Phase diagrams and critical behavior of Ising square lattices with nearest-, next-nearest-, and third-nearest-neighbor couplings

Monte Carlo simulations are used to study the location and nature of phase boundaries for Ising square lattices with antiferromagnetic coupling $J_{\mathrm{NN}}$ between nearest neighbors and additional interactions $J_{\mathrm{NNN}}$ between next-nearest neighbors and $J_{3\mathrm{N}\mathrm{N}}$ between third-nearest neighbors. Results in zero magnetic field are obtained for a wide range of R=$J_{\mathrm{NNN}/J_{\mathrm{NN}}}$ and R’=$J_{3\mathrm{N}\mathrm{N}/J_{\mathrm{NN}}}$. In addition to the c(2×2) and (2×1) phases, which also occur for R’=0, we find new (4×4) and (4×2) ordered states, for R’≠0, which are separated from the disordered state by lines of first-order transitions. The nonuniversal critical behavior of the (2×1) phase is studied using the block-distribution method and finite-size scaling. The possible existence of incommensurate phases is also explored.