Magnetotransport properties of the $\alpha$-T$_3$ model

Using the well-known Kubo formula, we evaluate magnetotransport quantities like the collisional and Hall conductivities of the $\alpha$-T$_3$ model. The collisional conductivity exhibits a series of peaks at strong magnetic field. Each of the conductivity peaks for $\alpha=0$ (graphene) splits into two in presence of a finite $\alpha$. This splitting occurs due to a finite phase difference between the contributions coming from the two valleys. The density of states is also calculated to explore the origin of the splitting of conductivity peaks. As $\alpha$ approaches $1$, the right split part of the conductivity peak comes closer to the left split part of the next conductivity peak. At $\alpha=1$, they merge with each other to produce a new series of the conductivity peaks. On the other hand, the Hall conductivity undergoes a smooth transition from $\sigma_{yx}=2(2n+1)e^2/h$ to $\sigma_{yx}=4ne^2/h$ with $n=0,1,2,...$ as we tune $\alpha$ from $0$ to $1$. For intermediate $\alpha$, we obtain the Hall plateaus at values $0,2,4,6,8,...$ in units of $e^2/h$.