The anisotropic 3D Ising model

Abstract

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 /k B T c)sinh(2(Jx + Jy )/k B T c)) = 1. This expression is in perfect agreement with a mathematically rigorous result (k B T c/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.