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

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)].