Interface tension of the square lattice Ising model with next-nearest-neighbour interactions

In a recent letter, Zandvliet (Europhys. Lett., 73 (2006) 747) presented a simple derivation of an analytical expression for the interface free energy in the (10) direction of the Ising model on a square lattice with nearest- and next-nearest-neighbour couplings, reproducing the famous exact Onsager formula in the special case of only nearest-neighbour interactions. By comparing the resulting transition temperatures, determined as the point where the interface tension vanishes, with previous numerical results in the literature, support for the validity of the new analytical formula in the general case was claimed. Guided by the fact that Zandvliet's simple, but rather heuristic derivation neglects overhang configurations and bubble excitations completely, we show that his approach is equivalent to the classic solid-on-solid (SOS) approximation which is known to reproduce accidentally the exact interface tension along one of the two main axes in the case of only nearest-neighbour interactions. In the limiting situation where only next-nearest-neighbour interactions are considered, we prove analytically that such a coincidence no longer holds. To assess the accuracy of Zandvliet's formula for the general model we have performed a careful computer simulation study using multicanonical and cluster Monte Carlo techniques combined with finite-size scaling analyses. Our results for the hitherto unknown interface tension and the transition temperatures show that the analytical formula yields fairly good approximations but, in general, is not exact.