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. J_x and J_y ) are small compared to the third exchange energy (J_z ), the following asymptotically exact equation for the critical temperature is derived, sinh(2J_z /k _B T _c)sinh(2(J_x  + J_y )/k _B T _c)) = 1. This expression is in perfect agreement with a mathematically rigorous result (k _B T _c/J_z  = 2[ln(J_z /(J_x  + J_y ))−ln(ln(J_z /(J_x  + J_y )) + 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 ((H_x  + H_y )/H_z ) = 10⁻² to ∼0.76 at ((H_x  + H_y )/H_z ) = 10⁻²⁰.

Keywords:

• Phase transitions and critical phenomena
• Critical point effects
• Specific heats
• Short-range order