The Landau symmetry breaking states are short-range entangled states that break the symmetry of the Hamiltonian. The group theory provides a systematic understanding of those symmetry breaking phases. We find that short-range entangled states can belong to different quantum phases even if they do not break any symmetry. Those short-range entangled phases are called Symmetry Protected Trivial/Topological (SPT) phases.
Topological insulator and topological superconductors for free fermions are examples of SPT phases. The entanglement point of view of SPT phases allows us to obtain a systematic understanding of interacting SPT phases in term of group cohomology theory. Many new SPT phases for interacting systems are identified, which include new interacting topological insulator/superconductors for interacting bosons.