The topology and geometry of Bloch electrons

We review the early development of electronic topological and geometric theory over a period of twenty some years, and explore two important applications of topological Chern numbers in condensed matter physics. The first is the quantum Hall effect, where the Hall conductivity can be written as a Chern number topological invariant under insulating conditions; its exact quantization found in experiment will be explained. The second is adiabatic pumping, which describes the adiabatic current response of Bloch bands and is closely related to electronic polarization. The topological Chern number is the integral of the Berry curvature over the Brillouin zone, wherein the latter has its own physical significance. We then describe the effect of Berry curvature on electron dynamics, including the anomalous velocity and orbital magnetization. We also generalize this theory to multi-band situations, which enables us to study spin transport phenomena. Finally, we demonstrate how to obtain an effective quantum theory by re-quantizing the semiclassical model. In the non-relativistic limit, the Pauli-Schrödinger equation can be seen as an equivalent quantum theory of Dirac electrons in the positive energy spectrum, where spin-orbit coupling is found to be a geometric effect.