Response functions of a quantum fractal system: The Wannier-Azbel-Hofstadter problem

It is shown how the dynamic response of lattice fermions in a magnetic field (the Wannier-Azbel-Hofstadter'' system) may be calculated. Details are given for electromagnetic perturbations {delta}{bold A}({bold r},{ital t}){similar to}{delta}{bold A}({bold Q},{omega})exp({ital i}{bold Q}{center dot}{bold r}{minus}{ital i}{omega}{ital t}). Linear-response theory is valid for periodic boundary conditions, and can be justified in general for rational filling. Analytic results are given for the various current-current responses. Applications are briefly discussed.