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Chinese Physics Letters

Critical Point of Ising Films with Different Growth Directions

Huai‐Yu Wang, Yun‐Song Zhou, D. L. Lin

Published Jan 1, 2000• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0256-307x/17/1/019

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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.
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.
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.
The susceptibility of simple-cubic-lattice ferromagnetic Ising films of nnn two-dimensional lattice layers is studied by extrapolation of high-temperature series expansions (to eleventh and twelfth orders) for n=3,4,…,14n=3,4,\dots{},14n=3,4,…,14 layers with periodic boundary conditions, and n=3,4,…,10n=3,4,\dots{},10n=3,4,…,10 layers with free-surface conditions. The corresponding surface susceptibility series for sc, bcc, and fcc lattices are analyzed in the light of universality hypotheses. On the basis of finite-size scaling theory explicit scaling functions are constructed describing the crossover from two-dimensional to three-dimensional critical behavior in terms of the ratio $\frac{n}{\ensuremath{\xi}(T)}$, where $\ensuremath{\xi}\ensuremath{\sim}\ensuremath{\Delta}{T}^{\ensuremath{-}\ensuremath{\nu}}$ is the bulk correlation length.
The Niemeijer-Van Leeuwen cumulant expansion is used in the renormalization-group analysis of the critical-point behavior of the Ising model. Second- and third-order results are obtained for the simple cubic and square lattices, respectively, for a variety of cell sizes, using a parameter-dependent renormalization transformation introduced by Kadanoff and Houghton. Although the results in two dimensions are in general good, there is no evidence of any convergence, with the best values for the critical exponents and critical temperature being given by the second-order theory. In three dimensions the values found for the critical temperature and magnetic scaling index for a cell of 27 spins are reasonable, but the corresponding thermal scaling index is quite poor.
Ising lattices consisting of n=2,3,4,and5n=2, 3, 4, \mathrm{and} 5n=2,3,4,and5 interacting plane square lattice layers are studied by hightemperature series expansions. Specifically, seven to nine terms of the zero-field susceptibility expansion have been obtained for (a) free-surface boundary conditions (in which each surface spin interacts with only five nearest neighbors); and (b) periodic boundary conditions (in which all spins interact equivalently with six nearest neighbors). Estimates of the critical temperatures Tc(n){T}_{c}(n)Tc​(n) obtained by ratio and Pad\'e approximant techniques are presented. These results are consistent with the conjectures that ${T}_{c}(\ensuremath{\infty})\ensuremath{-}{T}_{c}(n)$ varies with thickness nnn, as ${n}^{\ensuremath{-}\ensuremath{\lambda}}$ with $\ensuremath{\lambda}=1$ in case (a) and $\ensuremath{\lambda}=\frac{1}{{\ensuremath{\nu}}_{c}}\ensuremath{\simeq}1.56$ in case (b); but other, somewhat larger, values of $\ensuremath{\lambda}$ are not excluded.
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.