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

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