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Finite-size scaling in systems with long-range interaction

The finite-size critical properties of the ${\cal O}(n)$ vector $ϕ^4$ model, with long-range interaction decaying algebraically with the interparticle distance $r$ like $r^{-d-σ}$, 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. $0<σ<2$ and it turns out to be exponential in case of short-range interaction i.e. $σ=2$. The results are valid for arbitrary dimension $d$, between the lower ($d_<=σ$) and the upper ($d_>=2σ$) critical dimensions.