Classifying fractionalization:  symmetry classification of gapped Z2 spin liquids in two dimensions

 

Quantum number fractionalization is a remarkable property of topologically ordered states of matter, such as the fractional quantum Hall liquids, and certain quantum spin liquid states. For a given type of topological order, there are generally many ways to fractionalize the quantum numbers of a given symmetry. What does it mean to have different types of fractionalization? Are different types of fractionalization a universal property that can be used to distinguish phases of matter? In this talk, I will answer these questions, focusing on a simple class of topologically ordered phases, namely two-dimensional gapped Z2 spin liquids, and I will present a symmetry classification of these phases.