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 $r^{ -d-\sigma}$, 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 $\sqrt\epsilon$, where $\epsilon=2\sigma-d$ up to ${ \cal O}(\epsilon^{ 3/2})$. The thermodynamic functions are found to be functions the scaling variable $z=RL^{ 2-\eta-\epsilon/2}U^{ -1/2}$, 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.
Categories: cond-mat.stat-mech