The finite-size critical properties of the O(n) vector ϕ 4 model, with longrange interaction decaying algebraically with the interparticle distance r like r −d−σ, are investigated. The system is confined to a finite geometry subject to periodic boundary condition. Special attention is paid to the finite-size correction to the bulk susceptibility above the critical temperature Tc. We show that this correction has a power-law nature in the case of pure long-range interaction i.e. 0 < σ < 2 and it turns out to be exponential in case of shortrange interaction i.e. σ = 2. The results are valid for arbitrary dimension d, between the lower (d< = σ) and the upper (d> = 2σ) critical dimensions.
There are problems with defining the thermodynamic limit of systems with long-range interactions; as a result, the thermodynamic behavior of these types of systems is anomalous. In the present work, we review some concepts from both extensive and nonextensive thermodynamic perspectives. We use a model, whose Hamiltonian takes into account spins ferromagnetically coupled in a chain via a power law that decays at large interparticle distance
r as $1∕{r}^{\ensuremath{\alpha}}$ for $\ensuremath{\alpha}\ensuremath{\geqslant}0$. Here, we review old nonextensive scaling. In addition, we propose a Hamiltonian scaled by $2\frac{{(N∕2)}^{1\ensuremath{-}\ensuremath{\alpha}}\ensuremath{-}1}{1\ensuremath{-}\ensuremath{\alpha}}$ that explicitly includes symmetry of the lattice and dependence on the size
N of the system. The approach enabled us to improve upon previous results. A numerical test is conducted through Monte Carlo simulations. In the model, periodic boundary conditions are adopted to eliminate surface effects.
We present analytical results for the finite-size scaling in
d-dimensional
O(N) systems with strong anisotropy where the critical exponents (e.g., ${\ensuremath{\nu}}_{\ensuremath{\Vert}}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$) depend on the direction. Prominent examples are systems with long-range interactions, decaying with the interparticle distance
r as ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ with different exponents $\ensuremath{\sigma}$ in corresponding spatial directions, systems with space-``time'' anisotropy near a quantum critical point, and systems with Lifshitz points. The anisotropic properties involve also the geometry of the systems. We consider
O(N) systems in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, confined to a
d-dimensional layer with geometry ${L}^{m}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\infty}}^{n};m+n=d$ and periodic boundary conditions across the finite
m dimensions. The arising difficulties are avoided using a technique of calculations based on the analytical properties of the generalized Mittag-Leffler functions.
The present review is devoted to the problems of finite-size scaling due to the presence of long-range interaction decaying at large distance as 1/rd+σ, σ>0. The attention is focused mainly on the renormalization group results in the framework of [Formula: see text]-theory for systems with fully finite (block) geometry under periodic boundary conditions. Some bulk critical properties and Monte Carlo results are also reviewed. The role of the cut-off effects as well as their relation with those originating from the long-range interaction is additionally discussed. Special attention is paid to the description of an adequate mathematical technique that allows for the treatment of the long-range and short-range interactions equally. The review concludes with a short discussion of some open problems.
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 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.