We report explicit results for the zero-field correlation length, xi (T;L), of a spherical-model ferromagnet confined to geometry L d-d' * infinity d' (d>2, d'<or=2) and subjected to 'twisted' boundary conditions. In the region of first-order phase transition (T<T c ), xi under 'twisted' boundary conditions obeys the same scaling law as under periodic boundary conditions, though with a different amplitude. In the 'core' region (T approximately=T c ), the scaling behaviour changes radically as one moves from one set of boundary conditions to the other-affected greatly by the 'pinning' of the ground-state wavevector k 0 of the system.
Finite-size effects in the correlation function G(R,T;L) of a spherical-model ferromagnet, confined to geometry ${\mathrm{L}}^{\mathrm{d}\mathrm{\ensuremath{-}}\mathrm{d}\ensuremath{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathrm{d}\ensuremath{'}}$ (2d4,d\ensuremath{'}\ensuremath{\leqslant}2) and subjected to twisted boundary conditions, are analyzed. Focusing our attention on the region of first-order phase transition (T
Tc), we examine the influence of the twist parameter \ensuremath{\tau} on the function G(R,T;L) in different regimes of the distance parameter \ensuremath{\varepsilon} (=R/L). We find that the functional dependence of the function G on the parameter \ensuremath{\varepsilon} is highly anisotropic, in that it depends very differently on the components, ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\perp}}}$ and ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\parallel}}}$, of \ensuremath{\varepsilon} pertaining to the finite and the infinite directions, respectively. While the long-range order in the direction of ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\perp}}}$ is significantly altered by the parameter \ensuremath{\tau}, it is curtailed very seriously in the direction of ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\parallel}}}$. In fact, for ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\parallel}}}$\ensuremath{\gg}1, the qualitative features of the correlation function acquire characteristics of a d\ensuremath{'}-dimensional bulk system, though the correlation length \ensuremath{\xi} appearing therein still pertains to the actual, finite-sized system. The net result is that long-range order in the direction of ${\mathbf{R}}_{\mathrm{\ensuremath{\parallel}}}$ persists only to distances small in comparison with \ensuremath{\xi} which, for d\ensuremath{'}2, is known to be O{L(L/a${)}^{(\mathrm{d}\mathrm{\ensuremath{-}}2)/(2\mathrm{\ensuremath{-}}\mathrm{d}\ensuremath{'})}$}, a being the lattice constant.
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
c(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
Tc), our results are formally the same as in other cases with d>2. The ``core'' region (T\ensuremath{\simeq}
Tc), 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.
Finite-size effects in the correlation function G(R,T;L)\ensuremath{\equiv}〈\ensuremath{\Phi}(0)\ensuremath{\cdot}\ensuremath{\Phi}(R)〉 of a spherical-model ferromagnet, confined to geometry ${\mathit{L}}^{\mathit{d}\mathrm{\ensuremath{-}}\mathit{d}\ensuremath{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathit{d}\ensuremath{'}}$ (d\ensuremath{\ge}4, ${\mathit{d}}^{\ensuremath{'}}$\ensuremath{\le}2) and subjected to twisted boundary conditions, are analyzed. Focusing attention on the region of first-order phase transition (T
Tc), we examine the influence of the twist parameter \ensuremath{\tau} on the function G(R,T;L) in different regimes of the distance parameter \ensuremath{\varepsilon}(=R/L). Two complementary methodologies are employed: (i) dimensional regularization involving the limit d\ensuremath{\rightarrow}4 as approached from the mean-field regime at short separations \ensuremath{\varepsilon}\ensuremath{\ll}1, and (ii) \ensuremath{\zeta}-function regularization at comparably larger separations \ensuremath{\varepsilon}=O(1) using generators from more simplified classes of sums so as to properly handle the singular features of the correlation function at d=4. Results following from the two methodologies are found to be completely consistent with one another.
We propose a finite-size scaling hypothesis for the ``singular'' part of the free-energy density,
f(s)(T,H;L), and the correlation function, G(R,T,H;L), of a system with O(n) symmetry (n\ensuremath{\ge}2), confined to geometry ${L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\infty}}^{d\mathcal{'}}$ (d>4, d'\ensuremath{\le}2) and subjected to periodic boundary conditions. On the basis of this hypothesis, we predict finite-size effects in the zero-field susceptibility, \ensuremath{\chi}(T;L), the correlation length, \ensuremath{\xi}(T;L), and the ``singular'' part of the specific heat,
c(s)(T;L), in the region of first-order phase transition (T
Tc) as well as the region of second-order phase transition (T\ensuremath{\simeq}
Tc). It turns out that in both these regions the system in zero field can be described by a single, unifying variable v\ensuremath{\sim}${L}^{2(d\mathrm{\ensuremath{-}}d\mathcal{'})/(4\mathrm{\ensuremath{-}}d\mathcal{'})}$t\ifmmode \tilde{}\else \~{}\fi{}, where t\ifmmode \tilde{}\else \~{}\fi{} is an appropriate temperature parameter of the system. To test the various predictions, a detailed analytical study is carried out in the case of the spherical model of ferromagnetism (n=\ensuremath{\infty}), and all predictions are seen to be fully borne out. For general n, we compare our findings with those of the previous authors, wherever possible, and in each case the comparison turns out to be perfect.
A scaling hypothesis is set up for the magnetization m(T,H;L), susceptibility \ensuremath{\chi}(T,H;L), and correlation length \ensuremath{\xi}(T,H;L) of a finite-sized system, with O(n) symmetry (n\ensuremath{\ge}2) and long-range interactions decaying as 1/${r}^{d+\ensuremath{\sigma}}$ (0<\ensuremath{\sigma}<2), confined to geometry ${L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\infty}}^{d\mathcal{'}}$ (\ensuremath{\sigma}<d<2\ensuremath{\sigma},d'\ensuremath{\le}\ensuremath{\sigma}) and subjected to periodic boundary conditions. Finite-size effects are predicted, in the region of a first-order phase transition (T
Tc) as well as in the region of a second-order phase transition (T\ensuremath{\simeq}
Tc), for different regimes of the parameters H and L. To test these predictions, a detailed analytical study is carried out in the case of the spherical model of ferromagnetism (n=\ensuremath{\infty}), and all predictions based on the scaling hypothesis are seen to be fully borne out. In situations where L\ensuremath{\gg}\ensuremath{\xi}, finite-size corrections to standard bulk values of the various physical quantities pertaining to these models are found to vary as (\ensuremath{\xi}/L${)}^{d+\ensuremath{\sigma}}$, rather than as ${e}^{\mathrm{\ensuremath{-}}L/\ensuremath{\xi}}$, which is characteristic of models with short-range interactions.
We study the correlation function G(R,T;L) and the correlation length \ensuremath{\xi}(T;L) in a finite spherical model if size ${L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\infty}}^{d\mathcal{'}}$ under periodic boundary conditions and emphasize the role of the variable L/\ensuremath{\xi}(L), rather than L/\ensuremath{\xi}(\ensuremath{\infty}), as a natural scaling variable of the system throughout the transition region, including temperatures below
Tc(\ensuremath{\infty}). We obtain a variety of finite-size effects, some of which may have validity for all O(n) models with n>2.