Resonant frequencies and spatial correlations in frustrated arrays of Josephson type nonlinear oscillators

We present a theoretical study of resonant frequencies and spatial correlations of Josephson phases in frustrated arrays of Josephson junctions. Two types of one-dimensional arrays, namely, the diamond and sawtooth chains, are discussed in detail. For these arrays in the linear regime the Josephson phase dynamics is characterized by multiband dispersion relation omega(k), and the lowest band becomes completely flat at a critical value of frustration, f = fc. In a strongly nonlinear regime such critical value of frustration determines the crossover from non-frustrated (0 < f < f(c) to frustrated (f(c) < f < 1) regimes. The crossover is characterized by the thermodynamic spatial correlation functions of phases on vertices, (phi(i) , i.e. C-p (i - j) = < cos[p(phi(i) - phi(j))]> displaying the transition from long- to short-range spatial correlations. We find that higher-order correlations functions, e.g. p = 2 and p = 3, restore the long-range behavior deeply in the frustrated regime, f similar or equal to 1. Monte-Carlo simulations of the thermodynamics of frustrated arrays of Josephson junctions are in good agreement with analytical results. We also outline the extension of our results to the case of kagome lattice, prototypical 2D frustrated lattice, and other higher dimensional lattices.