Exponential localization of Wannier functions in insulators

TL;DR

This paper proves that Wannier functions in time-reversal symmetric insulators are exponentially localized, confirming a long-standing conjecture, and clarifies the conditions under which they can be real.

Contribution

It establishes the equivalence between the existence of analytic quasi-Bloch functions and zero Chern numbers, proving exponential localization for insulators and excluding it for Chern insulators.

Findings

Wannier functions are exponentially localized in time-reversal symmetric insulators.

Chern insulators cannot have exponentially localized Wannier functions.

Explicit conditions for the reality of Wannier functions are provided.

Abstract

The exponential localization of Wannier functions in two or three dimensions is proven for all insulators that display time-reversal symmetry, settling a long-standing conjecture. Our proof relies on the equivalence between the existence of analytic quasi-Bloch functions and the nullity of the Chern numbers (or of the Hall current) for the system under consideration. The same equivalence implies that Chern insulators cannot display exponentially localized Wannier functions. An explicit condition for the reality of the Wannier functions is identified.