Ordering in two-dimensional Ising models with competing interactions

We study the 2D Ising model on a square lattice with additional non-equal diagonal next-nearest neighbor interactions. The cases of classical and quantum (transverse) models are considered. Possible phases and their locations in the space of three Ising couplings are analyzed. In particular, incommensurate phases occurring only at non-equal diagonal couplings, are predicted. We also analyze a spin-pseudospin model comprised of the quantum Ising model coupled to XY spin chains in a particular region of interactions, corresponding to the Ising sector’s super-antiferromagnetic (SAF) ground state. The spin-SAF transition in the coupled Ising-XY model into a phase with co-existent SAF Ising (pseudospin) long-range order and a spin gap is considered. Along with destruction of the quantum critical point of the Ising sector, the phase diagram of the Ising-XY model can also demonstrate a re-entrance of the spin-SAF phase. A detailed study of the latter is presented. The mechanism of the re-entrance, due to interplay of interactions in the coupled model, and the conditions of its appearance are established. Applications of the spin-SAF theory for the transition in the quarter-filled ladder compound NaV2O5 are discussed.