| This dissertation concerns fundamental problems on the dynamics of lattice electrons in strong magnetic fields. The 'lattice' here can be an electric potential, or even a magnetic flux array. The major effect of the applied magnetic field is to split an energy band into several subbands. Unlike usual Bloch bands, the subbands here are based on magnetic translation symmetry, symmetry of translations combined with gauge transformations. We develop a theory of semiclassical dynamics to describe the motion of electrons in these bands. The most important difference from the usual dynamics for Bloch bands is the appearance of Berry curvature in electron velocity. This term leads to a correction to the Onsager quantization formula for cyclotron motion and is essential in interpreting the Hofstadter spectrum. The organization of this dissertation is as follows: it begins with a review of the basic concepts of magnetic translation symmetry (Chap. 1), it is then followed by a specific calculation of the band structure for a two dimensional 'magnetic' lattice (Chap. 2). In Chap. 3, we investigate the electron dynamics for general magnetic Bloch bands. Its connection to the complex structure of the Hofstadter spectrum is made in Chap. 4. |
