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arXiv:cond-mat/0406664 · Journal of Theoretical and Computational Chemistry

CONSTRUCTING THE CRITICAL CURVE FOR A SYMMETRIC TWO-LAYER ISING MODEL

Mehrdad Ghaemi, Behrouz Mirza, Gholamabbas Parsafar

Published Jun 1, 2004• 13 citations
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Abstract

A numerical method based on the transfer matrix method is developed to calculate the critical temperature of two-layer Ising ferromagnet with a weak inter-layer coupling. The reduced internal energy per site has been accurately calculated for symmetric ferromagnetic case, with the nearest neighbor coupling K1=K2=K (where K1 and K2 are the nearest neighbor interaction in the first and second layers, respectively) with inter-layer coupling J. The critical temperature as a function of the inter-layer coupling [Formula: see text], is obtained for very weak inter-layer interactions, ξ<0.1. Also a different function is given for the case of the strong inter-layer interactions (ξ>1). The importance of these relations is due to the fact that there is no well tabulated data for the critical points versus J/K. We find the value of the shift exponent ϕ=γ is 1.74 for the system with the same intra-layer interaction and 0.5 for the system with different intra-layer interactions.

DOI
10.1142/s0219633604000994

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A new finite-size scaling approach based on the transfer matrix method is developed to calculate the critical temperature of anisotropic two-layer Ising ferromagnet, on strips of r wide sites of square lattices. The reduced internal energy per site has been accurately calculated for the ferromagnetic case, with the nearest neighbor couplings Kx, Ky (where Kx and Ky are the nearest neighbor interactions within each layer in the x and y directions, respectively) and with inter-layer coupling Kz, using different size-limited lattices. The calculated energies for different lattice sizes intersect at various points when plotted versus the reduced temperature. It is found that the location of the intersection point versus the lattice size can be fitted on a power series in terms of the lattice sizes. The power series is used to obtain the critical temperature of the unlimited two-layer lattice. The results obtained, are in good agreement with the accurate values reported by others.
A new finite-size scaling approach based on the transfer matrix method is developed to calculate the critical temperature of an anisotropic two-layer Ising ferromagnet on strips of r -wide sites of square lattices. The reduced internal energy per site has been accurately calculated for the ferromagnetic case, with the nearest-neighbor couplings K x and K y (where K x and K y are the nearest-neighbor interactions within each layer in the x and y directions, respectively) and with interlayer coupling K z, using different size-limited lattices. The calculated energies for different lattice sizes intersect at various points when plotted versus the reduced temperature. It is found that the location of the intersection point versus the lattice size can be fitted to a power series in terms of the lattice sizes. The power series is used to obtain the critical temperature of the unlimited two-layer lattice. The results obtained are in good agreement with the accurate values reported by others.
Within the framework of the transfer matrix formalism, a variational method is elaborated with a constraint related to the fundamental role of the correlation length near a critical point. The method is applied to a two-layer ferromagnetic film, made of two horizontal Ising planes interacting through a vertical coupling. The critical curve of the model, as well as the correlation length above and near T c are calculated. The critical exponent nu obtained agrees with the expected two-dimensional exact value. For the two particular values of the vertical coupling, for which there are numerical estimates of T c through series analysis or the Monte Carlo method, an essential agreement with these estimates is found. Analytical expressions are given in the weak and strong vertical coupling regimes.
A new graphical method is developed to calculate the critical temperature of 2- and 3-dimensional Ising models as well as that of the 2-dimensional Potts models. This method is based on the transfer matrix method and using the limited lattice for the calculation. The reduced internal energy per site has been accurately calculated for different 2-D Ising and Potts models using different size-limited lattices. All calculated energies intersect at a single point when plotted versus the reduced temperature. The reduced temperature at the intersection is 0.4407, 0.2746, and 0.6585 for the square, triangular, and honeycombs Ising lattices and 1.0050, 0.6309, and 1.4848 for the square, triangular, and honeycombs Potts lattices, respectively. These values are exactly the same as the critical temperatures reported in the literature, except for the honeycomb Potts lattice. For the two-dimensional Ising model, we have shown that the existence of such an intersection point is due to the duality relation. The method is then extended to the simple cubic Ising model, in which the intersection point is found to be dependent on the lattice sizes. We have found a linear relation between the lattice size and the intersection point. This relation is used to obtain the critical temperature of the unlimited simple cubic lattice. The obtained result, 0.221(2), is in a good agreement with the accurate value of 0.22165 reported by others.
An antiferromagnetic triangular Ising model with ferromagnetic next nearest neighbor interactions is studied by means of a transfer matrix method and a finite size scaling. Using the Roomany-Wyld approximation, it cannot be concluded definitely whether the system has only one critical point or not. By studying decays of the correlation functions of each sublattice, it is found that there is a finite temperature region where the correlation functions of each sublattice show a power law decay, namely, in this region every temperature is a critical temperature. The results by means of the present transfer matrix method are in good agreement with that by Monte Carlo methods, namely, the discrepancy between both results, reported recently, can be got rid of.
A finite-size scaling approach based on the transfer matrix method is developed to calculate the critical temperature and critical exponent of the symmetric and the asymmetric two-layer three-state Potts models. For similar intralayer interactions our calculation of the shift exponent $\ensuremath{\varphi}$ confirm some scaling arguments which predict $\ensuremath{\varphi}=\ensuremath{\gamma}$, where $\ensuremath{\gamma}$ is the susceptibility exponent. For unequal intralayer interactions we have obtained $\ensuremath{\varphi}=0.5$ which differ from the prediction $\ensuremath{\varphi}=\ensuremath{\gamma}∕2$ of a generalized mean-field theory.