoiding L\'evy flights on a 1D lattice, we compute the anomalous dimension \eta, the correlation length exponent \nu, and the susceptibility exponent \gamma across a wide range of LR decay parameters \sigma. Our results provide strong numerical evidence that supports Sak's scenario. They identify the crossover at \sigma^* = 1 and demonstrate the continuity of critical exponents across this point, with strong corrections to scaling. The study also reveals deviations from Flory-type scaling predictions and discusses the limitations of effective dimension approaches in general. These findings clarify the nature of the LR-SR crossover in low-dimensional systems and open avenues for exploring criticality in disordered and complex networks.