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arXiv:cond-mat/0510834 · Journal of Physics A Mathematical and General

Generalized Mittag–Leffler functions in the theory of finite-size scaling for systems with strong anisotropy and/or long-range interaction

Hassan Chamati, N. S. Tonchev

Published Dec 21, 2005• 26 citations
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Abstract

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.

DOI
10.1088/0305-4470/39/3/001

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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.
We present analytical results for the finite-size scaling in ddd-dimensional O(N)O(N)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 rrr 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)O(N)O(N) systems in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, confined to a ddd-dimensional layer with geometry ${L}^{m}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\infty}}^{n};m+n=d$ and periodic boundary conditions across the finite mmm dimensions. The arising difficulties are avoided using a technique of calculations based on the analytical properties of the generalized Mittag-Leffler functions.
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.
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.
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.
Small systems consisting of particles interacting with long-range potentials exhibit enormous size effects. The Tsallis conjecture [Tsallis, Fractals 3, 541 (1995)FRACEG0218-348X10.1142/S0218348X95000473], valid for translationally invariant systems with long-range interactions, states a well-known scaling relating different sizes. Here we propose to generalize this conjecture to systems with this symmetry broken, by adjusting one parameter that determines an effective distance to compute the strength of the interaction. We apply this proposal to the one-dimensional Ising model with ferromagnetic interactions that decay as 1/r^{1+σ} in the region where the model has a finite critical temperature. We demonstrate the convenience of using this generalization to study finite-size effects, and we compare this approach with the finite-size scaling theory.