Universal characterizing topological insulator and topological semi-metal with Wannier functions

The nontrivial evolution of Wannier functions (WF) for the occupied bands is a good starting point to understand topological insulator. By modifying the definition of WFs from the eigenstates of the projected position operator to those of the projected modular position operator, we are able to extend the usage of WFs to Weyl metal where the WFs in the old definition fails because of the lack of band gap at the Fermi energy. This extension helps us to universally understand topological insulator and topological semi-metal in a same framework. Another advantage of using the modular position operators in the definition is that the higher dimensional WFs for the occupied bands can be easily obtained. We show one of their applications by schematically explaining why the winding numbers $\nu_{3D}=\nu_{2D}$ for the 3D topological insulators of DIII class presented in Phys. Rev. Lett. 114, 016801(2015).