Dissipative and Measurement-Induced Phases in Many-Body Quantum Systems

Open quantum many-body systems are an important setting that allows us to study the interplay between dissipation and interactions, contributing to a more accurate description of real-life phenomena and experiments, and providing new features to be exploited both on theoretical and technological grounds. This thesis deals with the study of such systems across a wide range of settings. Beyond providing the theoretical tools to analyze and understand this framework, the thesis also addresses some important and modern concepts. A key goal is to show how dissipation enables the realization of new phases of matter, such as quantum synchronization and time-crystal phases. To achieve this, various theoretical tools will be employed, ranging from path integral formalism to a Lindblad master equation approach. These same tools will also allow us to pursue another important goal: studying the dynamics of many-body systems subject to continuous measurements. This will lead to observing the emergence of a recently discovered phenomenon, known as measurement-induced phase transitions, and will provide a natural framework for addressing the complexity arising from the stochastic nature of quantum measurements. This thesis constitutes a modern and comprehensive view of the field of open quantum many-body systems, with a special focus on novel phases of matter emerging from interactions with an external environment or with a measuring device.