arXiv:cond-mat/0406665 · Iranian Journal of Physics Research

Size Reduction of The Transfer Matrix of Two-Dimensional Ising and Potts Models

Mehrdad Ghaemi, Gholamabbas Parsafar

Published Jun 27, 2004• 1 citation

Abstract

A new algebraic method is developed to reduce the size of the transfer matrix of Ising and three-state Potts ferromagnets, on strips of width r sites of square and triangular lattices. This size reduction has been set up in such a way that the maximum eigenvalues of both the reduced and original transfer matrices became exactly the same. In this method we write the original transfer matrix in a special blocked form in such a way that the sums of row elements of a block of the original transfer matrix be the same. The reduced matrix is obtained by replacing each block of the original transfer matrix with the sum of the elements of one of its rows. Our method results in significant matrix size reduction which is a crucial factor in determining the maximum eigenvalue.

DOI: 10.48550/arxiv.cond-mat/0406665

Related papers

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

Mehrdad Ghaemi, Gholamabbas Parsafar, Mahmud Ashrafizaadeh

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.

arXiv:physics/0406135The Journal of Physical Chemistry BSep 27, 200115 citations

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

Mehrdad Ghaemi, Mohammad Ghannadi, Behrouz Mirza

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.

The Journal of Physical Chemistry BDec 18, 200212 citations

Finite-size scaling corrections in two-dimensional Ising and Potts ferromagnets

S L A de Queiroz

Finite-size corrections to scaling of critical correlation lengths and free energies of Ising and three-state Potts ferromagnets are analysed by numerical methods, on strips of width N sites of square, triangular and honeycomb lattices. Strong evidence is given that the amplitudes of the `analytical' correction terms, N -2 , are identically zero for triangular and honeycomb Ising systems. For Potts spins, our results are broadly consistent with this lattice-dependent pattern of cancellations, though for correlation lengths non-vanishing (albeit rather small) amplitudes cannot be entirely ruled out.

arXiv:cond-mat/9912090Journal of Physics A Mathematical and GeneralJan 24, 200027 citations

Transfer Matrix and Finite-Size Scaling for the Ising Model on Two- and Three-Dimensional Lattices

Takehiko Oguchi, Y‐h. Taguchi

The transfer matrices for the ferromagnetic Ising model on the two- and three-dimensional lattices with finite size are formulated using the screw method introduced by Kramers and Wannier, and the largest and next largest eigenvalues of matrices are obtained by a computer. Then the specific heats and magnetic susceptibilities are calculated for each system with various sizes. By use of the finite-size scaling and Roomany-Wyld approximations, the critical exponents of the infinite-size systems are evaluated. They are in quite good agreement with the reliable ones.

Progress of Theoretical Physics SupplementJan 1, 19868 citations

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

Mehrdad Ghaemi, Mohammad Ghannadi, Behrouz Mirza

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.

arXiv:cond-mat/0406669Jun 28, 20040 citations

The Calculations of 3-Dimensional Ising Model of Finite Width

Bin Cao, Wei -Shih Yang, Chunxiang Cui

In order to calculate 3-dimensional Ising model, we develop a method to build a much smaller transfer matrix containing the largest eigenvalue from the original 2 N 1 N 2 ×2 N 1 N 2 matrix V 1 . Firstly, the transfer matrix V 1 is written as the linear combination of several basic vectors. Secondly, we divide the basic vectors into several subgroups. The multiplication of a basic vector and V 1 can be written as the linear combination of basic vectors from the same subgroup. Finally, we use a new transfer matrix V 2 to describe the relationship between basic vectors of the same subgroup. V 2 is much smaller than the original transfer matrix and contains the largest eigenvalue of V 1 . We use this method to calculate the specific heat per atom C pa and the magnetic momentum per atom M pa . The results show that there exists a pair of temperature and magnetic field intensity where the specific heat gets to its maximum value. When N 1 N 2 increases, the maximum value of specific heat becomes larger.

Sep 25, 20160 citations