Bloch Bundles, Marzari-Vanderbilt Functional and Maximally Localized Wannier Functions

We consider a periodic Schrodinger operator and the composite Wannier functions corresponding to a relevant family of its Bloch bands, separated by a gap from the rest of the spectrum. We study the associated localization functional introduced in Marzari and Vanderbilt (Phys Rev B 56:12847-12865, 1997) and we prove some results about the existence and exponential localization of its minimizers, in dimension . The proof exploits ideas and methods from the theory of harmonic maps between Riemannian manifolds.