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The Journal of Physical Chemistry

Calculation of the critical temperature for two-dimensional and three-dimensional Ising models and for two-dimensional Potts models using the transfer matrix method

Mehrdad Ghaemi, Mahmud Ashrafizaadeh, Gholamabbas Parsafar

Published Jan 1, 2001• 0 citations
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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.

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Critical temperatures of the three- and four-state Potts models on the kagome lattice

Seung Ki Baek, H. Mäkelä, Petter Minnhagen +1 more

The value of the internal energy per spin is independent of the strip width for a certain class of spin systems on two-dimensional infinite strips. It is verified that the Ising model on the kagome lattice belongs to this class through an exact transfer-matrix calculation of the internal energy for the two smallest widths. More generally, one can suggest an upper bound for the critical coupling strength K(c)(q) for the q-state Potts model from exact calculations of the internal energy for the two smallest strip widths. Combining this with the corresponding calculation for the dual lattice and using an exact duality relation enables us to conjecture the critical coupling strengths for the three- and four-state Potts models on the kagome lattice. The values are K(c)(q=3)=1.0565094269290 and K(c)(q=4)=1.1493605872292, and the values can, in principle, be obtained to an arbitrary precision. We discuss the fact that these values are in the middle of earlier approximate results and furthermore differ from earlier conjectures for the exact values.

arXiv:1106.1296•Physical Review E•Jun 6, 2011•5 citations
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.
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.
The value of the internal energy per spin is independent of the strip width for a certain class of spin systems on two-dimensional infinite strips. It is verified that the Ising model on the kagome lattice belongs to this class through an exact transfer-matrix calculation of the internal energy for the two smallest widths. More generally, one can suggest an upper bound for the critical coupling strength K(c)(q) for the q-state Potts model from exact calculations of the internal energy for the two smallest strip widths. Combining this with the corresponding calculation for the dual lattice and using an exact duality relation enables us to conjecture the critical coupling strengths for the three- and four-state Potts models on the kagome lattice. The values are K(c)(q=3)=1.0565094269290 and K(c)(q=4)=1.1493605872292, and the values can, in principle, be obtained to an arbitrary precision. We discuss the fact that these values are in the middle of earlier approximate results and furthermore differ from earlier conjectures for the exact values.