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Magnetic properties of the $α$-$T_3$ model: magneto-optical conductivity and the Hofstadter butterfly

The $α$-$T_3$ model extrapolates between the pseudospin $S=1/2$ honeycomb lattice of graphene and the pseudospin $S=1$ dice lattice via parameter $α$. We present calculations of the magnetic properties of this hybrid pseudospin model, namely the absorptive magneto-optical conductivity and the Hofstadter butterfly spectra. In the magneto-optics curves, signatures of the hybrid system corollary a doublet structure present in the peaks, resulting from differing Landau level energies in the $K$ and $K^{\prime}$ valleys. In the Hofstadter spectra, we detail the evolution of the Hofstadter butterfly as it changes its periodicity by a factor of three as we vary between the two limiting cases of the $α$-$T_3$ model.