Recursion Relations and Fixed Points for Ferromagnets with Long-Range Interactions

Recent calculations on ferromagnets with n-component spins and with long-range forces (the interaction between spin decays as 1r^d+σ, where d is the dimensionality of the system and σ>0) have given exponents which are apparently discontinuous at σ=2, i.e., when the transition to short-range interactions is made. By solving Wilson's exact recursion relations to order ε²(ε=2σ-d) we show that the exponents η, γ, and φ are continuous functions of σ and that there exists a region of long-range potentials defined by 2>σ>2-ηSR where the exponents assume their short-range values.