Symmetry-protected topological phases with charge and spin symmetries: response theory and dynamical gauge theory in 2D, 3D and the surface of 3D

Understand

A large class of symmetry-protected topological phases (SPT) in boson / spin systems have been recently predicted by the group cohomology theory.

• In this work, we consider SPT states at least with charge symmetry (U(1) or Z$_N$) or spin $S^z$ rotation symmetry (U(1) or Z$_N$) in 2D, 3D, and the surface of 3D.
• If both are U(1), we apply external electromagnetic field / `spin gauge field' to study the charge / spin response.
• For the SPT examples we consider (i.e.