Calculation of the Critical Temperature for the Anisotropic Two-Layer Ising Model Using the Transfer Matrix Method

Abstract

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 K x , 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 inter-layer 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 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.

Calculation of the Critical Point for Two-Layer Ising and Potts Models Using Cellular Automata

“…For example, in order to compute the fourth digit of K c in the twolayer Ising model, it is sufficient to increase the number of time step up to 300000 steps and draw the graph <m> vs. K . The calculated K c is 0.3108 that is in good agreement with other numerical method [14].…”

Cellular Automata Simulation of Two-Layer Ising and Potts Models

“…Ghaemi et al have used the transfer matrix method to construct the critical curve for a symmetric two-layer Ising model. In another work (Ghaemi et al, 2003), they have used this method to get the critical temperature for the anisotropic two-layer Ising model. Such calculations are limited to lattice with the width 5 cells in each layer and the critical point is obtained by the extrapolation approach.…”

“…The configuration energy for this model may be defined (Ghaemi et al, 2003) where * indicates the periodic boundary conditions (eqs 1,2), and Kx and Ky are the nearestneighbor interactions within each layer in the x and y directions, respectively, and Kz is the interlayer coupling. Therefore, the configuration energy per spin is…”

“…2). Then, the obtained results have been fitted in order to obtain a general ansatz equation for the critical point for the anisotropic two-layer Potts model in terms of inter-and intralayer couplings (ξ and ) as follow, (Ghaemi et al, 2003). b From the corner transfer matrix renormalization group method (Li et al, 2001) (Mardani et al, 2005) Table 2.…”

Obtaining critical point and shift exponent for the anisotropic two-layer Ising and Potts models: Cellular automata approach

“…Ghaemi et al [10] have used the transfer matrix method to construct the critical curve for a symmetric two-layer Ising model. In another work [11], they have used this method to get the critical temperature for the anisotropic twolayer Ising model. Such calculations are limited to lattices with a width of five cells in each layer and the critical point is obtained by the extrapolation approach.…”