Scaling behavior for finite O(n) systems with long-range interaction

Abstract

A detailed investigation of the scaling properties of the fully finite ${\cal O}(n)$ systems with long-range interaction, decaying algebraically with the interparticle distance $r$ like $r^{-d-σ}$, below their upper critical dimension is presented. The computation of the scaling functions is done to one loop order in the non-zero modes. The results are obtained in an expansion of powers of $\sqrtε$, where $ε=2σ-d$ up to ${\cal O}(ε^{3/2})$. The thermodynamic functions are found to be functions the scaling variable $z=RL^{2-η-ε/2}U^{-1/2}$, where $R$ and $U$ are the coupling constants of the constructed effective theory, and $L$ is the linear size of the system. Some simple universal results are obtained.