Approximate calculations for the two-dimensional Ising model

Yoshiaki Tanaka, Norikiyo Uryû

In the Ising model on a helical lattice with period M (in the limit M →∞ of this lattice, the model corresponds to the two-dimensional Ising model), the exact simultaneous equations to calculate physical quantities are derived. The zero field magnetic susceptibility χ 0 , and also the specific heat C are obtained numerically for the system of M ≤11. Furthermore, in the Ising model with up to the next nearest neighbor interactions on the helical lattice, χ 0 and C are calculated for the system of M ≤9. The results obtained are an excellent approximation to the exact χ 0 and C of the two-dimensional systems.

Franz Wegner

Michel Droz, Sheldon L. Katz, J. D. Gunton

The recursion equations corresponding to the one-hypercube approximation of Kadanoff are studied in more detail for the Wilson-Fisher $n$-vector model in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and the $d$-dimensional Ising model. The free energy for the one-dimensional Ising model is solved exactly in this approximation and is shown to give the correct singular behavior near zero temperature and the exact free energy in zero field. The corrections to scaling exponents are shown to be exact to first order in $\ensuremath{\epsilon}$ for the Wilson-Fisher model and are given for the $d=2,3, \mathrm{and} 4$ dimensional Ising models. The amplitude of the specific-heat divergence is calculated for the two-dimensional Ising model and is shown to be in reasonable agreement with the exact result.

Henk W. J. Blöte, G. Kamieniarz

Jian Wang

We calculated the autorelaxation time for the two-dimensional kinetic Ising model with Glauber dynamics using a high-temperature series-expansion technique. The series is analyzed using a differential Pad\'e approximant. The exponent z for critical slowing down is found to be z=2.207\ifmmode\pm\else\textpm\fi{}0.008.

R. Shankar