Clarify whether the CPA-predicted instability corresponds to a phase transition or a crossover in the 2D XY spin glass with binary disorder

Determine whether the instability of the ferromagnetic (synchronized) state predicted by the coherent potential approximation for the two-dimensional XY spin glass with nearest-neighbor binary ±1 couplings at any positive concentration c of negative bonds corresponds to a genuine ferromagnetic-to-paramagnetic phase transition or merely a crossover.

Background

The paper studies a Kuramoto model without intrinsic frequencies on a 2D lattice, which is equivalent to gradient descent on the energy of the corresponding XY spin glass. In discussing prior results on the XY spin glass with binary (±1) couplings, the authors note that coherent potential approximation (CPA) predicts instability of the ferromagnetic state for any c>0 concentration of negative bonds.

However, the precise nature of this loss of stability—whether it signifies a true phase transition to a paramagnetic state or represents a gradual crossover—has not been established. Clarifying this point is important for understanding the phase structure and metastability landscape relevant to both XY spin glasses and the associated frustrated Kuramoto dynamics.