Hall quantization and optical conductivity evolution with variable Berry   phase in $\alpha$-$T_3$ model
  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. 
E. Illes, J. P. Carbotte and E. J. Nicol
cond-mat.mes-hall