Fig. 3

Critical correlation strength \(\xi _{\textrm{crit}}\) as a function of detuning \(\Delta \omega\) and exchange coupling J. Phase diagram showing the threshold \(\xi _{\textrm{crit}} = \Delta \omega / (2\Gamma _\theta )\) for the onset of PT symmetry breaking as a function of oscillator detuning and interaction strength. The coherence decay rate \(\Gamma _\theta\) depends on the statistical phase and temperature, thereby modulating the location of the exceptional point. For bosonic systems, this expression recovers the analytical result derived in Ref.¹⁷; for fractional statistics, the phase-dependent \(\Gamma _\theta\) reshapes the threshold contour.