A brief description of the role of symmetry in phase transitions is given. Quantum phase transitions (QPT) in algebraic models are discussed, both at the semi-classical and quantal level. The newly introduced concept of excited states quantum phase transitions (ESQPT) is presented.
Abstract Pronounced structural changes within individual configurations (Type I QPT), superimposed on an abrupt crossing of these configurations (Type II QPT), define the notion of intertwined quantum phase transitions (QPTs). We discuss and present evidence for such a scenario in finite Bose and Bose-Fermi systems. The analysis is based on algebraic models with explicit configuration mixing, where the two types of QPTs describe shape-phase transitions in-between different dynamical symmetries and shape-coexistence with crossing.