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Physical Review A

Critical temperature of (d+1)-dimensional Ising films

D. L. Lin, Hao Che, Yu Xia

Published Aug 1, 1992• 26 citations
Publisher

Abstract

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 ${\mathit{T}}_{\mathit{c}}$(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 ${\mathit{T}}_{\mathit{c}}^{(\mathit{m})}$(l) that [${\mathit{T}}_{\mathit{c}}$(l)-${\mathit{T}}_{\mathit{c}}$]/${\mathit{T}}_{\mathit{c}}$=b/(l+a) in the limit l\ensuremath{\rightarrow}\ensuremath{\infty}. Comparisons with existing results for d=2 are discussed.

DOI
10.1103/physreva.46.1805

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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.
The critical temperatures Kc and the nearest-neighbour correlation functions Uc have been calculated for the d-dimensional hypercubic Ising ferromagnet for d=4 to d=30, by generalizing a recent theory reported for d=3. The results are of relevance for Monte Carlo Renormalization Group work involving critical properties in random fields, which calls for a knowledge of Kc in the limit of zero field. As d becomes large, Kc and Uc are found to approach the spherical model values. Polynomial expressions in 1/d are fitted to the calculated values, using the analogous spherical model expressions as guide. Comparison is made with the corresponding expressions of Fisher and Gaunt. Uc/(1−Kc), independent of d for the spherical model, is investigated; it is seen to be approximately independent of lattice structure for d=2 and for d=3.
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.