BERRY PHASE, HYPERORBITS, AND THE HOFSTADTER SPECTRUM - SEMICLASSICALDYNAMICS IN MAGNETIC BLOCH BANDS

We have derived a set of semiclassical equations for electrons in magn etic Bloch bands. The velocity and energy of magnetic Bloch electrons are found to be modified by the Berry phase and magnetization. This se miclassical approach is used to study general electron transport in a de or ac electric field. We also find a close connection between the c yclotron orbits in magnetic Bloch bands and the energy subbands in the Hofstadter spectrum. Based on this formalism, the pattern of band spl itting, the distribution of Hall conductivities, and the positions of energy subbands in the Hofstadter spectrum can be understood in a simp le and unified picture.