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

TL;DR

This paper introduces a universal framework using modified Wannier functions with modular position operators to characterize both topological insulators and semi-metals, including Weyl metals, providing new insights into their topological properties.

Contribution

It extends the definition of Wannier functions with modular position operators to universally describe topological insulators and semi-metals, overcoming limitations of previous definitions.

Findings

Unified understanding of topological insulators and semi-metals.

Application to Weyl metals where traditional Wannier functions fail.

Explanation of winding number equivalence in 3D topological insulators.

Abstract

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).