The critical properties of two-dimensional oscillator arrays

We present a renormalization group study of two-dimensional oscillator arrays, with dissipative, short-range interactions. We consider the case of non-identical oscillators, with distributed intrinsic frequencies within the array and study the steady-state properties of the system. In two dimensions no macroscopic mutual entrainment is found but, for identical oscillators, critical behaviour of the Berezinskii–Kosterlitz–Thouless (BKT) type is shown to be present. We then discuss the stability of BKT order in the physical case of distributed quenched random frequencies. In order to do that, we show how the steady-state dynamical properties of the two-dimensional array of non-identical oscillators are related to the equilibrium properties of the XY model with quenched randomness, which has already been studied in the past. We propose a novel set of recursion relations to study this system within the Migdal–Kadanoff (MK) renormalization group scheme, by means of a discrete clock state formulation. For identical oscillators, we compute the phase diagram in the presence of random dissipative coupling, at finite values of the clock state parameter. Possible experimental applications in two-dimensional arrays of microelectromechanical oscillators are briefly suggested. This work is devoted to Professor David Sherrington, in occasion of his 65th birthday .