Conformal Invariance of the Ising Model in Three Dimensions

We investigate a critical Ising-like model in the curved geometry $S^2\times R^1$ obtained by a conformal mapping of the infinite 3D space $R^3$. The incompatibility of regular lattices with this geometry is avoided by use of the anisotropic limit of the lattice Ising model, which renders one of the space coordinates continuous. We determine magnetic and energylike correlation lengths of this model by means of a cluster Monte Carlo algorithm. From these data, and the assumption of conformal invariance, we obtain the magnetic and temperature scaling dimensions as $X_h=0.5178\mathrm{}\left(12\right)$ and $X_t=1.423\mathrm{}\left(19\right)$, respectively. These numbers are in a good agreement with the existing results for the 3D Ising universality class.