Floquet systems with Hall effect: Topological properties and phase transitions

We study the quantum topological properties of Floquet (time-periodic) systems exhibiting Hall effects due to perpendicular magnetic and electric fields. The systems are charged particles periodically kicked by a one-dimensional cosine potential in the presence of such fields, where the magnetic field is perpendicular also to the kicking direction. We consider parameter values including small kicking strength, which enables the quantum evolution to be described by effective Floquet Hamiltonians. We also assume the semiclassical regime of a small scaled Planck constant. In the case of an electric field parallel to the kicking direction, one observes, as the electric-field strength is varied, a series of topological phase transitions, causing changes in the Chern numbers of quasienergy (QE) bands. These transitions are due to band degeneracies which, in an initial wide range of electric-field strengths, are shown to occur when the energy of the classical separatrix is approximately equal to the average QE of the degenerating bands. For larger electric-field strengths, the degeneracies reflect changes in the classical phase-space structure. In the case of an electric field not parallel to the kicking direction and satisfying some resonance conditions, we show that the topological properties of the QE bands are characterized by universal (electric-field independent) Chern numbers. These same numbers also characterize the QE band spectrum in the first case, at the end of the basic electric-field interval where the topological phase transitions take place. The two cases above are known to exhibit significantly different dynamical rates, both classical and quantum. This is well reflected by the different topological properties in the two cases.