Exact results for a finite-sized spherical model of ferromagnetism at the borderline dimensionality 4

We report exact results on the zero-field susceptibility χ(T;L), the correlation length ξ(T;L), and the “singular” part of the specific heat ${\mathrm{c}}^{\mathrm{}\left(\mathrm{s}\right)\mathrm{}}$ (T;L) of a finite-sized spherical model of ferromagnetism subjected to periodic boundary conditions. We take the total dimensionality of the system d to be 4 and deal with the geometry ${\mathrm{L}}^{4\mathrm{-}\mathrm{d}\prime }$×${\mathrm{\infty }}^{\mathrm{d}\prime }$ (d′⩽2). In the region of first-order phase transition (T${\mathrm{T}}_{\mathrm{c}}$), our results are formally the same as in other cases with d>2. The “core” region (T≃${\mathrm{T}}_{\mathrm{c}}$), however, is characterized by the appearance of factors involving lnL, which appear only when d=4. The relationship between these results and the corresponding ones following from the hyperscaling regime as d→4- or from the mean-field regime as d→4+ is explored, and a formulation in terms of the finite-size scaling theory is presented.