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Computational Methods in Science and Technology

The algorithm for the partition function of the generalised Ising model on a square lattice

Piotr Kozłowski, G. Kamieniarz, G. Musiał

Published Jan 1, 1997• 0 citations
Publisher

Abstract

The paper presents the algorithm and description of a procedure gim written in the FORTRAN 90 language, for calculation of the partition function of the generalised Ising model on a square lattice.The procedure is available at the Poznań Supercomputing and Networking Center on a Cray J916 supercomputer in numerical procedure library mestat.

DOI
10.12921/cmst.1997.03.01.87-90

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The paper presents the algorithm and description of a procedure gim written in the FORTRAN 90 language, for calculation of the partition function of the generalised Ising model on a square lattice.The procedure is available at the Poznań Supercomputing and Networking Center on a Cray J916 supercomputer in numerical procedure library mestat.
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.
The numerical transfer matrix for the partition function of discrete lattice models is generalized to allow the calculation of the density of states \ensuremath{\Omega}(E), and the restricted density of states \ensuremath{\Omega}(E,M). Given \ensuremath{\Omega}(E,M) the partition function is expressed as a polynomial in the variables x=${\mathit{e}}^{\mathrm{\ensuremath{\beta}}\mathit{h}}$ and y=${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}}$. These algorithms are illustrated with calculations for the Ising model on finite square lattices. The zeros of the partition function are examined in both the complex x and y planes. Finite size scaling analysis of the zeros leads to very accurate estimates for the critical temperature and critical exponents.
Abstract The partition function of the square lattice Ising model on the rectangle with open boundary conditions in both directions is calculated exactly for arbitrary system size L × M and temperature. We start with the dimer method of Kasteleyn, McCoy and Wu, construct a highly symmetric block transfer matrix and derive a factorization of the involved determinant, effectively decomposing the free energy of the system into two parts, F ( L , M ) = F s t r i p ( L , M ) + F s t r i p r e s ( L , M ) , where the residual part F s t r i p r e s ( L , M ) contains the nontrivial finite- L contributions for fixed M . It is given by the determinant of a M / 2 × M / 2 matrix and can be mapped onto an effective spin model with M Ising spins and long-range interactions. While F s t r i p r e s ( L , M )