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Physical review. B, Condensed matter

Critical temperature of two coupled Ising planes

Alessandro Pelizzola

Published May 1, 1995• 7 citations
Publisher

Abstract

It is shown that the method proposed by Wosiek [Phys. Rev. B 49, 15 023 (1994)] for the calculation of the order-disorder critical temperature, which turns out to be exact when the critical temperature is determined by symmetry properties like self-duality, is, when applied to a system of two coupled Ising planes, the lowest-order approximation in a hierarchy which appears to converge to the exact result.

DOI
10.1103/physrevb.51.12005

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It is shown that the method proposed by Wosiek [Phys. Rev. B 49, 15 023 (1994)] for the calculation of the order-disorder critical temperature, which turns out to be exact when the critical temperature is determined by symmetry properties like self-duality, is, when applied to a system of two coupled Ising planes, the lowest-order approximation in a hierarchy which appears to converge to the exact result.
We have found a simple criterion which allows for the straightforward determination of the order-disorder critical temperatures. The method reproduces exactly known results for the two-dimensional Ising, Potts, and Z(N5) models. It also works for the Ising model on the triangular lattice. For systems that are not self-dual, our proposition remains an unproven conjecture. It predicts ${\mathrm{\ensuremath{\beta}}}_{\mathit{c}}$=0.2656. . . for the two coupled layers of Ising spins. The critical temperature of the three-dimensional Ising model is related to the free energy of the two-layer Ising system.
We studied a new and simple method which gives successive approximate expressions for the magnetization (or any order parameter) of a random Ising model as a function of temperature, site, or bond dilution, frustration, magnetic field,... To obtain the various approximations a trace is performed on finite clusters increasing in size. In particular, we investigated a honeycomb Ising model with random nearest-neighbor interactions and calculated sequences of critical temperatures, percolation thresholds,... which converge towards exact known results.
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 interface method is used to derive the order-disorder critical temperature of the two-dimensional Ising model on a rectangular lattice with ferromagnetic interactions in the x direction and antiferromagnetic interactions in the y direction. Employing this approximation in two different ways gives different critical temperatures Tc{T}_{c}Tc​ for nonzero uniform magnetic fields H. However, both of the phase boundaries Tc{T}_{c}Tc​(H) reduce to the exact results available at H=0 and T=0. .AE