Exceptional points and phase transitions in non-Hermitian nonlinear binary systems

A recent study [R. Hanai et al., Phys. Rev. Lett. 122, 185301 (2019)] highlighted a first-order-like dissipative phase transition in a two-component quantum system with an exceptional point coinciding with the phase boundary endpoint. Here, we show a disparity between the exceptional point and the endpoint which is closely connected to the stability of solutions. We present a general phase diagram describing different phases in a generic nonlinear binary system. The phase transition may occur also in the regime of weak coupling between the modes, which was excluded previously. In a certain range of parameters, the system converges to a limit cycle, which vanishes at the exceptional point. Our results emphasize the connection between phase transitions, bistability, and exceptional points of non-Hermitian nonlinear systems in general, providing insight into strongly coupled light-matter systems in particular.