Finite-size scaling for the effective correlation length and the pair correlation function of the mean spherical model is studied in the case of general geometry Ld−d′×∞d′, periodic boundary conditions, and long-range interactions decaying like R−d−σ at large distances R, with arbitrary real parameters 0<σ≤2, σ≤d≤2σ, d′≥0. The analytical technique used is based on integral transformations with kernels of Mittag–Leffler type. It makes it possible to easily generalize a number of results available only for short-range interactions or in special cases of the Ld−d′×∞d′ geometry. The effective correlation length is identified at arbitrary temperatures from the finite-size large-distance asymptotic behavior of the pair correlation function. It is explicitly shown that the finite-size scaling functions are not singular in the ε-expansion when ε→0+ neither at d=σ+ε, nor at d=2σ−ε. Moreover, the applicability of the ε-expansion is tested in the specific finite-size case of d′=σ±ε. The definition of the scaled field variable and the finite-size scaling in the regime of the first-order phase transition are also considered.
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.
The difficulties arising in the investigation of finite-size scaling in d -dimensional O ( n ) systems with strong anisotropy and/or long-range interaction, decaying with the interparticle distance r as r − d −σ (0 < σ ⩽ 2), are discussed. Some integral representations aiming at the simplification of the investigations are presented for the classical and quantum lattice sums that take place in the theory. Special attention is paid to a more general form allowing to treat both cases on an equal footing and in addition cases with strong anisotropic interactions and different geometries. The analysis is simplified further by expressing this general form in terms of a generalization of the Mittag–Leffler special functions. This turned out to be very useful for the extraction of asymptotic finite-size behaviours of the thermodynamic functions.
A scaling hypothesis is set up for the correlation function G(R,T;L) of a finite-sized system, with O(n) symmetry (n\ensuremath{\ge}2), confined to geometry ${L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\infty}}^{d\mathcal{'}}$ (2d4, d'\ensuremath{\le}2) and subjected to periodic boundary conditions. Finite-size effects in the correlation length \ensuremath{\xi}(T;L) are predicted in the region of first-order phase transition (T
Tc) as well as the region of second-order phase transition (T\ensuremath{\simeq}
Tc); for the correlation function, the predictions relate mostly to the regime T
Tc. To test these predictions, a detailed analytical study is carried out in the case of the spherical model and all predictions are seen to be fully borne out. In the process we find that, though a single correlation length \ensuremath{\xi} governs the scaling of the variable R in all directions, the functional dependence of the correlation function on the scaled parameter R/\ensuremath{\xi} is highly anisotropic, in that it depends very differently on the components, ${\mathrm{R}}_{\ensuremath{\perp}}$ and
R?, of R pertaining to the finite and the infinite dimensions, respectively. Most importantly, while long-range order at temperatures below
Tc prevails all the way in the direction of ${\mathrm{R}}_{\ensuremath{\perp}}$ it is severely limited in the direction of
R?. In fact, for
R?\ensuremath{\gg}L, the qualitative features of the correlation function become characteristic of a d'-dimensional bulk system, though \ensuremath{\xi} still pertains to the actual, finite-sized system; the net result is that long-range order in the direction of
R? persists only to distances small in comparison with \ensuremath{\xi} which, for d'2, is found to be O(L(L/a${)}^{(d\mathrm{\ensuremath{-}}2)/(2\mathrm{\ensuremath{-}}d\mathcal{'})}$), a being the lattice constant.
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.
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 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.