The 2D Ising square lattice with nearest- and next-nearest-neighbor interactions

An analytical expression for the boundary free energy of the Ising square lattice with nearest- and next-nearest-neighbor interactions is derived. The Ising square lattice with anisotropic nearest- neighbor (J(x) and J(y)) and isotropic next-nearest-neighbor (J(d)) interactions has an order-disorder phase transition at a temperature T = T-c given by the condition e(-2Jx/kbTc) + e(-2Jy/ kbTc) + e(-2(Jx+ Jy)/ kbTc) (2 - e(-4Jd/ kbTc)) = e (4Jd/ kbTc). The critical line that separates the ordered (ferromagnetic) phase from the disordered ( paramagnetic) phase is in excellent agreement with series expansion, finite scaling of transfer matrix and Monte Carlo results. For a vanishing next-nearest-neighbor interaction Onsager's famous result, i.e. sinh(2J(x)/k(b)T(c)) sinh(2J(y)/k(b)T(c)) = 1, is recaptured.