Berry phase effects on electronic properties

Ever since its discovery the notion of Berry phase has permeated through all branches of physics. Over the past three decades it was gradually realized that the Berry phase of the electronic wave function can have a profound effect on material properties and is responsible for a spectrum of phenomena, such as polarization, orbital magnetism, various (quantum, anomalous, or spin) Hall effects, and quantum charge pumping. This progress is summarized in a pedagogical manner in this review. A brief summary of necessary background is given and a detailed discussion of the Berry phase effect in a variety of solid-state applications. A common thread of the review is the semiclassical formulation of electron dynamics, which is a versatile tool in the study of electron dynamics in the presence of electromagnetic fields and more general perturbations. Finally, a requantization method is demonstrated that converts a semiclassical theory to an effective quantum theory. It is clear that the Berry phase should be added as an essential ingredient to our understanding of basic material properties.