The Strange World of Fractionalized Quantum Numbers in Quantum Matter

Within the Landau paradigm we define a phase or a state of matter by a local order parameter, as in crystals, magnets, and superconductors. However, due to frustration generated by geometry and interactions, some materials do not order, even at the lowest temperatures, and enter a new state of matter called a quantum spin liquid. These states are identified by long range quantum entanglement and a new type of topological order with fractionalized excitations. I will give an overview of the theoretical developments in this field and discuss recent experiments that point to the existence of such fractionalized excitations.