Journal of Physics C Solid State Physics

On the critical behaviour of anisotropic systems

Ann Bruce

Published Jun 21, 1974• 61 citations

Abstract

The critical properties of d dimensional systems with a quadratic anisotropic term in the Hamiltonian are investigated with the aid of a renormalization group analysis based upon the Wilson perturbation expansion in epsilon =4-d. For anisotropic cubic systems with a nonzero four-spin coupling constant, u, it is found that the asymptotic critical behaviour is always one with vanishing quadratic anisotropy. In systems with lattice anisotropy a simple renormalization group argument leads to results in agreement with the universality hypothesis.

DOI: 10.1088/0022-3719/7/12/003

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Journal of Physics C Solid State Physics

On the critical behaviour of anisotropic systems

Ann Bruce

Published Jun 21, 1974• 61 citations

Publisher

Abstract

DOI: 10.1088/0022-3719/7/12/003

Related papers

Critical behaviour and cubic anisotropy

Thomas Nattermann, Steffen Trimper

The influence of the cubic anisotropy both in the quadratic and quartic part of an n-component 'spin' Hamiltonian is examined in the framework of a parquet-graph summation. For values of the anisotropy parameter f larger than a critical value, f - , a first-order transition occurs which corresponds to the appearance of complex fixed points in the renormalization group approach. On the other hand for f<f - a first- or second-order transition occurs depending on f, n and the values of the bare coupling constants. For second-order transitions a quantity m(n,f) plays the role of an effective number of components, m(n,0)=n. For m>4 and m<-8 the system has a strongly cubic behaviour whereas 0<m<4 a weakly cubic behaviour is observed; the latter changes to pure isotropic behaviour for f to 0. For -8<m<0 the behaviour is strongly or weakly cubic depending on the values of the bare coupling constants.The influence of the cubic anisotropy both in the quadratic and quartic part of an n-component 'spin' Hamiltonian is examined in the framework of a parquet-graph summation. For values of the anisotropy parameter f larger than a critical value, f - , a first-order transition occurs which corresponds to the appearance of complex fixed points in the renormalization group approach. On the other hand for f<f - a first- or second-order transition occurs depending on f, n and the values of the bare coupling constants. For second-order transitions a quantity m(n,f) plays the role of an effective number of components, m(n,0)=n. For m>4 and m<-8 the system has a strongly cubic behaviour whereas 0<m<4 a weakly cubic behaviour is observed; the latter changes to pure isotropic behaviour for f to 0. For -8<m<0 the behaviour is strongly or weakly cubic depending on the values of the bare coupling constants.

Journal of Physics A Mathematical and GeneralDec 1, 197555 citations

Effect of Anisotropy on the Critical Behaviour of Three- Dimensional Heisenberg Ferromagnets

R. Shanker, R. A. Singh

The anisotropic nearest-neighbour Heisenberg model for the simple cubic lattice has been investigated by interpolating the anisotropy between the Ising and isotropic Heisenberg limits via general spin high-temperature series expansions of the zero-field suspectibility. This is done by estimating the critical temperature (Tc(3)) and the susceptibility exponent γ from the analysis of the series by the Ratio and Pade approximants methods. It is noted that Tc(3) varies with anisotropy while γ is almost the same for the anisotropic system, and a jump in it occurs for the isotropic case in agreement with the universality hypothesis. The effect of anisotropy on the susceptibility is also shown. Further, it is seen that estimates of γ for the two extreme limits agree well with those of previous theoretical as well as experimental investigations. In addition, critical temperatures have been summarised in a relation, and expressions for the magnetisation have been derived.The anisotropic nearest-neighbour Heisenberg model for the simple cubic lattice has been investigated by interpolating the anisotropy between the Ising and isotropic Heisenberg limits via general spin high-temperature series expansions of the zero-field suspectibility. This is done by estimating the critical temperature (Tc(3)) and the susceptibility exponent γ from the analysis of the series by the Ratio and Pade approximants methods. It is noted that Tc(3) varies with anisotropy while γ is almost the same for the anisotropic system, and a jump in it occurs for the isotropic case in agreement with the universality hypothesis. The effect of anisotropy on the susceptibility is also shown. Further, it is seen that estimates of γ for the two extreme limits agree well with those of previous theoretical as well as experimental investigations. In addition, critical temperatures have been summarised in a relation, and expressions for the magnetisation have been derived.

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The renormalization group approach is applied to the study of the short-time critical behaviour of the d -dimensional n -component anisotropic cubic spin systems with long-range interaction of the form p σ s p s - p in momentum space. Firstly, the system is quenched from a high temperature to the critical temperature and then relaxes to equilibrium within the model A dynamics. The asymptotic scaling laws and the initial slip exponents θ' and θ of the order parameter and the response function, respectively, are calculated to the second order in = 2σ- d . For 1⩽ d <2σ and n > n c , the cubic anisotropy affects the short-time critical behaviour.The renormalization group approach is applied to the study of the short-time critical behaviour of the d -dimensional n -component anisotropic cubic spin systems with long-range interaction of the form p σ s p s - p in momentum space. Firstly, the system is quenched from a high temperature to the critical temperature and then relaxes to equilibrium within the model A dynamics. The asymptotic scaling laws and the initial slip exponents θ' and θ of the order parameter and the response function, respectively, are calculated to the second order in = 2σ- d . For 1⩽ d <2σ and n > n c , the cubic anisotropy affects the short-time critical behaviour.

Journal of Physics A Mathematical and GeneralFeb 19, 200112 citations

Critical properties of high-spin Ising models on anisotropic lattices

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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.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 PhysicsNov 1, 20061 citation

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).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/0410629Journal of Experimental and Theoretical PhysicsSep 1, 20043 citations

Universality in the partially anisotropic three-dimensional Ising lattice

M. A. Yurishchev

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 1Using 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

Jan 1, 20043 citations

\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).