The anisotropic 3D Ising model

An expression for the free energy of an (001) oriented domain wall of the anisotropic 3D Ising model is derived. The order--disorder transition takes place when the domain wall free energy vanishes. In the anisotropic limit, where two of the three exchange energies (e.g. Jx and Jy) are small compared to the third exchange energy (Jz), the following asymptotically exact equation for the critical temperature is derived, sinh(2Jz/kBTc)sinh(2(Jx + Jy)/kBTc)) = 1. This expression is in perfect agreement with a mathematically rigorous result (kBTc/Jz = 2[ln(Jz/(Jx + Jy))-ln(ln(Jz/(Jx + Jy)) + O(1)]-1) derived earlier by Weng, Griffiths and Fisher (Phys. Rev. 162, 475 (1967)) using an approach that relies on a refinement of the Peierls argument. The constant that was left undetermined in the Weng et al. result is estimated to vary from ∼0.84 at ((Hx + Hy)/Hz) = 10-2 to ∼0.76 at ((Hx + Hy)/Hz) = 10-20