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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical temperature of Ising films with cubic lattices

D. L. Lin, Hao Che, Wei Lai, Thomas F. George

Published Mar 1, 1994• 12 citations
Publisher

Abstract

Following the approach recently proposed by Lin, Che, and Xia [Phys. Rev. A 46, 1805 (1992)], we calculate the critical temperature ${\mathit{T}}_{\mathit{c}}$ for Ising films with lattice structures other than simple cubic by means of the variational cumulant expansion (VCE) of the free energy of the spin system. The critical temperature is derived in terms of the ratio of two successive cumulants to an arbitrary order of accuracy. Since the cumulant of any order m can be calculated analytically by a graphic method, analytical expressions of ${\mathit{T}}_{\mathit{c}}$ can be derived, in principle, as functions of the number L of spin layers in the film to an arbitrary order of accuracy in the VCE. We describe the graphic method in great detail in this paper, and calculate ${\mathit{T}}_{\mathit{c}}$ up to the fourth-order cumulant for films of body-centered-cubic lattice and to the third order for films of face-centered-cubic lattice. Our results show that in the limit of L\ensuremath{\rightarrow}\ensuremath{\infty}, ${\mathit{T}}_{\mathit{c}}$ approaches the corresponding bulk value rather quickly with increasing m in both cases.

DOI
10.1103/physreve.49.2155

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Following the approach recently proposed by Lin, Che, and Xia [Phys. Rev. A 46, 1805 (1992)], we calculate the critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ for Ising films with lattice structures other than simple cubic by means of the variational cumulant expansion (VCE) of the free energy of the spin system. The critical temperature is derived in terms of the ratio of two successive cumulants to an arbitrary order of accuracy. Since the cumulant of any order m can be calculated analytically by a graphic method, analytical expressions of Tc{\mathit{T}}_{\mathit{c}}Tc​ can be derived, in principle, as functions of the number L of spin layers in the film to an arbitrary order of accuracy in the VCE. We describe the graphic method in great detail in this paper, and calculate Tc{\mathit{T}}_{\mathit{c}}Tc​ up to the fourth-order cumulant for films of body-centered-cubic lattice and to the third order for films of face-centered-cubic lattice. Our results show that in the limit of L\ensuremath{\rightarrow}\ensuremath{\infty}, Tc{\mathit{T}}_{\mathit{c}}Tc​ approaches the corresponding bulk value rather quickly with increasing m in both cases.
With the variational cumulant expansion (VCE) method, the thermodynamic behaviors of antiferromagnetic Heisenberg films in simple cubic lattices are studied analytically. From the analytic properties of the free energy, in principle we are able to calculate analytically the critical temperatures and the thermodynamic functions, to any order cumulant as the functions of the number of L (the hyperlayers in the hyperfilm). Explicit expressions for up to the fourth order are given. A comparison with the existing results for 3-dimensional system is given. The effective range of the interaction is obtained from numerical results.
An alternative approach is proposed to treat the critical point for general Ising spin systems. From the analytic properties of the free energy, we are in principle able to calculate analytically the critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​(l) for (d+1)-dimensional Ising films to any order cumulant as a function of the number l of hyperlayers in the hyperfilm. Explicit expressions up to the fourth order are given. It is shown from the general expression for Tc(m){\mathit{T}}_{\mathit{c}}^{(\mathit{m})}Tc(m)​(l) that [Tc{\mathit{T}}_{\mathit{c}}Tc​(l)-Tc{\mathit{T}}_{\mathit{c}}Tc​]/Tc{\mathit{T}}_{\mathit{c}}Tc​=b/(l+a) in the limit l\ensuremath{\rightarrow}\ensuremath{\infty}. Comparisons with existing results for d=2 are discussed.
The variational-cumulant-expansion (VCE) method is adopted to calculate the Heisenberg model. The free energy of the model is expanded to the third order, and the analytical formulas for the critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ are given for each order. The first order of VCE corresponds to the mean-field theory, the second order gives the same result for Tc{\mathit{T}}_{\mathit{c}}Tc​ as the 1/z expansion, and the third order gives a more complicated correction. As a comparison, the results of the two-dimensional (2D) Ising model for VCE to the seventh order are shown. We carry out a trial to predict the critical temperature of the infinite order of the VCE, and get a very accurate result for the 2D Ising model. A prediction of Tc{\mathit{T}}_{\mathit{c}}Tc​ for the Heisenberg model is also made.
The critical temperature T c of a spin-1/2 Ising film of cubic structures is calculated by the variational cumulant expansion method for three directions of growth. The results from different growth directions are analysed and compared with each other. In the present model, the T c values depend largely on the number of nearest neighbors in a monolayer for films with the same number of monolayers but grown in different directions. For sc, bcc and fcc structures, the highest T c is found along the (100), (110), and (111) direction, respectively.
The critical temperatures of the Ising model in square and simple cubic lattices are derived with the variational-cumulant expansion approach to the seventh order. An extrapolating method is proposed to predict the critical temperature of infinite series according to the finite order expansion, at which a weighted fitting method is used. For the 2-dimensional Ising model, the critical temperature obtained by this method is in a deviation of 0.04% compared with the exact value, and the estimated results are robust for different order prediction. The critical temperature for the 3D Ising model is predicted as 4.490.