Dynamics of electrons in crystalline solids: Wannier functions, Berry curvature and related notions

Crystalline solids are solids in which the ionic cores of the atoms are arranged periodically. The dynamics of a test electron in a crystalline solid can be conveniently analyzed by using the Bloch-Floquet transform, while the localization properties of electrons are better described by using Wannier functions. The latter can also be obtained by minimizing a suitable localization functional, yielding a convenient numerical algorithm. Macroscopic transport properties of electrons in crystalline solids are derived, by using adiabatic theory, from the analysis of a perturbed Hamiltonian, which includes the effect of external macroscopic or slowly-varying electromagnetic potentials. The geometric Berry phase and its curvature play a prominent role in the corresponding effective dynamics.