Floquet engineering of low-energy dispersions and dynamical localization in a periodically kicked three-band system

Much having learned about Floquet dynamics of pseudospin-$1/2$ system namely, graphene, we here address the stroboscopic properties of a periodically kicked {three-band fermionic system such as $\alpha$-T$_3$ lattice. This particular model provides an interpolation between graphene and dice lattice via the continuous tuning of the parameter $\alpha$ from 0 to 1.} In the case of dice lattice ($\alpha=1$), we reveal that one can, in principle, engineer various types of low energy dispersions around some specific points in the Brillouin zone by tuning the kicking parameter in the Hamiltonian along a particular direction. Our analytical analysis shows that one can experience different quasienergy dispersions for example, Dirac type, semi-Dirac type, gapless line, absolute flat quasienergy bands, depending on the specific values of the kicking parameter. Moreover, we numerically study the dynamics of a wave packet in dice lattice. The quasienergy dispersion allows us to understand the instantaneous structure of wave packet at stroboscopic times. We find a situation where absolute flat quasienergy bands lead to a complete dynamical localization of the wave packet. {Aditionally, we calculate the quasienergy spectrum numerically for $\alpha$-T$_3$ lattice. A periodic kick in a perpendicular (planar) direction breaks (preserves) the particle-hole symmetry for $0<\alpha<1$. Furthermore, it is also revealed that the dynamical localization of wave packet does not occur at any intermediate $\alpha \ne 0,\,1$.}