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