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arXiv:cond-mat/0303323 · International Journal of Modern Physics C

NEW ALGORITHM FOR THE COMPUTATION OF THE PARTITION FUNCTION FOR THE ISING MODEL ON A SQUARE LATTICE

Krzysztof Malarz, M. S. Magdoń-Maksymowicz, A.Z. Maksymowicz, B. Kawecka-Magiera, KRZYSZTOF KUŁAKOWSKI

Published Jun 1, 2003• 3 citations
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Abstract

A new and efficient algorithm is presented for the calculation of the partition function in the S=±1 Ising model. As an example, we use the algorithm to obtain the thermal-dependence of the magnetic spin susceptibility of an Ising antiferromagnet for a 8×8 square lattice with open boundary conditions. The results agree qualitatively with the prediction of the Monte Carlo simulations and the experimental data, and they are better than the mean field approach results. For the 8×8 lattice, the algorithm reduces the computation time by nine orders of magnitude.

DOI
10.1142/s012918310300484x

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A new and efficient algorithm is presented for the calculation of the partition function in the S=±1 Ising model. As an example, we use the algorithm to obtain the thermal-dependence of the magnetic spin susceptibility of an Ising antiferromagnet for a 8×8 square lattice with open boundary conditions. The results agree qualitatively with the prediction of the Monte Carlo simulations and the experimental data, and they are better than the mean field approach results. For the 8×8 lattice, the algorithm reduces the computation time by nine orders of magnitude.
Parallel algorithm of partition function calculation of two-dimensional Ising model for systems with a finite number of spins was developed. Within a method of complete enumeration by using MPI technology with subsequent optimization of a parallel code time of calculations was reduced considerably. Partition function was calculated for systems of 16, 25, 36 Ising spins. Based on the obtained results, main thermodynamic and magnetic values dependences (such as heat capacity, magnetic susceptibility, mean square magnetization) for ferromagnetic and antiferromagnetic interactions was investigated. The analysis of a different configurations contribution showed, that states with the minimum energy have essential influence on dependences of thermodynamic values. Comparison with the results obtained by the Wang Landau algorithm was performed.
The exact partition functions of the Ising model on L × L square lattices with free boundary conditions are evaluated up to L = 22 for the first time by using a microcanonical transfer matrix and counting all possible spin states. The total number of states is 2 22×22 = 2 484 ≈ 5.0 × 10 145 for L = 22. Using the partition function zeros in the complex temperature plane of the Ising model on L × L square lattices with free boundary conditions, we analyze the critical behavior of the square-lattice Ising antiferromagnet and compare them with the results for periodic boundary conditions (the most popular boundary conditions). Our results show clearly that free boundary conditions are much more efficient than periodic boundary conditions in the study of an antiferromagnet.
We study the square-lattice Ising model with nearest-neighbor ( J 1 ) and next nearest-neighbor ( J 2 ) interactions, using the Wang–Landau Monte Carlo algorithm and the partition function zeros in the complex temperature plane (Fisher zeros). Here, the competition between J 1 and J 2 interactions is defined by the coupling ratio R = J 2 / J 1 . For J 2 > 0, the critical behavior is trivial and is the same as the square-lattice Ising model with only J 1 interaction. On the other hand, the new phase of superantiferromagnet appears for J 2 < 0. However, the nature of the transition into the superantiferromagnetic phase is still controversial in the literature. To resolve the controversy, we estimate the critical temperature T c ( R ) and thermal scaling exponent y t ( R ) from the Fisher zeros, as a function of R . The estimated values of T c ( R ) agree well with the results in the literature. We obtain the simple critical behavior with a fixed value of the thermal scaling exponent, y t ( R ) = 1, for and J 2 < 0 (Ising antiferromagnet). For and J 2 < 0 (Ising superantiferromagnet), however, it is shown that y t ( R ) continuously varies depending on R values, and the estimated values of y t ( R ) are clearly less than that of the first-order phase transition, y t = 2.
We have calculated the low-temperature series expansions of the spontaneous magnetization and the zero-field susceptibility of the square-lattice ferromagnetic Ising model with first-neighbour interaction J1 and second-neighbour interaction J2 to the 30th and 26th order respectively by computer. Our results extend the previous calculations by Lee and Lin to six more orders. We use the Padé approximants to estimate the critical exponents and the critical temperature for different ratios of R = J2/J1. The estimated critical temperature as a function of R agrees with the estimation by Oitmaa from high-temperature series expansions.
The partition functions of the two- and three-dimensional finite Ising models in the presence of a magnetic field have been calculated with use of a high-speed computer. Periodic boundary conditions were imposed. All the numbers of configurations for the 3×3×3, 5×5, and 4×6 lattices are tabulated. The distribution of zeros of the partition functions in the complex magnetic-field plane has been obtained both for the ferro- and antiferromagnetic cases. The zeros in the ferromagnetic cases are distributed on the unit circle according to the Yang-Lee theorem, and the dependence on temperature of the distribution functions has been studied. In the antiferromagnetic case most of zeros are on the negative real axis, but a certain number of complex roots appear in the left half of the plane. The magnetization and the magnetic susceptibility have been calculated as functions of temperature.