Title: Spectral Theory of Non-Markovian Dissipative Phase Transitions

Abstract: To date, dissipative phase transitions (DPTs) have mostly been studied for quantum systems coupled to idealized Markovian (memoryless) environments, where the closing of the Liouvillian gap constitutes a hallmark. Here, we extend the spectral theory of DPTs to arbitrary non-Markovian systems and present a general and systematic method to extract their signatures, which is fundamental for the understanding of realistic materials and experiments such as in the solid-state, cold atoms, cavity or circuit QED. We first illustrate our theory to show how memory effects can be used as a resource to control phase boundaries in a model exhibiting a first-order DPT, and then demonstrate the power of the method by capturing all features of a challenging second-order DPT in a two-mode Dicke model for which previous attempts had fail up to now.