Critical Behavior of Systems With Long-Range Interaction  in Restricted Geometry

Chamati, H.; Tonchev, N. S.

doi:10.1142/s0217984903006232

Cited by 12 publications

(13 citation statements)

References 59 publications

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“…t ≃ 0. The function F , the reader is invited to consult references [1,3,5,15,18], where the finite-size scaling predictions are investigated in great details. Let us not that at this level, the anisotropy of the scaling behaviour in Eq.…”

Section: Finite-size Computationsmentioning

confidence: 99%

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

Chamati¹,

Tonchev²

2005

J. Phys. A: Math. Gen.

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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.

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Section: Finite-size Computationsmentioning

confidence: 99%

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

Chamati¹,

Tonchev²

2005

J. Phys. A: Math. Gen.

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“…In the renormalization group (RG) theory of critical phenomena, the characteristic spatial length of the system can be introduced as a new relevant parameter, and so, based on this theory, the finite-size effects can be used to determine both the critical parameters and critical exponents [10][11][12]. The general RG ideas have been applied in the context of confined systems with LR interactions in several works [13][14][15][16][17][18]. With this theory, the knowledge of the scaling function, the critical exponents, and critical parameters allows understanding and predicting the behavior of a finite system but only in the proximity of the critical temperature.…”

Section: Introductionmentioning

confidence: 99%

Size effects in finite systems with long-range interactions

Loscar

Horowitz

2018

Phys. Rev. E

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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.

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“…In contrast to the theory of finite-size scaling in isotropic systems (see e.g. [3,4]) and weakly anisotropic systems (see e.g [5] and refs. therein) the theory of finite -size scaling in strongly anisotropic systems (see [6,7,8,9,10,11,12,13] ) is still a field where the lack of results obtained in the framework of simplified and analytically tractable models are noticeable.…”

Section: Introductionmentioning

confidence: 99%

Finite-size scaling in anisotropic systems

Tonchev

2007

Phys. Rev. E

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We present analytical results for the finite-size scaling in d-dimensional O(N) systems with strong anisotropy where the critical exponents (e.g., nu{ ||} and nu{ perpendicular}) depend on the direction. Prominent examples are systems with long-range interactions, decaying with the interparticle distance r as r{-d-sigma} with different exponents 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-->infinity limit, confined to a d-dimensional layer with geometry L{m} X infinity {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.

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Critical Behavior of Systems With Long-Range Interaction in Restricted Geometry

Cited by 12 publications

References 59 publications

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

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

Size effects in finite systems with long-range interactions

Finite-size scaling in anisotropic systems

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