Spherical Model with Long-Range Ferromagnetic Interactions

The behavior of the spherical model of a ferromagnet with an interaction energy between the magnetic spins which varies with distance as 1r^d+σ (where d is the dimensionality of the lattice and σ>0) is analyzed. It is shown that the model exhibits a ferromagnetic transition in one and two dimensions, providing 0<σ<d. (The usual spherical model with nearest-neighbor interactions does not have a transition in one and two dimensions.) The critical-point behavior is investigated. It is found that the singularities in the specific heat and susceptibility are dependent on σ and d, but the behavior of the magnetization is independent of σ and d. In three dimensions the susceptibility diverges as (T-Tc)^-γ, where γ=1 for 0<σ<3/2, γ=σ(3-σ) for 3/2<σ<2 and γ=2 for σ>2. The asymptotic form of the spin-spin correlation function Γ(r) is studied in the neighborhood of the critical temperature Tc. At T=Tc, Γ(r) decays for large r as 1r^d-σ. Several two-dimensional models with long-range interactions falling off as 1r² in certain directions only are also investigated.