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We consider the $\alpha$-$T_3$ model which provides a smooth crossoverbetween the honeycomb lattice with pseudospin $1/2$ and the dice lattice withpseudospin $1$ through the variation of a parameter $\alpha$. We study thedynamics of a wave packet representing a quasiparticle in the $\alpha$-T$_3$model with zero and finite transverse magnetic field. For zero field, it isshown that the wave packet undergoes a transient $zitterbewegung$ (ZB). Variousfeatures of ZB depending on the initial pseudospin polarization of the wavepacket have been revealed. For an intermediate value of the parameter $\alpha$i.e. for $0<\alpha<1$ the resulting ZB consists of two distinct frequencieswhen the wave packet was located initially in $rim$ site. However, the wavepacket exhibits single frequency ZB for $\alpha=0$ and $\alpha=1$. It is alsounveiled that the frequency of ZB corresponding to $\alpha=1$ gets exactly halfof that corresponding to the $\alpha=0$ case. On the other hand, when theinitial wave packet was in $hub$ site, the ZB consists of only one frequencyfor all values of $\alpha$. Using stationary phase approximation we findanalytical expression of velocity average which can be used to extract theassociated timescale over which the transient nature of ZB persists. On thecontrary the wave packet undergoes permanent ZB in presence of a transversemagnetic field. Due to the presence of large number of Landau energy levels theoscillations in ZB appear to be much more complicated. The oscillation patterndepends significantly on the initial pseudospin polarization of the wavepacket. Furthermore, it is revealed that the number of the frequency componentsinvolved in ZB depends on the parameter $\alpha$.

journal of physics. condensed matter

Dynamics of a quasiparticle in the $\alpha$-T$_3$ model: Role of  pseudospin polarization and transverse magnetic field on  \textbf{\textit{zitterbewegung}}