Frequency dependent magneto-optical conductivity in the generalized   $\alpha - T_3$ model
  We have studied a generalized three band crossing model in 2D, the generalized $\alpha - T_3$ lattice, ranging from the pseudospin-1 Dirac equation through a quadratic+flat band touching to the pseudospin-1/2 Dirac equation. A general method is presented to determine the operator form of the Green's function, being gauge and representation independent. This yields the Landau level structure in a quantizing magnetic field and the longitudinal and transversal magneto-optical conductivities of the underlying system Although the magneto-optical selection rules allow for many transitions between Landau levels, the dominant one stems from exciting a particle from/to the flat band to/from a propagating band. The Hall conductivity from each valley is rational (not quantized at all), in agreement with Berry phase considerations, though their sum is always integer quantized. 
\'Aron D\'aniel Kov\'acs, Gyula D\'avid, Bal\'azs D\'ora, J\'ozsef   Cserti
cond-mat.mes-hall