Driven Hofstadter Butterflies and Related Topological Invariants

The properties of the Hofstadter butterfly, a fractal, self similar spectrum of a two dimensional electron gas, are studied in the case where the system is additionally illuminated with monochromatic light. This is accomplished by applying Floquet theory to a tight binding model on the honeycomb lattice subjected to a perpendicular magnetic field and either linearly or circularly polarized light. It is shown how the deformation of the fractal structure of the spectrum depends on intensity and polarization. Thereby, the topological properties of the Hofstadter butterfly in presence of the oscillating electric field are investigated. A thorough numerical analysis of not only the Chern numbers but also the $W_{3}$-invariants gives the appropriate insight into the topology of this driven system. This includes a comparison of a direct $W_3$-calculation to the method based on summing up Chern numbers of the truncated Floquet Hamiltonian.