Enforced symmetry breaking: the symmetry constraint in topological phases*

The symmetry and topology are two very important aspects of condensed matter physics. In recent years, the interplay between symmetry and topology have brought us great advances in understanding the phases of matter in nature, among which two are celebrated: the Symmetry Protected Topological phase and Symmetry Enriched Topological phase. In the usual classification and construction theory, the symmetry is assumed to be compatible with the underlying topological order with long range entanglement structure. However, there could exsist situations that the symmetry are always incompatible with the underlying long range entanglement structure, i.e., topological order, which leads to a interesting phenomenon---enforced symmetry breaking by topological order. aIn this talk, we will review the concept of enforced symmetry breaking by the invertible topological order, and then generalize to other systems.