Out-of-equilibrium dynamics in round-trip and dissipation protocols

The study of out-of-equilibrium dynamics in quantum many-body systems has gained renewed interest due to advances in controlling nanoscale systems. In particular, under- standing how correlations evolve during transitions through critical points is crucial for both theoretical models and experimental applications. This thesis investigates the scaling behaviors that emerge when slowly traversing classical and quantum critical points, with a special focus on the Kibble-Zurek mechanism. By employing a unified renormalization group framework, we explore the dynamics of slow round-trip variations of system parame- ters, contrasting these with standard one-way KZ protocols. Additionally, the thesis examines the behavior of open quantum systems, where dissi- pative dynamics are modeled using Lindblad master equations. The effects of both local and uniform dissipation mechanisms are analyzed, especially in the context of phase tran- sitions and interactions with thermal environments. We also investigate quantum quench protocols, providing insight into their effects on system evolution in the presence of thermal baths. This work offers a comprehensive view of the dynamic scaling behaviors and dissi- pation in many-body quantum systems, revealing new insights into both equilibrium and out-of-equilibrium processes.