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Universality in the partially anisotropic three-dimensional Ising lattice

M. A. Yurishchev

Published Jan 1, 2004• 3 citations
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Abstract

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

Related papers

Universality in the partially anisotropic three-dimensional Ising lattice

M. A. Yurishchev

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).

arXiv:cond-mat/0410629•Journal of Experimental and Theoretical Physics•Sep 1, 2004•3 citations

Anisotropic simple-cubic Ising lattice: extended phenomenological renormalization-group treatment

M. A. Yurishchev

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 $Δ=J'/J$.

arXiv:cond-mat/0312555•Dec 21, 2003•0 citations

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

M. A. Yurishchev

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).

arXiv:cond-mat/9612219•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Apr 1, 1997•16 citations

Hyperuniversality of a fully anisotropic three-dimensional Ising model

M. A. Yurishchev

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.

arXiv:cond-mat/9408099•Physical review. B, Condensed matter•Nov 1, 1994•20 citations

Critical properties of high-spin Ising models on anisotropic lattices

M. A. Yurishchev

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.

Journal of Experimental and Theoretical Physics•Nov 1, 2006•1 citation

The critical temperature of a fully anisotropic three-dimensional Ising model

M. A. Yurishchev

The critical temperature of a three-dimensional Ising model on a simple cubic lattice with different coupling strengths along all three spatial directions is calculated via the transfer matrix method and a finite-size scaling for L*L* infinity clusters (L=2 and 3). The results obtained are compared with available calculations. An exact analytical solution is found for the 2*2* infinity Ising chain with fully anisotropic interactions (arbitrary J x , J y and J z ).

arXiv:cond-mat/0203609•Journal of Physics Condensed Matter•Oct 25, 1993•4 citations
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
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 J→=(J′,J′,J)\mathop J\limits^ \to = (J',J',J)J→​=(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=Aχ(4)Aκ/Aχ2,Y1=Aκ′′/AχU = A_{\chi ^{(4)} } A_\kappa /A_\chi ^2 ,Y_1 = A_{\kappa ''} /A_\chiU=Aχ(4)​Aκ​/Aχ2​,Y1​=Aκ′′​/Aχ​ , and Y2=Aκ(4)/Aχ(4)Y_2 = A_{\kappa ^{(4)} } /A_{\chi ^{(4)} }Y2​=Aκ(4)​/Aχ(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 [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 J⃗=(J′,J′,J){\vec J}=(J',J',J)J=(J′,J′,J) are obtained. Universality of both fundamental critical exponents yty_tyt​ and yhy_hyh​ is confirmed. We show also that the critical finite-size scaling amplitude ratios Aχ(4)Aκ/Aχ2A_{χ^{(4)}}A_κ/A_χ^2Aχ(4)​Aκ​/Aχ2​, Aκ′′/AχA_{κ^{''}}/A_χAκ′′​/Aχ​, and Aκ(4)/Aχ(4)A_{κ^{(4)}}/A_{χ^{(4)}}Aκ(4)​/Aχ(4)​ are independent of the lattice anisotropy parameter Δ=J′/JΔ=J'/JΔ=J′/J.
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.
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.