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Physical review. B, Condensed matter

Sparse-matrix analysis of spin-(1/2) Ising systems

Norman H. Fuchs, S. Gartenhaus

Published Jun 1, 1985• 9 citations
Publisher

Abstract

We describe an extension of an approximation scheme proposed earlier for calculating the critical properties of spin-(1/2) Ising systems in terms of the duo-diagonal, sparse matrices of Kramers and Wannier. The computational framework is refined and extended to accommodate a variety of two-, three-, and higher-dimensional lattices as well as interactions other than nearest neighbor. The relation between the underlying symmetries of the lattice and the structure of its associated phase diagram is clarified. Making use of a probabilistic interpretation for the components of the eigenvectors associated with the maximum eigenvalue of the duo-diagonal matrices, it is shown that this makes practical the computation of various thermodynamic quantities---as well as correlation functions---in terms of the expectation values of appropriate operators taken between these eigenvectors. When combined with the techniques of finite-size scaling the present scheme is confirmed to be effective for extracting the critical-point properties of the infinite lattice.

DOI
10.1103/physrevb.31.7261

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Physical Review•May 15, 1950•11 citations
We describe an extension of an approximation scheme proposed earlier for calculating the critical properties of spin-(1/2) Ising systems in terms of the duo-diagonal, sparse matrices of Kramers and Wannier. The computational framework is refined and extended to accommodate a variety of two-, three-, and higher-dimensional lattices as well as interactions other than nearest neighbor. The relation between the underlying symmetries of the lattice and the structure of its associated phase diagram is clarified. Making use of a probabilistic interpretation for the components of the eigenvectors associated with the maximum eigenvalue of the duo-diagonal matrices, it is shown that this makes practical the computation of various thermodynamic quantities---as well as correlation functions---in terms of the expectation values of appropriate operators taken between these eigenvectors. When combined with the techniques of finite-size scaling the present scheme is confirmed to be effective for extracting the critical-point properties of the infinite lattice.
An approximation for calculating the thermodynamic properties of spin-\textonehalf{} Ising models is described. The method is based on the formulation of Kramers and Wannier for the square lattice, and involves expressing the free energy in terms of the limiting maximum eigenvalue of a matrix sequence which represents the contribution from a single spin of the lattice. The application of this formulation to various two- and three-dimensional lattices shows that many of these are characterized by an identically structured matrix sequence with exactly two nonvanishing elements in each row and column. To explore the potential of using this similarity and its simplicity for analyzing Ising models we consider the two-dimensional square lattice in an external field and approximate the doubly infinite lattice by a semi-infinite one consisting of $n (\ensuremath{\le}10)$ infinitely long rows. Calculations are carried out and compared with (1) Onsager's formula for the free energy, (2) the Onsanger-Yang formula for the spontaneous magnetization, and (3) values calculated from series. Except for the region around the critical point where the correlation length exceeds nnn lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.
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.
We study the S=1 Ising model, equivalent to the three-state lattice-gas model, with nearest-neighbor, pairwise interactions on a two-dimensional, triangular lattice. We pay particular attention to the antiferromagnetic phase diagrams. We show its relation to other well-studied models (S=(1/2 Ising, Blume-Capel, Blume-Emery-Griffiths), classify the ground states, and calculate finite-temperature phase diagrams using transfer matrices and finite-size scaling for infinite strips of three and six sites width. The phase diagrams are quite complicated, with surfaces of first- and second-order transitions that intersect along lines of multicritical points of various kinds, providing a rich laboratory for studying a number of first-order phase transitions, critical and multicritical phenomena within the framework of one single model.
We obtain an approximation of the phase diagram of an Ising model with both nearest-neighbor and next-nearest-neighbor interactions on the square lattice. We use the Fisher zeros of the partition functions of a sequence of finite-sized systems along with various extrapolation methods to obtain phase transition points. In addition, we obtain an approximate value of the correlation length critical exponent nu. Our results are compared to previous results for this system using a wide variety of other approaches.
The partition function ZZZ of a two-dimensional Ising lattice is evaluated by expressing it in the form $Z=\ensuremath{\Sigma}{i}^{}\ensuremath{\lambda}_{i}^{}{}_{}{}^{\mathrm{nm}}$ where ${\ensuremath{\lambda}}_{i}$ are the eigenvalues of a 2n{2}^{n}2n-dimensional matrix M, nnn is the number of rows and mmm the number of columns of the lattice. M is a generalization of the V-shaped matrix studied by Kramers and Wannier in connection with the same problem. It is shown that M can be expressed in terms of 2n{2}^{n}2n-dimensional representations of 2n2n2n-dimensional orthogonal matrices. The eigenvalues of M are determined for nnn large by making use of the known relations between the eigenvalues of a 2n2n2n-dimensional orthogonal matrix and the eigenvalues of its 2n{2}^{n}2n-dimensional representative.The lattice considered differs only slightly from that treated by Onsager and Kaufman. In the lattice considered here, the boundary effects are eliminated by introducing an interaction between the last lattice point of each row and the first point of the next row. The matrix V considered by Onsager and Kaufman is, except for the boundary effects, equivalent to the matrix Mn{\mathbf{M}}^{n}Mn. ZZZ is shown to differ from the partition function calculated by Onsager and Kaufman only in order ${n}^{\ensuremath{-}2}$.