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arXiv:cond-mat/0303030 · Modern Physics Letters B

CRITICAL BEHAVIOR OF SYSTEMS WITH LONG-RANGE INTERACTION IN RESTRICTED GEOMETRY

Hassan Chamati, N. S. Tonchev

Published Oct 10, 2003• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s0217984903006232

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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.
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.
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 detailed investigation of the scaling properties of the fully finite O(n)\mathcal{O}(n)O(n) systems, under periodic boundary conditions, with long-range interaction, decaying algebraically with the interparticle distance r like ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}},$ below their upper critical dimension, is presented. The computation of the scaling functions is done to one loop order in the nonzero modes. The results are obtained in an expansion of powers of $\sqrt{\ensuremath{\varepsilon}},$ where $\ensuremath{\varepsilon}=2\ensuremath{\sigma}\ensuremath{-}d$ up to $\mathcal{O}({\ensuremath{\varepsilon}}^{3/2}).$ The thermodynamic functions are found to depend upon the scaling variable ${z=RU}^{\ensuremath{-}1/2}{L}^{2\ensuremath{-}\ensuremath{\eta}\ensuremath{-}\ensuremath{\varepsilon}/2},$ where R and U are the coupling constants of the constructed effective theory, and L is the linear size of the system. Some simple universal results are obtained.
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 (TTc{T}_{c}Tc​) as well as in the region of a second-order phase transition (T\ensuremath{\simeq}Tc{T}_{c}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 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+σ1/r^{d+σ}1/rd+σ, where ddd is the spatial dimension and the long-range parameter σ>0σ>0σ>0. Classical and quantum systems are considered.
Abstract A number of numerical calculations, like Monte Carlo or transfer matrix calculations, are performed with systems in which the size in several or all dimensions is finite. To extrapolate the results to the infinite system, it is thus necessary to understand how the infinite size limit is reached. In particular in a system in which the forces are short range, no phase transition can occur in a finite volume, or in a geometry in which the size is infinite only in one dimension. This indicates that the infinite-size extrapolation is somewhat non-trivial. In this chapter, the problem is analysed in the case of second-order phase transitions, in the framework of the N-vector model. The existence of a finite-size scaling is established, extending renormalization group (RG) arguments to this new situation. Then, finite volume geometry and cylindrical geometry, in which the size is finite in all dimensions except one, are distinguished. It is explained how to adapt the methods used in the case of infinite systems to calculate the new universal quantities appearing in finite-size effects, for example, in d = 4−ϵ or d = 2+ϵ dimensions. Special properties of the commonly used periodic boundary conditions are emphasized.