Calculation of the Critical Temperature for 2- and 3-Dimensional Ising Models and for 2-Dimensional Potts Models Using the Transfer Matrix Method

Abstract

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.

Study of two dimensional anisotropic Ising models via a renormalization group approach

“…Table 1 presents results of reduced critical temperatures and exponents from the solution of Eqs. (14) and (19)- (21). According to Table 1, increasing the number of sites of the 2D anisotropic triangular lattice gives more accurate values (19 sites result in an error of 1.5%.)…”

“…On the basis of Table 2, the reduced critical temperature decreases with increasing the size of the lattice. The deviation of the reduced critical temperature from the exact value [20,21] is 2.26% for 25 sites.…”

“…There are 2, 3, 4 and 5 interaction links for the coarse-graining schemes using 4, 9, 16 and 25 spins, respectively. Increasing the number of interaction links as well as the number of spins in the blocks of the square lattice leads us to approach the exact value (i.e., 0.442) in the thermodynamic limit [20,21]. For the square lattice, when the spin coupling interaction in one direction is much bigger than in the other direction, the 2D Ising model is converted into the …”

“…An analytical method was developed to calculate the critical temperature of 2D [20] and 3-D Ising models as well as that of the 2D Potts models by Ghaemi et al [21]. A new finite-size scaling approach based on the transfer matrix method was also developed by Ghaemi et al to calculate the critical temperature of an anisotropic two-layer Ising ferromagnet on strips of r-wide sizes of square lattices [22].…”

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

“…There is no exact solution for Potts model and a lot of numerical and simulation methods have been used to obtain the critical point for the Potts model [32]. In recent work, the CA method is used for the two-layer 3-state Potts model to obtain the critical point.…”