Upper and Lower Critical Decay Exponents of Ising Ferromagnets with Long-range Interaction

TL;DR

This study uses advanced Monte Carlo simulations to precisely determine the critical decay exponents in 2D long-range Ising models, revealing boundaries between different universality classes and proposing improved analysis techniques.

Contribution

It introduces effective physical quantities to accurately identify critical decay exponents, resolving discrepancies in previous studies.

Findings

Lower critical decay exponent is σ=1.

Upper critical decay exponent is σ=7/4.

Proposed combined Binder ratios improve finite-size scaling analysis.

Abstract

We investigate the universality class of the finite-temperature phase transition of the two-dimensional Ising model with the algebraically decaying ferromagnetic long-range interaction, $J_{ij} = |\vec{r}_i -\vec{r}_j|^{-(d+\sigma)}$, where $d$ (=2) is the dimension of the system and $\sigma$ the decay exponent, by means of the order-$N$ cluster-algorithm Monte Carlo method. In particular, we focus on the upper and lower critical decay exponents, the boundaries between the mean-field-universality, intermediate, and short-range-universality regimes. At the critical decay exponents, it is found that the critical amplitude of the standard Binder ratio of magnetization exhibits the extremely slow convergence as a function of the system size. We propose more effective physical quantities, the combined Binder ratio and the self-combined Binder ratio, both of which cancel the leading finite-size corrections of the conventional Binder ratio. Utilizing these techniques, we clearly demonstrate that in two dimensions the lower and upper critical decay exponents are $\sigma = 1$ and 7/4, respectively, contrary to the recent Monte Carlo and the renormalization-group studies [M. Picco, arXiv:1207.1018; T. Blanchard, et al., Europhys. Lett. 101, 56003 (2013)].