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The $\alpha$-$T_3$ model is characterized by a variable Berry phase thatchanges continuously from $\pi$ to $0$. We take advantage of this property tohighlight the effects of this underlying geometrical phase on a number ofphysical quantities. The Hall quantization of the two limiting cases isdramatically different - a relativistic series is associated with a Berry phaseof $\pi$ and a non-relativistic series is associated with the other limit. Westudy the quantization of the Hall plateaux as they continuously evolve from arelativistic to a non-relativistic regime. Additionally, we describe twophysical quantities that retain knowledge of the Berry phase, in the absence ofa motion-inducing magnetic field. The variable Berry phase of the$\alpha$-$T_3$ model allows us to explicitly describe the Berry phasedependence of the dynamical longitudinal optical conductivity and of theangular scattering probability.

Hall quantization and optical conductivity evolution with variable Berry phase in $\alpha$-$T_3$ model