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

TL;DR

This paper investigates how the variable Berry phase in the $\

Contribution

It introduces a detailed analysis of the $\

Findings

Hall quantization varies continuously with Berry phase

Optical conductivity depends explicitly on Berry phase

Angular scattering probability retains Berry phase information

Abstract

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.