Critical finite-size-scaling amplitudes of a fully anisotropic three-dimensional Ising model

Yurishchev, M. A.

doi:10.1103/physreve.55.3915

Cited by 16 publications

(23 citation statements)

References 29 publications

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“…(4) or (6) was supposed to be valid for all systems in a given universality class [1,5] including anisotropic lattice systems provided that an appropriate rescaling of the lattice spacings (or length L) is performed [5]. This appears to be consistent with existing studies of finite-size effects in anisotropic systems where it was stated that isotropy can be restored asymptotically by an anisotropic scale transformation [11,13,16,18,19,20,21,23,30].…”

supporting

confidence: 72%

“…Such special cases with a diagonal matrix A are d = 2 or d = 3 spin models on sc cubic lattices with only NN couplings J x = J y [3,13,23] or J x = J y = J z [18], respectively.…”

Section: Nonuniversal Parameters (Rather Than D Parameters) Within Tmentioning

confidence: 99%

“…There have been several studies of this problem in the past [5,13,15,16,17,18]. We shall only briefly comment on the more complicated case of anisotropic confined systems with non-periodic boundary conditions [17,19,20,21,22,23].…”

mentioning

confidence: 99%

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Nonuniversal finite-size scaling in anisotropic systems

Chen

Dohm²

2004

Phys. Rev. E

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We study the bulk and finite-size critical behavior of the O(n) symmetric ϕ 4 theory with spatially anisotropic interactions of non-cubic symmetry in d < 4 dimensions. In such systems of a given (d, n) universality class, two-scale factor universality is absent in bulk correlation functions, and finite-size scaling functions including the Privman-Fisher scaling form of the free energy, the Binder cumulant ratio and the Casimir amplitude are shown to be nonuniversal. In particular it is shown that, for anisotropic confined systems, isotropy cannot be restored by an anisotropic scale transformation.PACS numbers: 05.70. Jk, A basic tenet in the physics of critical phenomena is the notion of a universality class. It is characterized by the dimensionality d of the system and by the number n of the components of the order parameter. (See, e.g., the review article [1].) Within a certain (d, n) universality class, the universal quantities (critical exponents, amplitude ratios and scaling functions) are independent of microscopic details, such as the particular type of (finiterange or van der Waals type) interactions or the lattice structure [2]. This implies that a given universality class includes both spatially isotropic and anisotropic systems.Once the universal quantities of a universality class are known the asymptotic critical behavior of very different systems (e.g., fluids and magnets) is believed to be known completely provided that only two nonuniversal amplitudes A 1 and A 2 are given. This property is known as two-scale factor universality or hyperuniversality [1,3,4]. In terms of the singular part of the reduced bulk free energy densitywith W (0) = 1 and t = (T − T c )/T c ≪ 1, this property can be stated asThus the amplitude ξ 0 = (Q/A 1 ) 1/d of the correlation length ξ = ξ 0 t −ν at zero ordering field h is not an independent amplitude but is universally related to A 1 . The validity of two-scale factor universality has been established by the renormalization-group (RG) theory on the basis of an isotropic Hamiltonian with short-range interactions below the upper critical dimension d * = 4 [4] but no general proof has been given for the anisotropic case.In this paper we study the critical behavior of systems with a spatial anisotropy of non-cubic symmetry within a given (d, n) universality class. An example is an Ising ferromagnet with an isotropic nearest-neighbor (NN) coupling J > 0 and an anisotropic next-nearestneighbor (ANNN) coupling J ′ on a simple-cubic lattice. In some range of J ′ /J this model has the same type of critical behavior as the ordinary (J ′ = 0) Ising model. We shall show that for such systems Eq. (2) must be generalized towhere 0 , A 2 whose ratios are also nonuniversal. Note that there still exists a unique critical exponent ν(d, n) that is identical for isotropic and anisotropic systems within the same (d, n) universality class [6,7,8,9,10].A different type of critical behavior exists in the socalled strongly anisotropic systems [11,12,13,14] where not only amplitudes depend on t...

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supporting

confidence: 72%

“…Such special cases with a diagonal matrix A are d = 2 or d = 3 spin models on sc cubic lattices with only NN couplings J x = J y [3,13,23] or J x = J y = J z [18], respectively.…”

Section: Nonuniversal Parameters (Rather Than D Parameters) Within Tmentioning

confidence: 99%

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Nonuniversal finite-size scaling in anisotropic systems

Chen

Dohm²

2004

Phys. Rev. E

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“…Using formula (8) we then obtain expressions for the free energy densities f Li (i = 1, 2, 3) contained in (17). Subsequent calculations using these formulas show that the curves F (K) do not intersect the abscissa and thus the equation F (K) = 0 has no real solutions (see Fig.…”

Section: Ising Modelmentioning

confidence: 99%

“…Formulas for the derivatives of the inverse correlation length and the free energy with respect to h, expressed in terms of the eigenvalues and eigenvectors of the transfer matrices, suitable for programming are given in [16,17].…”

Section: Equations For the Phenomenological Renormalization Groupmentioning

confidence: 99%

Improved phenomenological renormalization schemes

Yurishchev

2000

J. Exp. Theor. Phys.

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An analysis is made of various methods of phenomenological renormalization based on finite-size scaling equations for inverse correlation lengths, the singular part of the free energy density, and their derivatives. The analysis is made using two-dimensional Ising and Potts lattices and the three-dimensional Ising model. Variants of equations for the phenomenological renormalization group are obtained which ensure more rapid convergence than the conventionally used Nightingale phenomenological renormalization scheme. An estimate is obtained for the critical finite-size scaling amplitude of the internal energy in the threedimensional Ising model. It is shown that the two-dimensional Ising and Potts models contain no finite-size corrections to the internal energy so that the positions of the critical points for these models can be determined exactly from solutions for strips of finite width. It is also found that for the two-dimensional Ising model the scaling finite-size equation for the derivative of the inverse correlation length with respect to temperature gives the exact value of the thermal critical exponent.

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Size-driven dimensional crossover in the quasi-two-dimensional Ising model

Lee

2004

Solid State Communications

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Critical finite-size-scaling amplitudes of a fully anisotropic three-dimensional Ising model

Cited by 16 publications

References 29 publications

Nonuniversal finite-size scaling in anisotropic systems

Nonuniversal finite-size scaling in anisotropic systems

Improved phenomenological renormalization schemes

Size-driven dimensional crossover in the quasi-two-dimensional Ising model

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