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

Abstract

We report exact results on the zero-field susceptibility \ensuremath{\chi}(T;L), the correlation length \ensuremath{\xi}(T;L), and the ``singular'' part of the specific heat ${\mathrm{c}}^{(\mathrm{s})}$ (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 ${\mathrm{L}}^{4\mathrm{\ensuremath{-}}\mathrm{d}\ensuremath{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathrm{d}\ensuremath{'}}$ (d\ensuremath{'}\ensuremath{\leqslant}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 dg2. The ``core'' region (T\ensuremath{\simeq}${\mathrm{T}}_{\mathrm{c}}$), 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\ensuremath{\rightarrow}4- or from the mean-field regime as d\ensuremath{\rightarrow}4+ is explored, and a formulation in terms of the finite-size scaling theory is presented.