Interface free energy and critical line for the ising model with nearest and next-nearest-neighbor interactions

A method due to Müller-Hartmann and Zittartz is used to calculate the free energy of a diagonal interface in an Ising model on a square lattice with interactions between nearest and next-nearest neighbors. Instead of solving the full bulk problem, one only takes into account a simple subset of interface configurations. The subset, which is somewhat different from that considered by Müller-Hartmann and Zittartz, is chosen in such a way that exact results for the interface free energy are reproduced if the system has only nearest or only next-nearest-neighbor interactions. When the critical line of ferromagnetic transitions is obtained by demanding that the interface free energy as a function of both couplings vanish, a surprisingly accurate approximation is obtained.