Fig. 5: The influence of long-range interactions on the exceptional points of the trimer.

a–c The real parts of the eigenfrequencies \({\tilde{\omega }}_{{{{{{\mathrm{n}}}}}}}^{\prime}\), as a function of the coupling strength g [Eq. (26)]. d–f The imaginary parts, corresponding to the real parts in a–c. Dashed lines: the exceptional points denote the border between broken and unbroken \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric phases. The results for the first, second and third eigenfrequencies \({\tilde{\omega }}_{{{{{{\mathrm{n}}}}}}}^{\prime}\) are denoted by the thin green, medium orange and thick cyan lines respectively [see the legend in a, which applies to the whole figure]. In the first, second, and third columns, the second-nearest neighbor coupling constant h = κ/4, κ/2, and 3κ/4, respectively.