Two-dimensional Ising model with competing interactions and its application to clusters and arrays of π -rings and adiabatic quantum computing

We study planar clusters consisting of loops including a Josephson π junction (π -rings). Each π -ring carries a persistent current and behaves as a classical orbital moment. The type of particular state associated with the orientation of orbital moments at the cluster depends on the interaction between these orbital moments and can be easily controlled, i.e., by a bias current or by other means. We show that these systems can be described by the two-dimensional Ising model with competing nearest-neighbor and diagonal interactions and investigate the phase diagram of this model. The characteristic features of the model are analyzed based on the exact solutions for small clusters such as a five-site square plaquette as well as on a mean-field-type approach for the infinite square lattice of Ising spins. The results are compared with spin patterns obtained by Monte Carlo simulations for the 100×100 square lattice and with experiment. We show that the π -ring clusters may be used as a special type of superconducting memory elements. The obtained results may be verified in experiments and are applicable to adiabatic quantum computing where the states are switched adiabatically with the slow change of coupling constants.