Statistical sum of a three-dimensional Ising model

Kurt Binder

Syû Ono, Yoshitomo Karaki, Masuo Suzuki +1 more

Bin Liu, Moshe Gitterman

The critical temperatures of the Ising model can be obtained by considering the elementary cells of the corresponding lattice, namely the square lattice in two dimensions and the cube in three dimensions. The configurations in the statistical sum of a cell are divided into nondegenerate and degenerate cases. At the critical temperature of the infinite lattice the contributions of these two groups of configurations are assumed to be equal. This conjecture reproduces the exact Onsager result for two dimensions and the numerical result for the three-dimensional Ising lattices. Although this conjecture is not exact, it gives insight into the nature of the transitions.

Franz Wegner

George Allen Baker Jr.

George Allen Baker Jr., Bernie G. Nickel, Melville S. Green +1 more

The coefficients in the Callan-Symanzik equations for a three-dimensional, continuous spin Ising model with an $\mathrm{exp}(\ensuremath{-}A{s}^{4}+B{s}^{2})$ spin-weight factor are expanded in the dimensionless, renormalized coupling constant. These series are summed by the Pad\'e-Borel method to yield the critical indices $\ensuremath{\gamma}=1.241\ifmmode\pm\else\textpm\fi{}0.002$, $\ensuremath{\eta}=0.02\ifmmode\pm\else\textpm\fi{}0.02$, $\ensuremath{\nu}=0.63\ifmmode\pm\else\textpm\fi{}0.01$, and ${\ensuremath{\Delta}}_{1}=0.49\ifmmode\pm\else\textpm\fi{}0.01$.