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arXiv:cond-mat/9612219 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

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

M. A. Yurishchev

Published Apr 1, 1997• 16 citations

Abstract

A fully anisotropic simple-cubic Ising lattice in the geometry of periodic cylinders n\ifmmode\times\else\texttimes\fi{}n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} is investigated by the transfer-matrix finite-size-scaling method. In addition to the previously obtained critical amplitudes of the inverse correlation lengths and singular part of the free energy [M. A. Yurishchev, Phys. Rev. B 50, 13 533 (1994)], the amplitudes of the usual (``linear'') and nonlinear susceptibilities and the amplitude of the second derivative of the spin-spin inverse correlation length with respect to the external field are calculated. The behavior of critical amplitude combinations (which, in accordance with the Privman-Fisher equations, do not contain in their composition the nonuniversal metric coefficients and geometry prefactor) are studied as a function of the interaction anisotropy parameters. A universality domain for the amplitude ratios is found in the quasi-one-dimensional regime of interactions in the system. In the case of a fully isotropic three-dimensional Ising model for which the high precision values of the critical coupling and critical-point free energy are available, improved estimates are obtained for the following four universal quantities: (1) the amplitude of spin-spin inverse correlation length, (2) the amplitude of singular part of the free energy, (3) the ratio of the amplitude of a second derivative of the spin-spin inverse correlation length with respect to the external field to the usual susceptibility amplitude, and (4) the ratio of the nonlinear susceptibility amplitude to the square of the linear susceptibility amplitude (i.e., for the finite-size counterpart of the four-point renormalized coupling constant).

DOI: 10.1103/physreve.55.3915

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Using transfer-matrix extended phenomenological renormalization-group methods the critical properties of spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths J = (J ′,J ′,J) are studied. Universality of both fundamental critical exponents yt and yh is confirmed. It is shown that the critical finite-size scaling amplitude ratios U = A χ (4)Aκ/A 2 χ, Y1 = A κ ′′/Aχ, and Y2 = A κ (4)/A χ (4) are independent of the lattice anisotropy parameter ∆ = J ′ /J. By this for the last above invariant of the threedimensional Ising universality class we give the first quantitative estimate: Y2 ≃ 2.013 (shape L × L × ∞, periodic boundary conditions in both transverse directions). PACS numbers: 05.50.+q, 05.70.Jk, 64.60.Fr, 75.10.Hk 1 1

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A fully anisotropic simple-cubic Ising lattice in the geometry of periodic cylinders n\ifmmode\times\else\texttimes\fi{}n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} is investigated by the transfer-matrix finite-size-scaling method. In addition to the previously obtained critical amplitudes of the inverse correlation lengths and singular part of the free energy [M. A. Yurishchev, Phys. Rev. B 50, 13 533 (1994)], the amplitudes of the usual (``linear'') and nonlinear susceptibilities and the amplitude of the second derivative of the spin-spin inverse correlation length with respect to the external field are calculated. The behavior of critical amplitude combinations (which, in accordance with the Privman-Fisher equations, do not contain in their composition the nonuniversal metric coefficients and geometry prefactor) are studied as a function of the interaction anisotropy parameters. A universality domain for the amplitude ratios is found in the quasi-one-dimensional regime of interactions in the system. In the case of a fully isotropic three-dimensional Ising model for which the high precision values of the critical coupling and critical-point free energy are available, improved estimates are obtained for the following four universal quantities: (1) the amplitude of spin-spin inverse correlation length, (2) the amplitude of singular part of the free energy, (3) the ratio of the amplitude of a second derivative of the spin-spin inverse correlation length with respect to the external field to the usual susceptibility amplitude, and (4) the ratio of the nonlinear susceptibility amplitude to the square of the linear susceptibility amplitude (i.e., for the finite-size counterpart of the four-point renormalized coupling constant).

For the fully anisotropic simple-cubic Ising lattice, the critical finite-size scaling amplitudes of both the spin-spin and energy-energy inverse correlation lengths and the singular part of the reduced free-energy density are calculated by the transfer-matrix method and a finite-size scaling for cyclic L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} clusters with L=3 and 4. Analysis of the data obtained shows that the ratios and the directional geometric means of the above amplitudes are universal.

Using transfer-matrix extended phenomenological renormalization-group methods [M.A.Yurishchev, Nucl. Phys. B (Proc. Suppl.) 83-84, 727 (2000); hep-lat/9908019; J. Exp. Theor. Phys. 91, 332 (2000); cond-mat/0108002] the improved estimates for the critical temperature of spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths

{\vec J}=(J',J',J)

are obtained. Universality of both fundamental critical exponents

y_t

and

y_h

is confirmed. We show also that the critical finite-size scaling amplitude ratios

A_{χ^{(4)}}A_κ/A_χ^2

,

A_{κ^{''}}/A_χ

, and

A_{κ^{(4)}}/A_{χ^{(4)}}

are independent of the lattice anisotropy parameter

Δ=J'/J

.

The coordinates of the critical points of spin-S Ising models with coupling constants J and J′ are calculated for 1/2 ≤ S ≤ 13/2. The calculations are performed for several values of S and Δ ≡ J′/J independently by using the phenomenological renormalization-group method or (approximate) self-duality. Numerical results combined with a mean-field analysis show that the critical coupling strength for Δ ∼ 1 (weakly anisotropic lattice) is K c () (Δ) = K c () (1)[1 + a(1 − Δ)], where a = (d − 1)/d is independent of S (d is the space dimension). Both free energy and internal energy are determined at the critical points. An extremum of the critical internal energy is found at Δ* ∈ (0, 1). The parameter Δ* can be used as a criterion that separates quasi-isotropic and quasi-one-dimensional regimes (Δ* < Δ ≤ 1 and Δ < Δ*, respectively). The finite-size scaling amplitudes A s and A e of the inverse spin-spin and energy-energy correlation lengths are estimated. Calculations show that the amplitudes A s and A e are independent of S within the accuracy of the adopted approximations. Moreover, their ratio A e/A s is independent of the anisotropy parameter Δ. These results support the Ising universality hypothesis.

Using transfer-matrix extended phenomenological renormalization-group methods, we study the critical properties of the spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths

\mathop J\limits^ \to = (J',J',J)

. The universality of both fundamental critical exponents y t and y h is confirmed. It is shown that the critical finite-size scaling amplitude ratios

U = A_{\chi ^{(4)} } A_\kappa /A_\chi ^2 ,Y_1 = A_{\kappa ''} /A_\chi

, and

Y_2 = A_{\kappa ^{(4)} } /A_{\chi ^{(4)} }

are independent of the lattice anisotropy parameter Δ=J′/J. For the Y 2 invariant of the three-dimensional Ising universality class, we give the first quantitative estimate Y 2≈2.013 (shape L×L×∞, periodic boundary conditions in both transverse directions).

Using transfer-matrix extended phenomenological renormalization-group methods the critical properties of spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths J = (J ′,J ′,J) are studied. Universality of both fundamental critical exponents yt and yh is confirmed. It is shown that the critical finite-size scaling amplitude ratios U = A χ (4)Aκ/A 2 χ, Y1 = A κ ′′/Aχ, and Y2 = A κ (4)/A χ (4) are independent of the lattice anisotropy parameter ∆ = J ′ /J. By this for the last above invariant of the threedimensional Ising universality class we give the first quantitative estimate: Y2 ≃ 2.013 (shape L × L × ∞, periodic boundary conditions in both transverse directions). PACS numbers: 05.50.+q, 05.70.Jk, 64.60.Fr, 75.10.Hk 1 1