Construction of real-valued localized composite Wannier functions for insulators

TL;DR

This paper proves the existence of real-valued, localized composite Wannier functions for insulators with time-reversal symmetry, providing a constructive algorithm for their computation in dimensions up to three.

Contribution

It introduces a method to construct smooth, real-valued, and localized Wannier functions for gapped periodic quantum systems with time-reversal symmetry, extending previous results.

Findings

Existence of a global smooth, real, and symmetric Bloch frame in 3D

Construction algorithm for localized Wannier functions

Applicability to broad class of gapped quantum systems

Abstract

We consider a real periodic Schr\"odinger operator and a physically relevant family of $m \geq 1$ Bloch bands, separated by a gap from the rest of the spectrum, and we investigate the localization properties of the corresponding composite Wannier functions. To this aim, we show that in dimension $d \leq 3$ there exists a global frame consisting of smooth quasi-Bloch functions which are both periodic and time-reversal symmetric. Aiming to applications in computational physics, we provide a constructive algorithm to obtain such a Bloch frame. The construction yields the existence of a basis of composite Wannier functions which are real-valued and almost-exponentially localized. The proof of the main result exploits only the fundamental symmetries of the projector on the relevant bands, allowing applications, beyond the model specified above, to a broad range of gapped periodic quantum systems with a time-reversal symmetry of bosonic type.