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The Journal of Physical Chemistry B

A New Exact Method for Solving the Two-Dimensional Ising Model

S. Ranjbar, Gholamabbas Parsafar

Published Aug 12, 1999• 9 citations
Publisher

Abstract

We have used the two-dimensional Ising model with a limited number of rows, but with the coordination number of four for each site, to set up the transfer matrix for the model. From the solution of such a matrix, the exact thermodynamic properties have been obtained for the model with a definite number of rows, n . We have solved the matrix for n ≤ 7 and n ≤ 10 in the presence and absence of a magnetic field, respectively. On the basis of such solutions, we have proposed an analytical expression for the partition function of the model with any number of rows in the absence of a magnetic field. The proposed expression becomes more accurate when n is larger, in such a way that it becomes very accurate for n ≥ 8 and is exact for n → ∞. Our results show that the singularity of the specific heat occurs only for the model with infinite number of rows. In the presence of a magnetic field, the solution to the matrixes is too complicated to propose a general analytical expression for the partition function of the model with any number of rows. However, our exact solution for the model with n = 4, 5, 6, and 7 reveals an important point that such results are independent of n when the field−spin interaction energy is almost equal to or larger than that of the spin−spin interaction.

DOI
10.1021/jp9902203

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We have used the two-dimensional Ising model with a limited number of rows, but with the coordination number of four for each site, to set up the transfer matrix for the model. From the solution of such a matrix, the exact thermodynamic properties have been obtained for the model with a definite number of rows, n . We have solved the matrix for n ≤ 7 and n ≤ 10 in the presence and absence of a magnetic field, respectively. On the basis of such solutions, we have proposed an analytical expression for the partition function of the model with any number of rows in the absence of a magnetic field. The proposed expression becomes more accurate when n is larger, in such a way that it becomes very accurate for n ≥ 8 and is exact for n → ∞. Our results show that the singularity of the specific heat occurs only for the model with infinite number of rows. In the presence of a magnetic field, the solution to the matrixes is too complicated to propose a general analytical expression for the partition function of the model with any number of rows. However, our exact solution for the model with n = 4, 5, 6, and 7 reveals an important point that such results are independent of n when the field−spin interaction energy is almost equal to or larger than that of the spin−spin interaction.
An example of a two-dimensional crystal in which each lattice point interacts with six neighbors is the so-called \triangular Ising lattice. In this work the transfer matrix method is used to nd the thermodynamic properties of the triangular Ising model with a limited number of rows, in the absence or presence of a magnetic eld. In the absence of a magnetic eld the results have been used to nd an analytical expression for the partition function of a model with any number of rows, from which the thermodynamic properties of the triangular planar lattice model are obtained as a polynomial in terms of x = e 2j tanh 2j=2 cosh2j, where J = jkT is the interaction energy of the two nearest neighbor spins. The degree of the polynomial depends on the number of rows in the model. The results are compared with the exact values. The calculated free energy is found to be very accurate, but the calculated internal energy and heat capacity show some small deviations from the exact values. In the presence of a magnetic eld the solution to the matrices is too complicated for a general analytical expression to be proposed for a model with any number of rows. However, our exact solution for the model with n = 4, 5, and 7 reveals an important point: that such results are independent of n when the eld-spin interaction energy is almost equal to or larger than that of the spin-spin interaction.
The Hamiltonian of the Ising model in one-, two- and three-dimensions has been analysed using unitary transformations and combinatorics. We have been able to obtain closed formulas for the eigenvalues of the Ising Hamiltonian for an arbitrary number of dimensions and sites. Although the solution provided assumes the absence of external magnetic fields an extension to include a magnetic field along the z-axis is readily extracted. Furthermore, generalisations to a higher number of spin components on each site are possible within this method. We made numerical comparisons with the partition function from the earlier analytical expressions known in the literature for one- and two-dimensional cases. We find complete agreement with these studies. (C) 2001 Elsevier Science B.V. All rights reserved.
The Ising model in a random magnetic field h is studied on a square lattice. h has the same value along a row, but varies randomly from row to row. h=\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty} with probability p/2 and is zero with probability 1-p. The quenched free energy F and specific heat C are calculated exactly. We find C\ensuremath{\sim}lnp for small p at the Onsager temperature Tc{T}_{c}Tc​. It is shown that for p\ensuremath{\ne}0, F(p,T) is not analytic in T at T=Tc{T}_{c}Tc​, i.e., it has a Griffiths-like singularity.
To solve the controversy, regarding the existence of an analytic solution to the 1-D Ising model with nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions in the presence of a magnetic field, we apply the transfer matrix method to solve the 1-D Ising model in the presence of a magnetic field, taking both NN and NNN interactions into account. We show that it is possible to write a transfer matrix only if the number of sites is even. Even in such a case, it is impossible to diagonalize the transfer matrix in an analytic form. Therefore, we employ a numerical method to obtain the eigenvalues of the transfer matrix. Moreover, the heat capacity, magnetization, and magnetic susceptibility versus temperature for different values of the competition factor (the ratio of NNN to NN interactions) are shown.
The partition function and the spontaneous magnetization of a two-dimensional (2D) Ising model with the next nearest interactions at zero magnetic field are derived analytically. At first, the transfer matrices are analyzed in three representations, i.e., Clifford algebraic representation, transfer tensor representation, and schematic representation, to inspect nontrivial topological structures in this system. The system is equivalent to a triangular Ising model plus an interaction along the z axis, so that the approaches developed for the 3D Ising models are modified to be appropriable for analytically solving the solution of the 2D Ising model with the next nearest interactions. The comparison with the exact solutions of other Ising lattices reveals that either the increase of the number of interactions in a unit cell or the presence/increase of topological contributions enhances the critical point of the Ising lattices. The results obtained in this work are helpful for understanding the physical properties of the 2D magnetic materials.
We use the transfer-matrix method to calculate density-of-states (DOS) functions for fully finite Ising-model systems. In this way, we obtain expressions for the corresponding partition functions for systems of sizes up to 12\ifmmode\times\else\texttimes\fi{}12 in zero field and up to 8\ifmmode\times\else\texttimes\fi{}8 in nonzero external magnetic field. We also establish a way of representing the DOS functions, in terms of the suitably scaled variables, that reveals a remarkably rapid convergence of these functions towards their thermodynamic limiting values as the system size increases. This rapid convergence implies that results obtained for systems of small sizes contain a surprisingly large amount of information about the behavior of infinite systems. In particular, we demonstrate that the finite-size-scaling-theory predictions are satisfied for small systems (treated exactly via this method) comparably well as for much larger systems (treated approximately via the Monte Carlo method). In addition, applying the method to the square Ising model in a nonzero field, we discover that the DOS functions for small systems display a regularity that one would expect to appear only for very large systems.