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7 A NEW LOOK AT THE 2D ISING MODEL FROM EXACT PARTITION FUNCTION ZEROS FOR LARGE LATTICE SIZES

Nelson A. Alves, J. R. Drugowich de Felício, Ulrich H. E. Hansmann

Published Aug 18, 2016• 7 citations
OpenAlex

Abstract

A general numerical method is presented to locate the partition function zeros in the complex β plane for large lattice sizes. We apply this method to the 2D Ising model and results are reported for square lattice sizes up to L = 64. We also propose an alternative method to evaluate corrections to scaling which relies only on the leading zeros. This method is illustrated with our data.

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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.

arXiv:cond-mat/0303323•International Journal of Modern Physics C•Jun 1, 2003•3 citations
A general numerical method is presented to locate the partition function zeros in the complex β plane for large lattice sizes. We apply this method to the 2D Ising model and results are reported for square lattice sizes up to L = 64. We also propose an alternative method to evaluate corrections to scaling which relies only on the leading zeros. This method is illustrated with our data.
The density of states for the three-dimensional Ising model is calculated with high precision by means of multicanonical simulations. This allows us to estimate the leading partition function zeros for lattice sizes up to L = 32. We have evaluated the critical exponent ν and the correction to scaling through an analysis of a multi-parameter fit and of the Bulirsch-Stoer (BST) extrapolation algorithm. The performance of the BST algorithm is also explored in case of the 2D Ising model, where the exact partition function zeros are known.
There is no an exact solution to three-dimensional (3D) finite-size Ising model (referred to as the Ising model hereafter for simplicity) and even two-dimensional (2D) Ising model with non-zero external field to our knowledge. Here by using an elementary but rigorous method, we obtain an exact solution to the partition function of the Ising model with NNN lattice sites. It is a sum of 2N2^N2N exponential functions and holds for DDD-dimensional (D=1,2,3,...D=1,2,3,...D=1,2,3,...) Ising model with or without the external field. This solution provides a new insight into the problem of the Ising model and the related difficulties, and new understanding of the classic exact solutions for one-dimensional (1D) (Kramers and Wannier, 1941) or 2D Ising model (Onsager, 1944). With this solution, the specific heat and magnetisation of a simple 3D Ising model are calculated, which are consistent with the results from experiments and/or numerical simulations. Furthermore, the solution here and the related approaches, can also be available to other models like the percolation and/or the Potts model.
Our aim is to investigate the critical behaviour of lattice spin models such as the three-dimensional Ising model in the thermodynamic limit. The exact partition functions (typically summed over the order of 1075 states) for finite simple cubic Ising lattices are computed using a transfer matrix approach. Q-state Potts model partition functions on two- and three-dimensional lattices are also computed and analysed. Our results are analysed as distributions of zeros of the partition function in the complex-temperature plane. We then look at sequences of such distributions for sequences of lattices approaching the thermodynamic limit. For a controlled comparison, we show how a sequence of zero distributions for finite 2d Ising lattices tends to Onsager’s thermodynamic solution. Via such comparisons, we find evidence to suggest, for example, a thermodynamic limit singular point in the behaviour of the specific heat of the 3d Ising model.
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.