Although we know much about Floquet dynamics of pseudospin-$1/2$ systems, namely graphene, we here address the stroboscopic properties of a periodically kicked three-band fermionic system such as the $\alpha \text{−}{\mathrm{T}}_3$ 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, the Dirac type, semi-Dirac type, gapless line, and/or 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 packets at stroboscopic times. We find a situation where absolute flat quasienergy bands lead to a complete dynamical localization of the wave packet. Additionally, we calculate the quasienergy spectrum numerically for the $\alpha \text{−}{\mathrm{T}}_3$ 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 a wave packet does not occur at any intermediate $\alpha \ne 0,1$.

中文翻译：

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周期性踢腿的三波段系统中低能量色散和动态定位的 Floquet 工程

尽管我们对赝自旋的 Floquet 动力学了解很多，$1/2$ 系统，即石墨烯，我们在这里解决了周期性踢动的三带费米子系统的频闪特性，例如 $\alpha \text{-}{\mathrm{吨}}_3$格子。这个特殊的模型通过参数的连续调整提供了石墨烯和骰子晶格之间的插值$\alpha$ 从 0 到 1。在骰子点阵的情况下 ($\alpha =1$)，我们揭示了原则上可以通过沿特定方向调整哈密顿量中的踢动参数，在布里渊区的某些特定点周围设计各种类型的低能量色散。我们的分析表明，人们可以经历不同的准能量色散，例如，狄拉克型、半狄拉克型、无间隙线和/或绝对平坦的准能带，具体取决于踢参数的具体值。此外，我们数值研究了骰子点阵中波包的动力学。准能量色散使我们能够了解频闪时间下波包的瞬时结构。我们发现绝对平坦的准能带导致波包完全动态定位的情况。此外，我们以数值方式计算准能谱$\alpha \text{-}{\mathrm{吨}}_3$格子。垂直（平面）方向的周期性反冲破坏（保持）粒子-孔对称性$0<\alpha <1$. 此外，还揭示了波包的动态定位不会发生在任何中间$\alpha \ne 0,1$.