Fig. 3: Disorder moves to additional variables.
a The order parameter r is depicted as a function of ∣K∣ for both real parts, i.e. the phase-variables (solid disk) and the other variables (the imaginary parts, open circles) for N = 128 and β = 0.01. As the yν are unbounded, we define phase-like variables θμ by a stereographic projection via \(\cos {\theta }_{\mu }:=\frac{1-{y}_{\mu }^{2}}{1+{y}_{\mu }^{2}}\) and \(\sin {\theta }_{\mu }:=\frac{2{y}_{\mu }}{1+{y}_{\mu }^{2}}\) for each μ and evaluate \(r=\left\vert \frac{1}{N}\mathop{\sum }_{\mu=1}^{N}{e}^{{{\rm{i}}}{\theta }_{\mu }}\right\vert\), in analogy to (1). b Complex locked states in the complex plane for N = 80 and ∣K∣ = 3.0 move with increasing α values from curves ① for α = 0 and ② for α = 1.5 to curve ③ for \(\alpha=\frac{\pi }{2}\). c Local angles φ of the curves around the origin are depicted as a function of α with gray solid guiding line indicating ∣φ(α)∣ = α as emerges for N = 2 up to corrections \({{\mathcal{O}}}(| K{| }^{-1})\).
