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arXiv:cond-mat/0312555

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

M. A. Yurishchev

Published Dec 21, 2003• 0 citations
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Abstract

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 $Δ=J'/J$.

DOI
10.48550/arxiv.cond-mat/0312555

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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.
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 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
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).
The Ising model is considered on a simple cubic lattice, with a coupling constant J along one axis and coupling constants J’ along the remaining two axes. The transfer-matrix technique and an extended phenomenological renormalization group theory [18, 19] are applied to obtain two-sided bounds on the critical temperature for the model with J′/J ≤1. The bounds monotonically converge with decreasing J′/J and provide improved estimates for the phase-transition temperature in anisotropic three-dimensional Ising model, as compared with those available from the literature.
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 a renormalized linked-cluster-expansion method, we have extended to order ${\ensuremath{\beta}}^{23}$ the high-temperature series for the susceptibility $\ensuremath{\chi}$ and the second-moment correlation length $\ensuremath{\xi}$ of the spin-1/2 Ising models on the sc and the bcc lattices. A study of these expansions yields updated direct estimates of universal parameters, such as exponents and amplitude ratios, which characterize the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}.$ Our best estimates for the inverse critical temperatures are ${\ensuremath{\beta}}_{c}^{\mathrm{sc}}=0.221654(1)$ and ${\ensuremath{\beta}}_{c}^{\mathrm{bcc}}=0.1573725(6).$ For the susceptibility exponent we get $\ensuremath{\gamma}=1.2375(6)$ and for the correlation length exponent $\ensuremath{\nu}=0.6302(4).$ The ratio of the critical amplitudes of $\ensuremath{\chi}$ above and below the critical temperature is estimated to be ${C}_{+}{/C}_{\ensuremath{-}}=4.762(8).$ The analogous ratio for $\ensuremath{\xi}$ is estimated to be ${f}_{+}{/f}_{\ensuremath{-}}=1.963(8).$ For the correction-to-scaling amplitude ratio we obtain ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$