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₍₍ between nearest neighbors and additional interactions J₍₍₍ between next-nearest neighbors and J₃₍₍ between third-nearest neighbors. Results in zero magnetic field are obtained for a wide range of R=J₍₍₍/₉₍₍ and R'=J₃₍₍/₉₍₍. In addition to the c (22) and (21) phases, which also occur for R'=0, we find new (44) and (42) ordered states, for R'0, which are separated from the disordered state by lines of first-order transitions. The nonuniversal critical behavior of the (21) phase is studied using the block-distribution method and finite-size scaling. The possible existence of incommensurate phases is also explored.