Anyon and Loop Braiding Statistics in Field Theories with a Topological $\Theta$-term

For gapped quantum many-body systems, the topological properties of the state are usually encoded by the exotic statistics between its excitations. In 2d, braiding statistics of quasi-particle excitations can be anyonic and uniquely determine the topological phase. This method is successfully applied in 2d to distinguish different Symmetry Protected Topological Phases. Recently, a generalized idea about braiding statistics of loop excitations in 3d gapped system was proposed. We demonstrate that the anyon statistics and three-loop statistics of various 2d and 3d topological phases can be derived using semiclassical Nonlinear Sigma Model field theories with a Topological $\Theta$-term. In our formalism, the braiding statistics has a natural geometric meaning: The braiding process of anyons or loops leads to a nontrivial field configuration in the space-time, which will contribute a braiding phase factor due to the $\Theta$-term. We also provide several physical pictures to understand the cyclic relation of the loop statistics.