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arXiv:cond-mat/0512146 · Physical Review E

Finite-size scaling in anisotropic systems

N. S. Tonchev

Published Mar 19, 2007• 10 citations
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Abstract

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.

DOI
10.1103/physreve.75.031110

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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., \ensuremathν\ensuremath∥{\ensuremath{\nu}}_{\ensuremath{\Vert}}\ensuremathν\ensuremath∥​ and \ensuremathν\ensuremath⊥{\ensuremath{\nu}}_{\ensuremath{\perp}}\ensuremathν\ensuremath⊥​) depend on the direction. Prominent examples are systems with long-range interactions, decaying with the interparticle distance rrr as r\ensuremath−d\ensuremath−\ensuremathσ{r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}r\ensuremath−d\ensuremath−\ensuremathσ with different exponents \ensuremathσ\ensuremath{\sigma}\ensuremathσ 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→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞ limit, confined to a ddd-dimensional layer with geometry Lm\ifmmode×\else\texttimes\fi\ensuremath∞n;m+n=d{L}^{m}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\infty}}^{n};m+n=dLm\ifmmode×\else\texttimes\fi\ensuremath∞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.
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.
The finite-size scaling functions of thermodynamic functions in anisotropic systems have been shown to be dependent on the spatial anisotropy [X.S. Chen and V. Dohm, Phys. Rev. E 70, 056136 (2004)]. Here we extend this study to the correlation length ξ‖ of the anisotropic O (n) symmetric φ4 model in an Ld−1 × ∞ cylindric geometry with periodic boundary conditions. We calculate the exact finite-size scaling function of correlation length ξ‖ for T ≥ Tc in 2 < d < 4 dimensions and in the limit n → ∞. The finite-size scaling function of ξ‖ is dependent on a normalized symmetric (d − 1) × (d − 1) matrix defined by the anisotropy matrix of anisotropic systems.
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 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 (TTc{T}_{c}Tc​) as well as the region of second-order phase transition (T\ensuremath{\simeq}Tc{T}_{c}Tc​); for the correlation function, the predictions relate mostly to the regime TTc{T}_{c}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?{\mathrm{R}}_{?}R?​, of R pertaining to the finite and the infinite dimensions, respectively. Most importantly, while long-range order at temperatures below Tc{T}_{c}Tc​ prevails all the way in the direction of ${\mathrm{R}}_{\ensuremath{\perp}}$ it is severely limited in the direction of R?{\mathrm{R}}_{?}R?​. In fact, for R?{R}_{?}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?{\mathrm{R}}_{?}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.
In the context of a phase transition, the critical singularity is characterized by a set of numbers, the critical exponents. It is widely believed that for transitions in systems with pair interactions on Bravais lattices (as opposed to hierarchical lattices), the critical exponents are determined solely by parameters d and n, representing the dimensionality of space and of the order parameter, respectively, and by \ensuremath{\sigma}, the interaction range. In this paper we consider a certain generalization of the Lifshitz singularity in the n-vector model in which a new parameter \ensuremath{\lambda}, describing a type of anisotropy of the interactions, is introduced. This is described by a model Hamiltonian in k space in which the leading term is of the form [${\mathrm{\ensuremath{\Sigma}}}_{\mathit{i}=1}^{\mathit{d}}$(ki2{\mathit{k}}_{\mathit{i}}^{2}ki2​${)}^{\mathrm{\ensuremath{\sigma}}/(2\ensuremath{\lambda})}$]\ensuremath{\lambda}. It should be noted that, as for the usual Lifshitz case, what would otherwise be the dominant k2{\mathit{k}}^{2}k2 term is supposed to have already been cancelled. For this model the parameters d, n, and \ensuremath{\sigma} are not sufficient to determine the value of the critical exponents to O(1/n), although they are sufficient for the first order in the \ensuremath{\epsilon} expansion.Calculations exact to O(1/n) show that, for a given set d, n, and \ensuremath{\sigma}, the exponents vary continuously with the parameter \ensuremath{\lambda} and include as a special case (for \ensuremath{\lambda}=1) the critical exponents for an anisotropic Lifshitz point. The critical exponents for this Lifshitz point differ markedly in value from those for the usually studied isotropic Lifshitz point, which is also a special case (for \ensuremath{\lambda}=\ensuremath{\sigma}/2) of our model. For \ensuremath{\sigma}=2, however, the two cases corresponding to \ensuremath{\lambda}=1 and \ensuremath{\sigma}/2 become degenerate. We provide another explicit example of the dependence of the critical exponents on \ensuremath{\lambda} by performing, in addition to the 1/n expansion, an expansion in \ensuremath{\sigma}-2. This helps to shed some light on the approach to the degeneracy at \ensuremath{\sigma}=2. The analysis of the Feynman integrals arising in the O(1/n) calculation provides interesting examples of the applications of hypergeometric functions; these integrals have themselves led to new results (published separately) in the theory of hypergeometric functions.
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.