Finite-size scaling in systems with long-range interaction

The finite-size critical properties of the $\mathcal{O}(n)$ vector ${\varphi }^4$ model, with long-range interaction decaying algebraically with the interparticle distance $r$ like $r^{-d-\sigma }$, are investigated. The system is confined to a finite geometry subject to periodic boundary condition. Special attention is paid to the finite-size correction to the bulk susceptibility above the critical temperature $T_c$. We show that this correction has a power-law nature in the case of pure long-range interaction i.e. $0<\sigma <2$ and it turns out to be exponential in case of short-range interaction i.e. $\sigma =2$. The results are valid for arbitrary dimension $d$, between the lower ($d_{<}=\sigma$) and the upper ($d_{>}=2\sigma$) critical dimensions.