From topological to dissipative many-body phases out of equilibrium

Out-of-equilibrium many-body physics is the fascinating study of complex systems that are subjected to driving, dissipation, or both. Through such procedures it is possible to generate exciting new dynamical phases of matter that have no static counterpart. In this thesis, we investigate the interplay between driving and dissipation in several paradigmatic many-body quantum systems. In particular, we analyze the impact of nonequilibrium phenomena on the stability of topological matter and light-matter interactions. We begin by addressing how to characterize the topological criticality of lower-dimensional systems whose parameters are periodically driven in time. We first introduce a curvature renormalization group method that extends the notions of Landau criticality to the realm of topological phase transitions. We show how this framework presents a simple and efficient way to map out topological phase diagrams of static and Floquet systems in any dimension. Furthermore, we illustrate its ability to precisely capture topological critical phenomena. We then apply it to prototypical static and Floquet topological systems to extract correlation lengths, critical exponents, scaling laws, and universality classes. In one dimension, we apply this methodology to the periodically driven Kitaev chain hosting Majorana end modes. As an interlude, we also examine the consequences of multifrequency driving, and reveal that it can be used as a valuable tool to dynamically control the number and the stability of the topological modes. In two dimensions, we analyze a Floquet-Chern insulator and the periodically driven Kitaev model on the honeycomb lattice. We also establish that periodic driving can induce very exotic topological semimetal phases hosting nodal loop gap closures. We demonstrate that the corresponding phase transitions belong to a different universality class than the ones described by Dirac low-energy theories, but both can coexist at multicritical points. We then discuss the physical origin of nodal loop gap closures and establish that they emerge as a consequence of enhanced symmetries introduced by the driving scheme. We separately analyze symmetry protection and provide concrete examples of how to induce it in both static and driven settings. In realistic settings, one has to often deal with open systems. To incorporate dissipation in our analysis, we re-examine the periodically driven Kitaev chain additionally subjected to a coupling with end reservoirs. In this context we discover that the Floquet-Majorana edge modes survive despite dissipation, and have strong signatures in stroboscopic observables that can reliably track their appearance and disappearance in different phases. Another challenge of contemporary physics is to understand the role of many-body interactions in driven-dissipative settings. As a first step in this direction, we present a very efficient numerical algorithm — the MultiConfigurational Time-Dependent Hartree method — tailored at reliably simulating the time-evolution of interacting many-body quantum systems. We illustrate its applications to several systems of ultracold atoms coupled with the radiation field of optical cavities that host intriguing light-matter phases stabilized by dissipation, such as superradiant phases, Mott insulating phases, and Tonks-Girardeau phases. We highlight how to calculate density distributions, correlation functions, and various order parameters for such setups, and how to simulate realistic single-shot experiments. We conclude this thesis by presenting a brief outlook on unresolved issues and new intriguing questions that have surfaced throughout our study