Mixed Wannier-Bloch functions for electrons and phonons in extended disordered systems

A major difficulty in the study of disordered systems arises from the inadequacy of the Bloch description of quasiparticle and collective excitations. The main alternative to the crystal momentum representation is provided by the Wannier picture. However, despite the success of the optimally localized Wannier functions in crystalline environments, their applicability to disordered systems is hindered by the lack of an explicit energy label, the latter being essential for interpreting spectroscopic data. We here introduce mixed Wannier-Bloch functions for studying electronic and vibrational spectra of extended disordered solids. These functions are designed to carry both spatial localization (Wannier character) and limited spectral broadening (Bloch character). Hence, they are best suited for associating specific spectral features to the underlying atomic-scale mechanisms. The new approach is demonstrated through applications to vitreous silica. The use of mixed Wannier-Bloch functions leads to a straightforward interpretation of the electronic spectrum of $v$-SiO$_2$ in terms of $s$ and $p$ atomic orbitals, and of the corresponding vibrational spectrum in terms of the molecular modes of the Si-O-Si bridge.