We demonstrate a new type of non-Hermitian phase transition in open systems far from thermal equilibrium, which can have place in the absence of an exceptional point. This transition takes place in coupled systems interacting with reservoirs at different temperatures. We show that the spectrum of energy flow through the system caused by the temperature gradient is determined by the $$\varphi^{4}$$ φ 4 -potential. Meanwhile, the frequency of the maximum in the spectrum plays the role of the order parameter. The phase transition manifests itself in the frequency splitting of the spectrum of energy flow at a critical point, the value of which is determined by the relaxation rates and the coupling strength. Near the critical point, fluctuations of the order parameter diverge according to a power law with the critical exponent that depends only on the ratio of reservoirs temperatures. The phase transition at the critical point has the non-equilibrium nature and leads to the change in the energy flow between the reservoirs. Our results pave the way to manipulate the heat energy transfer in the coupled out-of-equilibrium systems.

Strong coupling regime takes place in open hybrid systems consisting of two or more physical subsystems when the coupling strength between subsystems exceeds the relaxation rate. The relaxation arises due to the interaction of the system with environment. For this reason, it is usually believed that the enhancement of the interaction with environment inevitably leads to a transition of the system from the strong to weak coupling regime. In this paper, we refute this common opinion. We demonstrate the interaction of the coupled system with environment induces an additional coupling between the subsystems that contribute to retention the system in the strong coupling regime. We show that the environmental-induced coupling strength is proportional to the product of the Rabi coupling strength by the gradient of the density of states of the reservoir. There is a critical Rabi coupling strength above which the environmental-induced coupling ensures that the system remains in the strong coupling regime at any relaxation rate. In this case, the strong coupling regime takes place even when the relaxation rate is significantly above the Rabi coupling strength between the subsystems. The critical coupling depends on the gradient of the reservoir density of states. We demonstrate that managing this gradient can serve as an additional tool to control the properties of the coupled systems.