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We study planar clusters consisting of loops including a Josephson$\pi$-junction ($\pi$-rings). Each $\pi$-ring carries a persistent current andbehaves as a classical orbital moment. The type of particular state associatedwith the orientation of orbital moments at the cluster depends on theinteraction between these orbital moments and can be easily controlled, i.e. bya bias current or by other means. We show that these systems can be describedby the two-dimensional Ising model with competing nearest-neighbor and diagonalinteractions and investigate the phase diagram of this model. Thecharacteristic features of the model are analyzed based on the exact solutionsfor small clusters such as a 5-site square plaquette as well as on a mean-fieldtype approach for the infinite square lattice of Ising spins. The results arecompared with spin patterns obtained by Monte Carlo simulations for the 100$\times$ 100 square lattice and with experiment. We show that the $\pi$-ringclusters may be used as a new type of superconducting memory elements. Theobtained results may be verified in experiments and are applicable to adiabaticquantum computing where the states are switched adiabatically with the slowchange of coupling constants.

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