Details are presented of an extension of the size-scaling hypothesis to systems in which each element interacts equally with all others (systems for which the mean-field approximation is valid in the thermodynamic limit). A simple argument, which relates the large-size critical behavior of physical quantities with the upper critical dimensionality of the corresponding short-range system, already presented in a Letter, is here made precise and checked either analytically or numerically on several examples. In particular, the scaling form for the magnetization is explicitly derived in the case of the infinitely coordinated Ising model, and a numerical study is presented of the infinitely coordinated
XY-Ising quantum model in a transverse field, with its extension in the presence of an imaginary longitudinal field (a model exhibiting a Yang-Lee edge singularity).
We investigate a two-dimensional Ising model with long-range interactions that emerge from a generalization of the magnetic dipolar interaction in spin systems with in-plane spin orientation. This interaction is, in general, anisotropic whereby in the present work we focus on the isotropic case for which the model is found to be at its upper critical dimensionality. To investigate the critical behavior the temperature and field dependence of several quantities are studied by means of Monte Carlo simulations. On the basis of the Privman-Fisher hypothesis and results of the renormalization group the numerical data are analyzed in the framework of a finite-size scaling analysis and compared to finite-size scaling functions derived from a Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The obtained excellent agreement suggests that at least in the present case the concept of universal finite-size scaling functions can be extended to the upper critical dimensionality.
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.
The
n-component Ginzburg-Landau-Wilson model for a semi-infinite system is solved exactly at
T=Tc in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$. In the scaling regime the spin-spin correlation function is $G(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\rho}}, z, {z}^{\ensuremath{'}}, {T}_{c})=\mathrm{const}{{{[{\ensuremath{\rho}}^{2}+{(z\ensuremath{-}{z}^{\ensuremath{'}})}^{2}]}^{\ensuremath{-}1}\ensuremath{-}{[{\ensuremath{\rho}}^{2}+{(z+{z}^{\ensuremath{'}})}^{2}]}^{\ensuremath{-}1}}}^{\frac{(d\ensuremath{-}2)}{2}}$, for dimensionalities
d in the range
2<d<4, where
z, ${z}^{\ensuremath{'}}$ are the distances of the two spins from the surface and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}$ is their separation parallel to the surface. The critical exponents ${\ensuremath{\eta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ are $\frac{(d\ensuremath{-}2)}{2}$ and ($d\ensuremath{-}2$), respectively.
Although the fully connected Ising model does not have a length scale, we show that the critical exponents for thermodynamic quantities such as the mean magnetization and the susceptibility can be obtained using finite size scaling with the scaling variable equal to N, the number of spins. Surprisingly, the mean value and the most probable value of the magnetization are found to scale differently with N at the critical temperature of the infinite system, and the magnetization probability distribution is not a Gaussian, even for large N. Similar results inconsistent with the usual understanding of mean-field theory are found at the spinodal. We relate these results to the breakdown of hyperscaling and show that hyperscaling can be restored by increasing N while holding the Ginzburg parameter rather than the temperature fixed, or by doing finite size scaling at the pseudocritical temperature where the susceptibility is a maximum for a given value of N. We conclude that finite size scaling for the fully connected Ising model yields different results depending on how the mean-field limit is approached.