Dynamics of a quasiparticle in the $\alpha$-T$_3$ model

We study the dynamics of a wave packet representing a quasiparticle in the $\alpha$-T$_3$ model with zero and finite transverse magnetic field. For zero field, it is shown that the wave packet undergoes a transient zitterbewegung (ZB). Various features of ZB depending on the initial pseudospin polarization of the wave packet have been revealed. For an intermediate value of the parameter $\alpha$ i.e. for $0<\alpha<1$ the resulting ZB consists of two distinct frequencies when the wave packet was located initially in $rim$ site. However, the wave packet exhibits single frequency ZB for $\alpha=0$ and $\alpha=1$. It is also unveiled that the frequency of ZB corresponding to $\alpha=1$ gets exactly half of that corresponding to the $\alpha=0$ case. On the other hand, when the initial wave packet was in $hub$ site, the ZB consists of only one frequency for all values of $\alpha$. Using stationary phase approximation we find analytical expression of velocity average which can be used to extract the associated timescale over which the transient nature of ZB persists. On the contrary the wave packet undergoes permanent ZB in presence of a transverse magnetic field. Due to the presence of large number of Landau energy levels the oscillations in ZB appear to be much more complicated. The oscillation pattern depends significantly on the initial pseudospin polarization of the wave packet.