Berry Phase, Hyperorbits, and the Hofstadter Spectrum

We develop a semiclassical theory for the dynamics of electrons in a magnetic Bloch band, where the Berry phase plays an important role. This theory, together with the Boltzmann equation, provides a framework for studying transport problems in high magnetic fields. We also derive an Onsager-like formula for the quantization of cyclotron orbits, and we find a connection between the number of orbits and Hall conductivity. This connection is employed to explain the clustering structure of the Hofstadter spectrum. The advantage of this theory is its generality and conceptual simplicity.