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An analytic evaluation of Kane fermion magneto-optics in two and three dimensions

We calculate and present an analytic form of the magneto-optical conductivity for the gapped low-energy Kane model in two and three dimensions separately. The two-dimensional case maps onto the $α$-$\mathcal{T}_3$ model at a particular value of $α=1/\sqrt{3}$. In two dimensions, two chiral sectors exist, between which there are no optically activated transitions. In three dimensions, the extra dimension of dispersion mixes the two sectors so that intra- and inter-sector transitions can occur. The latter type of transition can be separated out via circular polarizations of light and shows a distinct signature in the transverse conductivity.