Finite-size scaling and dynamics in a two-dimensional lattice of identical oscillators with frustrated couplings

Abstract

A two-dimensional lattice of oscillators with identical (zero) intrinsic frequencies and Kuramoto type of interactions with randomly frustrated couplings is considered. Starting the time evolution from slightly perturbed synchronized states, we study numerically the relaxation properties, as well as properties at the stable fixed point which can also be viewed as a metastable state of the closely related XY spin glass model. According to our results, the order parameter at the stable fixed point shows generally a slow, reciprocal logarithmic convergence to its limiting value with the system size. The infinite-size limit is found to be close to zero for zero-centered Gaussian couplings, whereas, for a binary $\pm 1$ distribution with a sufficiently high concentration of positive couplings, it is significantly above zero. Besides, the relaxation time is found to grow algebraically with the system size. Thus, the order parameter in an infinite system approaches its limiting value inversely proportionally to $\ln t$ at late times $t$, similarly to that found in the model with all-to-all couplings [Daido, Chaos {\bf 28}, 045102 (2018)]. As opposed to the order parameter, the energy of the corresponding XY model is found to converge algebraically to its infinite-size limit.