Condensation and criticality of eigen microstates of phase fluctuations in Kuramoto model

The Kuramoto model is one of the most profound and classical models of coupled phase oscillators. Because of the global couplings between oscillators, its precise critical exponents can be obtained using the mean-field approximation (MFA), where the time average of the modulus of the mean-field is defined as the order parameter. Here, we further study the phase fluctuations of oscillators from the mean-field using the eigen microstate theory (EMT), which was recently developed. The synchronization of phase fluctuations is identified by the condensation and criticality of eigen microstates with finite eigenvalues, which follow the finite-size scaling with the same critical exponents as those of the MFA in the critical regime. Then, we obtain the complete critical behaviors of phase oscillators in the Kuramoto model. We anticipate that the critical behaviors of general phase oscillators can be investigated by using the EMT and different critical exponents from those of the MFA will be obtained.