Wannier functions, Bloch bundles and Marzari-Vanderbilt functional

Many properties of electrons in crystalline solids are described by Schroedinger operators with potentials whose periodicity reflects the periodic structure of the crystal. The construction of generalized eigenfunctions with exponential decay is crucial, both from a theoretical and from a numerical viewpoint. Through the Bloch-Floquet transform a band structure for the spectrum of the Hamiltonian naturally appears (Bloch bands) and in presence of a spectral gap, typical of insulators and semiconductors, to the relevant family of bands there is an associated Bloch bundle, whose rank is the number of relevant bands.

A natural basis for simple or multiband systems is given by composite Wannier functions, whose existence together with their exponential localization can be obtained using geometric obstruction theory whenever the Hamiltonian satisfies the time reversal symmetry (which ensures the triviality of the associated Bloch bundle and the existence of a smooth trivializing frame). This result largely extends partial results in case of a simple band or in one dimension.

In the talk we describe how to construct exponentially localized composite Wannier functions by variational methods, minimizing the localization functional introduced by Marzari and Vanderbilt. This functional can be transformed in one of calculus of variations type for maps from a torus to the unitary group. To study both existence and regularity of minimizers (and the corresponding localization property for the Wannier functions), it is very useful to exploit the connection with the theory of harmonic maps between Riemannian manifolds and the regularity theory for the corresponding elliptic systems. Here we aim to show existence and exponential localization for maximally localized Wannier functions (minimizers), in presence of the time reversal symmetry.

(joint with Gianluca Panati)