On the construction of composite Wannier functions

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.