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The Calculations of 3-Dimensional Ising Model of Finite Width

Bin Cao, Wei -Shih Yang, Chunxiang Cui

Published Sep 25, 2016• 0 citations
Publisher

Abstract

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.

DOI
10.11916/j.issn.1005-9113.2016.04.009

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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.
We have developed a method to change the transfer matrix V1 of 3D Ising model into a much smaller transfer matrix. V1 can be written as the linear combination of basic vectors. We discover that these basic vectors can be divided into several groups. The point product of one basic vector and V1 can be written as the linear combination of basic vectors from the same group. Based on that, we establish a new transfer matrix V2 to describe the relationship between basic vectors of the same group. The number of members from the same group is the order of V2. V2 is much smaller than V1 and contains the largest eigenvalue of V1. We use this method to calculate the specific heat and magnetic momentum per atom. The results show that there’s a pair of temperature and magnetic field strength that can make the specific heat get to its maximum value. When N1N2 increases, the maximum value of specific heat increases too.
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.
The eigenvalue problem of three-dimensional Ising model is solved in successive screw approximation on a computer. Suitable extrapolation of the results suggests that the maximum value of the specific heat of the simple cubic lattice tends to infinity, in proportion to the logarithm of the number of spins making up each layer. It does not seem unreasonable to consider that the temperature at which the specific heat has a maximum gradually approaches to the critical temperature predicted from the Padé approximant method as the number of spins in each layer increases.
We present the results of numerical calculations giving accurate estimates of the magnetization of the two-dimensional Ising model on a square lattice. Moreover, we argue that these results are strict lower bounds to the correct magnetization M(H,T)M(H,T)M(H,T). The estimates are obtained by dividing the infinite lattice into finite strips of width between two and nine spins and infinite length. The largest eigenvalue and the corresponding eigenvector of the transfer matrix are then obtained by an iterative process. The estimates of M(H,T)M(H,T)M(H,T) converge to the correct answer for the infinite lattice everywhere except for a small region in the $T\ensuremath{-}H$ plane. We also compute isotherms and critical isobar for the corresponding lattice gas. Finally, we propose a new approximation to the transfer matrix, exactly solvable in two dimensions for H=OH=OH=O, which reproduces exactly the critical-point behavior of the full Ising model.
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.
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.