On the construction of composite Wannier functions

TL;DR

This paper provides a constructive proof for the existence of smooth, periodic, and conjugation symmetric Bloch bases in low dimensions, ensuring the existence of real, exponentially localized composite Wannier functions, even under weak magnetic fields.

Contribution

It offers a constructive method to obtain smooth and symmetric Bloch bases and demonstrates their stability under weak magnetic perturbations.

Findings

Existence of smooth, periodic Bloch bases in 1D and 2D.

Construction of real, conjugation symmetric Wannier functions.

Persistence of localized bases under weak magnetic fields.

Abstract

We give a constructive proof for the existence of an $N$-dimensional Bloch basis which is both smooth (real analytic) and periodic with respect to its $d$-dimensional quasi-momenta, when $1\leq d\leq 2$ and $N\geq 1$. The constructed Bloch basis is conjugation symmetric when the underlying projection has this symmetry, hence the corresponding exponentially localized composite Wannier functions are real. In the second part of the paper we show that by adding a weak, globally bounded but not necessarily constant magnetic field, the existence of a localized basis is preserved.