Phase transitions in the Ising square lattice with next-nearest-neighbor interactions

An importance sampling Monte Carlo method has been used to study $L\times L$ Ising square lattices with periodic boundary conditions. The behavior of the thermal and magnetic properties was determined over a wide range of $R$, the ratio of next-nearest- to nearest-neighbor coupling. From these data we have extracted the $R$ dependence of the critical temperature and a range of other critical parameters. Within our experimental error the critical exponents are independent of $R$ but the critical amplitudes are not. The data do however suggest a change in $\alpha$ for $R$ slightly greater than ½. Variations in critical entropy and internal energy may be as large as those resulting from changes in lattice or spin dimensionalities. Our results are compared and contrasted with those obtained by a range of theoretical methods.