Explore emergent phenomena in strongly correlated quantum systems through this 47-minute lecture delivered as part of the II School on Emergent Phenomena in Non-Equilibrium Quantum Many-Body Systems. Learn from Thereza Paiva of UFRJ, Brazil, as she presents the fourth class in her series on strongly correlated systems, examining how collective behaviors and novel phases of matter emerge from the complex interactions between quantum particles. Delve into the theoretical frameworks and computational methods used to understand these phenomena, including how strong correlations between electrons lead to unexpected properties in materials such as high-temperature superconductors, quantum magnets, and Mott insulators. Gain insights into the latest research developments in this rapidly evolving field of condensed matter physics and quantum many-body theory, with particular emphasis on how these emergent properties arise from the underlying microscopic interactions in quantum systems.